Vestnik of Samara University. Natural Science SeriesVestnik of Samara University. Natural Science Series2541-75252712-8954Samara National Research University919510.18287/2541-7525-2020-26-4-15-24Research ArticleSYMMETRIC FINITE REPRESENTABILITY OF ℓp IN ORLICZ SPACESAstashkinS. V.<p>Doctor of Physical and Mathematical Sciences, professor, Head of the Department of Functional Analysis and Function Theory</p>astash56@mail.ruhttps://orcid.org/0000-0002-8239-5661Samara National Research University07122020264152417082021Copyright © 2021, Astashkin S.V.2021<p>It is well known that a Banach space need not contain any subspace isomorphic to a space ℓp (1 6 p ) or c0 (it was shown by Tsirelson in 1974). At the same time, by the famous Krivines theorem, every Banach space X always contains at least one of these spaces locally, i.e., there exist finite-dimensional subspaces of X of arbitrarily large dimension n which are isomorphic (uniformly) to ℓnp for some 1 6 p or cn0 . In this<br />case one says that ℓp (resp. c0) is finitely representable in X. The main purpose of this paper is to give a characterization (with a complete proof) of the set of p such that ℓp is symmetrically finitely representable in a separable Orlicz space.</p>ℓp-spacefinite representability of ℓp-spacessymmetric finite representability of ℓp-spacesOrlicz function spaceOrlicz sequence spaceMatuszewska-Orlicz indicesℓp-пространствофинитная представимость ℓp-пространствсимметричная
финитная представимость ℓp-пространствфункциональное пространство Орличапространство
последовательностей Орличаиндексы Матушевской — Орлича<h2 style="text-align: left; text-indent: 0pt; padding-left: 6pt;">Introduction</h2>
<p style="text-align: justify; line-height: 94%; text-indent: 14pt; padding-left: 6pt; padding-top: 3pt;">While a Banach space <span class="s34">X </span>need not contain any subspace isomorphic to a space <span class="s34">ℓ</span><span class="s143">p </span><span class="s145">(1 </span><span class="s41">� </span><span class="s34">p </span><span class="s44"></span><span class="s145">) </span>or <span class="s34">c</span><span class="s54">0</span> (as was shown by Tsirelson in [1]), it will always contain at least one of these spaces <span class="s45">locally</span>. This means that there exist finite-dimensional subsets of <span class="s34">X </span>of arbitrarily large dimension <span class="s34">n </span>which are isomorphic (uniformly)</p>
<p class="s144">p</p>
<p></p>
<p style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 6pt;">to <span class="s34">ℓ</span><span class="s143">n</span></p>
<p class="s50">0</p>
<p></p>
<p style="text-align: left; line-height: 14pt; text-indent: 0pt; padding-left: 3pt;">for some <span class="s145">1 </span><span class="s41">� </span><span class="s34">p </span><span class="s44"> </span>or <span class="s34">c</span><span class="s143">n</span>. This fact is the content of the famous result proved by Krivine in [2]</p>
<p style="text-align: left; line-height: 7pt; text-indent: 0pt; padding-left: 6pt;">(see also [3]). To state it we need some definitions.</p>
<p class="s144">i<span class="s50">=1</span></p>
<p></p>
<p style="text-align: left; line-height: 17pt; text-indent: 0pt; padding-left: 21pt;">Suppose <span class="s34">X </span>is a Banach space, <span class="s145">1 </span><span class="s41">� </span><span class="s34">p </span><span class="s41">� </span><span class="s44"></span>, and <span class="s44">{</span><span class="s34">z</span><span class="s146">i</span><span class="s44">}</span><span class="s55"></span></p>
<p style="text-align: left; text-indent: 0pt; padding-left: 2pt; padding-top: 9pt;">is a bounded sequence in <span class="s34">X</span>. The space <span class="s34">ℓ</span><span class="s143">p </span>is said</p>
<p class="s144">i<span class="s50">=1</span></p>
<p></p>
<p class="s44" style="text-align: left; line-height: 9pt; text-indent: 0pt; padding-left: 6pt;"><span class="p">to be </span><span class="s45">block finitely representable in </span>{<span class="s34">z</span><span class="s146">i</span>}<span class="s55"></span></p>
<p class="s34" style="text-align: left; line-height: 9pt; text-indent: 0pt; padding-left: 2pt;"><span class="p">if for every </span>n <span class="s44"> </span><span class="s147">N </span><span class="p">and </span> <span class="s145">0 </span><span class="p">there exist </span><span class="s145">0 = </span>m<span class="s52">0</span> m<span class="s52">1</span> . . . m<span class="s146">n</span></p>
<p class="s144">i<span class="s50">=</span>m<span class="s148">k</span><span class="s136"></span><span class="s149">1 </span><span class="s50">+1</span></p>
<p></p>
<p style="text-align: left; line-height: 15pt; text-indent: 0pt; padding-left: 6pt;">and <span class="s34"></span><span class="s146">i</span> <span class="s44"> </span><span class="s147">R </span>such that the vectors <span class="s34">u</span><span class="s146">k</span> <span class="s145">= </span><span class="s150"></span><span class="s151">m</span><span class="s152">k</span></p>
<p class="s34"><span class="s146">i</span>z<span class="s146">i</span><span class="p">, </span>k <span class="s145">= 1</span>, <span class="s145">2</span>, . . . , n<span class="p">, satisfy the inequality</span></p>
<p style="text-align: center; line-height: 11pt; text-indent: 0pt; padding-left: 18pt;">1 <span class="s153">n </span>1</p>
<p class="s34" style="text-align: left; line-height: 7pt; text-indent: 0pt; padding-left: 153pt;"><span class="s145">(1 + </span><span class="s145">)</span><span class="s123"></span><span class="s50">1</span><span class="s44">∥</span>a<span class="s44">∥</span><span class="s146">p</span> <span class="s41">� </span><span class="s154">1</span> <span class="s155"></span> a<span class="s146">k</span> u<span class="s146">k</span> <span class="s154">1</span></p>
<p class="s145" style="text-align: left; line-height: 7pt; text-indent: 0pt; padding-left: 8pt;"><span class="s41">� </span>(1 + <span class="s34"></span>)<span class="s44">∥</span><span class="s34">a</span><span class="s44">∥</span><span class="s146">p</span></p>
<p class="s144">k<span class="s50">=1</span></p>
<p></p>
<p class="s34" style="text-align: left; text-indent: 0pt; padding-left: 6pt; padding-top: 20pt;"><span class="p">for arbitrary </span>a <span class="s145">= (</span>a<span class="s146">k</span> <span class="s145">)</span><span class="s143">n</span></p>
<p class="s44" style="text-align: left; text-indent: 0pt; padding-left: 1pt; padding-top: 15pt;"> <span class="s147">R</span><span class="s143">n</span><span class="p">. In what follows,</span></p>
<p>1 1</p>
<p class="s154">1 <span class="s144">k</span><span class="s50">=1 </span>1<span class="s156">X</span></p>
<p class="s44">∥<span class="s34">a</span>∥<span class="s146">p</span> <span class="s145">:=</span></p>
<p>( <span class="s146">n</span></p>
<p></p>
<p></p>
<p class="s44" style="text-align: left; line-height: 18pt; text-indent: 0pt; padding-left: 25pt;">|<span class="s34">a</span><span class="s146">k</span>|<span class="s157">p</span></p>
<p class="s144" style="text-align: left; line-height: 11pt; text-indent: 0pt; padding-left: 9pt;">k<span class="s50">=1</span></p>
<p class="s158">)<span class="s50">1</span><span class="s144">/p</span></p>
<p class="s34" style="text-align: left; text-indent: 0pt; padding-left: 3pt; padding-top: 5pt;"><span class="p">if </span>p <span class="s44"></span>, <span class="p">and </span><span class="s44">∥</span>a<span class="s44">∥</span><span class="s159"></span></p>
<p></p>
<p class="s145" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 1pt;">:= max</p>
<p class="s144" style="text-align: left; line-height: 9pt; text-indent: 0pt; padding-left: 14pt;">k<span class="s50">=1</span>,<span class="s50">2</span>,...,n</p>
<p class="s44">|<span class="s34">a</span><span class="s146">k</span>|</p>
<p style="text-align: left; line-height: 13pt; text-indent: 0pt; padding-left: 6pt;">The space <span class="s34">ℓ</span><span class="s143">p</span>, <span class="s145">1 </span><span class="s41">� </span><span class="s34">p </span><span class="s41">� </span><span class="s44"></span>, is said to be <span class="s45">finitely representable </span>in <span class="s34">X </span>if for every <span class="s34">n </span><span class="s44"> </span><span class="s147">N </span>and <span class="s34"> </span><span class="s145">0 </span>there exist</p>
<p class="s144">k<span class="s50">=1</span></p>
<p></p>
<p class="s34" style="text-align: left; line-height: 13pt; text-indent: 0pt; padding-left: 6pt;">x<span class="s52">1</span>, x<span class="s52">2</span>, . . . , x<span class="s146">n</span> <span class="s44"> </span>X <span class="p">such that for any </span>a <span class="s145">= (</span>a<span class="s146">k</span> <span class="s145">)</span><span class="s143">n</span></p>
<p class="s44" style="text-align: left; line-height: 13pt; text-indent: 0pt; padding-left: 1pt;"> <span class="s147">R</span><span class="s143">n</span></p>
<p style="text-align: center; line-height: 8pt; text-indent: 0pt; padding-left: 18pt;">1 <span class="s153">n </span>1</p>
<p class="s34" style="text-align: left; line-height: 7pt; text-indent: 0pt; padding-left: 153pt;"><span class="s145">(1 + </span><span class="s145">)</span><span class="s123"></span><span class="s50">1</span><span class="s44">∥</span>a<span class="s44">∥</span><span class="s146">p</span> <span class="s41">� </span><span class="s154">1</span> <span class="s155"></span> a<span class="s146">k</span> x<span class="s146">k</span> <span class="s154">1</span></p>
<p class="s145" style="text-align: left; line-height: 7pt; text-indent: 0pt; padding-left: 8pt;"><span class="s41">� </span>(1 + <span class="s34"></span>)<span class="s44">∥</span><span class="s34">a</span><span class="s44">∥</span><span class="s146">p</span></p>
<p style="text-align: center; line-height: 1pt; text-indent: 0pt; padding-left: 18pt;">1 1</p>
<p><img src="Vestnik EN 2020_26_4/Image_083.png" alt="image" width="261" height="1" /></p>
<p class="s154" style="text-align: center; line-height: 18pt; text-indent: 0pt; padding-left: 25pt;">1 <span class="s144">k</span><span class="s50">=1 </span>1<span class="s156">X</span></p>
<p class="s160" style="text-align: left; line-height: 10pt; text-indent: 11pt; padding-left: 6pt; padding-top: 4pt;">1<span class="s161">The work was completed as a part of the implementation of the development program of the Scientific and Educational Mathematical Center the Volga Federal District, agreement no. 075-02-2021-1393.</span></p>
<p class="s40" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 34pt; padding-top: 2pt;">Astashkin S.V. Symmetric finite representability of <span class="s16">ℓ</span><span class="s162">p </span>in Orlicz spaces</p>
<p><img src="Vestnik EN 2020_26_4/Image_084.png" alt="image" width="636" height="1" /></p>
<p class="s40" style="text-align: left; line-height: 13pt; text-indent: 0pt; padding-left: 6pt;"><span class="s164">16 </span>Асташкин С.В. Симметричная финитная представимость <span class="s16">ℓ</span><span class="s162">p </span>в пространствах Орлича</p>
<p></p>
<p style="text-align: left; line-height: 17pt; text-indent: 0pt; padding-left: 6pt;">(alternatively, in the case <span class="s34">p </span><span class="s145">= </span><span class="s44"></span>, one might say that <span class="s34">c</span><span class="s54">0 </span>is finitely representable in <span class="s34">X</span>).</p>
<p class="s144">i<span class="s50">=1</span></p>
<p></p>
<p style="text-align: left; line-height: 13pt; text-indent: 0pt; padding-left: 21pt;">Clearly, if <span class="s34">ℓ</span><span class="s143">p </span>is block finitely representable in some sequence <span class="s44">{</span><span class="s34">z</span><span class="s146">i</span><span class="s44">}</span><span class="s55"></span></p>
<p class="s34" style="text-align: left; line-height: 13pt; text-indent: 0pt; padding-left: 1pt;"><span class="s44"> </span>X<span class="p">, then </span>ℓ<span class="s143">p </span><span class="p">is finitely representable</span></p>
<p style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 6pt;">in <span class="s34">X</span></p>
<p>. Therefore, the following famous result proved by Krivine in [2] (see also [3] and [4, Theorem 11.3.9])</p>
<p style="text-align: left; line-height: 14pt; text-indent: 0pt; padding-left: 6pt;">implies the finite representability of <span class="s34">ℓ</span><span class="s143">p </span>for some <span class="s145">1 </span><span class="s41">� </span><span class="s34">p </span><span class="s41">� </span><span class="s44"> </span>in any Banach space.</p>
<h4 style="text-align: left; text-indent: 0pt; padding-left: 21pt; padding-top: 4pt;">Theorem (Krivine)</h4>
<p class="s144">i<span class="s50">=1</span></p>
<p></p>
<p class="s44" style="text-align: left; line-height: 18pt; text-indent: 0pt; padding-left: 21pt;"><span class="p">Let </span>{<span class="s34">z</span><span class="s146">i</span>}<span class="s55"></span></p>
<p style="text-align: left; text-indent: 0pt; padding-left: 3pt; padding-top: 5pt;">be an arbitrary normalized sequence in a Banach space <span class="s34">X </span>such that the vectors <span class="s34">z</span><span class="s146">i</span> do not</p>
<p style="text-align: left; line-height: 11pt; text-indent: 0pt; padding-left: 6pt;">form a relatively compact set. Then <span class="s34">ℓ</span><span class="s143">p</span></p>
<p class="s144">i<span class="s50">=1</span></p>
<p></p>
<p class="s44" style="text-align: left; line-height: 13pt; text-indent: 0pt; padding-left: 3pt;"><span class="p">is block finitely representable in </span>{<span class="s34">z</span><span class="s146">i</span>}<span class="s55"> </span><span class="p">for some </span><span class="s34">p </span> <span class="s145">[1</span><span class="s34">, </span><span class="s145">]</span><span class="p">.</span></p>
<p style="text-align: justify; line-height: 12pt; text-indent: 14pt; padding-left: 6pt; padding-top: 6pt;">Here, we consider both Orlicz sequence and function spaces (see the next section for the definition) and in the separable case we give a characterization of the set of <span class="s34">p </span>such that <span class="s34">ℓ</span><span class="s143">p</span> is <span class="s45">symmetrically finitely representable </span>in such a space. To introduce the notion of symmetric finite representability, we need some more definitions.</p>
<p class="s144">k<span class="s50">=1</span></p>
<p></p>
<p class="s34" style="text-align: left; line-height: 13pt; text-indent: 0pt; padding-left: 21pt;"><span class="p">A sequence </span>y <span class="s145">= (</span>y<span class="s146">k</span> <span class="s145">)</span><span class="s55"></span></p>
<p class="s144">k<span class="s50">=1</span></p>
<p></p>
<p style="text-align: left; line-height: 13pt; text-indent: 0pt; padding-left: 2pt;">will be called a <span class="s45">copy </span>of a sequence <span class="s34">x </span><span class="s145">= (</span><span class="s34">x</span><span class="s146">k</span> <span class="s145">)</span><span class="s55"></span></p>
<p>if <span class="s34">x </span>and <span class="s34">y </span>have the same entries,</p>
<p class="s34" style="text-align: left; line-height: 13pt; text-indent: 0pt; padding-left: 6pt;"><span class="p">that is, there is a permutation </span> <span class="p">of the set of positive integers such that </span>y<span class="s153"></span><span class="s50">(</span><span class="s144">k</span><span class="s50">) </span><span class="s145">= </span>x<span class="s146">k </span><span class="p">for all </span>k <span class="s145">= 1</span>, <span class="s145">2</span>, . . .<span class="p">.</span></p>
<p class="s145" style="text-align: left; line-height: 11pt; text-indent: 0pt; padding-left: 21pt;"><span class="p">Given a measurable function </span><span class="s34">x</span>(<span class="s34">t</span>) <span class="p">on </span>[0<span class="s34">, </span>1]<span class="p">, we set</span></p>
<p class="s34">n<span class="s146">x</span><span class="s145">(</span> <span class="s145">) := </span>m<span class="s145">(</span><span class="s44">{</span>t <span class="s44"> </span><span class="s145">[0</span>, <span class="s145">) : </span><span class="s44">|</span>x<span class="s145">(</span>t<span class="s145">)</span><span class="s44">| </span> <span class="s44">}</span><span class="s145">)</span>, <span class="s145">0</span>.</p>
<p class="s34" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 6pt; padding-top: 3pt;"><span class="p">Here and in the sequel, </span>m <span class="p">denotes the Lebesgue measure. Functions </span>x<span class="s145">(</span>t<span class="s145">) </span><span class="p">and </span>y<span class="s145">(</span>t<span class="s145">) </span><span class="p">are called </span><span class="s45">equimeasurable</span></p>
<p class="s34" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 6pt;"><span class="p">if </span>n<span class="s146">x</span><span class="s145">(</span> <span class="s145">) = </span>n<span class="s146">y</span> <span class="s145">(</span> <span class="s145">) </span><span class="p">for each </span> <span class="s145">0</span><span class="p">.</span></p>
<p style="text-align: left; line-height: 60%; text-indent: 14pt; padding-left: 6pt; padding-top: 2pt;">Let <span class="s34">X </span>be a symmetric sequence space (see e.g. [5]), <span class="s145">1 </span><span class="s41">� </span><span class="s34">p </span><span class="s41">� </span><span class="s44"></span>. We say that <span class="s34">ℓ</span><span class="s143">p </span>is <span class="s45">symmetrically finitely representable </span>in <span class="s34">X </span>if for every <span class="s34">n </span><span class="s44"> </span><span class="s147">N </span>and each <span class="s34"> </span><span class="s145">0 </span>there exists an element <span class="s34">x</span><span class="s52">0 </span><span class="s44"> </span><span class="s34">X </span>such that for its</p>
<p class="s144">k<span class="s50">=1</span></p>
<p></p>
<p class="s34" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 6pt;"><span class="p">disjoint copies </span>x<span class="s146">k</span> <span class="p">, </span>k <span class="s145">= 1</span>, <span class="s145">2</span>, . . . , n<span class="p">, and for every </span><span class="s145">(</span>a<span class="s146">k</span> <span class="s145">)</span><span class="s143">n</span></p>
<p class="s44" style="text-align: left; line-height: 14pt; text-indent: 0pt; padding-left: 1pt;"> <span class="s147">R</span><span class="s143">n</span> <span class="p">we have</span></p>
<p style="text-align: center; line-height: 10pt; text-indent: 0pt; padding-left: 18pt;">1 <span class="s153">n </span>1</p>
<p class="s34" style="text-align: left; line-height: 7pt; text-indent: 0pt; padding-left: 153pt;"><span class="s145">(1 + </span><span class="s145">)</span><span class="s123"></span><span class="s50">1</span><span class="s44">∥</span>a<span class="s44">∥</span><span class="s146">p</span> <span class="s41">� </span><span class="s154">1</span> <span class="s155"></span> a<span class="s146">k</span> x<span class="s146">k</span> <span class="s154">1</span></p>
<p class="s145" style="text-align: left; line-height: 7pt; text-indent: 0pt; padding-left: 8pt;"><span class="s41">� </span>(1 + <span class="s34"></span>)<span class="s44">∥</span><span class="s34">a</span><span class="s44">∥</span><span class="s146">p</span></p>
<p style="text-align: center; line-height: 1pt; text-indent: 0pt; padding-left: 18pt;">1 1</p>
<p class="s154" style="text-align: center; line-height: 18pt; text-indent: 0pt; padding-left: 25pt;">1 <span class="s144">k</span><span class="s50">=1 </span>1<span class="s156">X</span></p>
<p style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 21pt; padding-top: 2pt;">Similar notion will be defined also in the function case. Let <span class="s34">X </span>be a symmetric function space on <span class="s145">[0</span><span class="s34">, </span><span class="s145">1]</span></p>
<p style="text-align: left; line-height: 16pt; text-indent: 0pt; padding-left: 6pt;">[5]. The space <span class="s34">ℓ</span><span class="s143">p </span>is <span class="s45">symmetrically finitely representable </span>in <span class="s34">X </span>if for every <span class="s34">n </span><span class="s44"> </span><span class="s147">N </span>and <span class="s34"> </span><span class="s145">0 </span>there exist</p>
<p class="s144">k<span class="s50">=1</span></p>
<p></p>
<p class="s145" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 6pt;"><span class="p">equimeasurable and disjointly supported on </span>[0<span class="s34">, </span>1] <span class="p">functions </span><span class="s34">u</span><span class="s146">i</span>(<span class="s34">t</span>)<span class="p">, </span><span class="s34">i </span>= 1<span class="s34">, </span>2<span class="s34">, . . . , n</span><span class="p">, such that for all </span>(<span class="s34">a</span><span class="s146">k</span> )<span class="s143">n</span></p>
<p class="s44" style="text-align: left; line-height: 13pt; text-indent: 0pt; padding-left: 1pt;"> <span class="s147">R</span><span class="s143">n</span></p>
<p class="s144" style="text-align: right; line-height: 3pt; text-indent: 0pt; padding-top: 2pt;">n</p>
<p>1</p>
<p class="s44"><span class="s145">(1 </span> <span class="s34"></span><span class="s145">)</span>∥<span class="s34">a</span>∥<span class="s146">p</span> <span class="s41">� </span><span class="s155"></span></p>
<p></p>
<p>1</p>
<p>1</p>
<p class="s144">i<span class="s50">=1</span></p>
<p>1</p>
<p></p>
<p class="s34">a<span class="s146">i</span>u<span class="s146">i</span><span class="s165">1</span></p>
<p style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 17pt;">1<span class="s166">X</span></p>
<p class="s145" style="text-align: left; text-indent: 0pt; padding-left: 1pt; padding-top: 6pt;"><span class="s41">� </span>(1 + <span class="s34"></span>)<span class="s44">∥</span><span class="s34">a</span><span class="s44">∥</span><span class="s146">p</span></p>
<p class="s34" style="text-align: left; line-height: 60%; text-indent: 14pt; padding-left: 6pt; padding-top: 2pt;"><span class="p">The set of all </span>p <span class="s44"> </span><span class="s145">[1</span>, <span class="s44"></span><span class="s145">] </span><span class="p">such that </span>ℓ<span class="s143">p </span><span class="p">is symmetrically finitely representable in </span>X <span class="p">(in both sequence and function cases) we will denote by </span><span class="s44">F </span><span class="s145">(</span>X<span class="s145">)</span><span class="p">.</span></p>
<p class="s144">N</p>
<p></p>
<p style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 21pt;">From the definition<span class="s167">2 </span>of the Matuszewska-Orlicz indices <span class="s34"></span><span class="s54">0</span></p>
<p class="s144">N</p>
<p></p>
<p style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 4pt;">and <span class="s34"></span><span class="s54">0</span></p>
<p class="s144">N</p>
<p></p>
<p style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 4pt;">(resp. <span class="s34"></span><span class="s55"></span></p>
<p class="s144">N</p>
<p></p>
<p style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 4pt;">and <span class="s34"></span><span class="s55"></span>) of an Orlicz</p>
<p class="s34" style="text-align: left; line-height: 7pt; text-indent: 0pt; padding-left: 6pt;"><span class="p">sequence space </span>ℓ<span class="s146">N </span><span class="p">(resp. an Orlicz function space </span>L<span class="s146">N</span> <span class="p">) it follows that </span><span class="s44">F </span><span class="s145">(</span>X<span class="s145">) </span><span class="s44"> </span><span class="s145">[</span><span class="s54">0</span> , <span class="s54">0</span> <span class="s145">] </span><span class="p">(resp. </span><span class="s44">F </span><span class="s145">(</span>X<span class="s145">) </span><span class="s44"></span></p>
<p class="s144">N N</p>
<p class="s34" style="text-align: left; line-height: 1pt; text-indent: 0pt; padding-left: 6pt;"><span class="s145">[</span><span class="s169">N </span>, <span class="s169">N</span> <span class="s145">]</span><span class="p">). The main purpose of this paper is to give a detailed proof of the opposite embedding for both</span></p>
<p class="s56" style="text-align: left; line-height: 6pt; text-indent: 0pt; padding-left: 15pt;"> </p>
<p style="text-align: left; text-indent: 0pt; padding-left: 6pt; padding-top: 2pt;">Orlicz sequence and function spaces. To this end, following the idea mentioned in [6, p. 140141] we will make use of the proof of Theorem 4.a.9 from [7].</p>
<p class="s145" style="text-align: left; line-height: 13pt; text-indent: 14pt; padding-left: 6pt;"><span class="p">Similar problems for Orlicz function spaces (and more generally symmetric spaces) on </span>(0<span class="s34">, </span><span class="s44"></span>) <span class="p">were</span></p>
<p style="text-align: left; line-height: 11pt; text-indent: 0pt; padding-left: 6pt;">considered in the paper [8].</p>
<p></p>
<ol id="l4">
<li>
<h2>Preliminaries</h2>
<ol id="l5">
<li style="text-align: left; text-indent: -39pt; padding-left: 45pt; padding-top: 9pt;">
<h3>Orlicz sequence spaces</h3>
<p style="text-align: left; line-height: 9pt; text-indent: 0pt; padding-left: 21pt; padding-top: 6pt;">A detailed information related to Orlicz sequence and function spaces see in monographs [911].</p>
<p class="s34" style="text-align: left; line-height: 17pt; text-indent: 0pt; padding-left: 21pt;"><span class="p">The Orlicz sequence spaces are a natural generalization of the </span>ℓ<span class="s143">p</span><span class="p">-spaces, </span><span class="s145">1 </span><span class="s41">� </span>p <span class="s41">� </span><span class="s44"></span>, <span class="p">which equipped</span></p>
<p style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 6pt;">with the usual norms</p>
<p></p>
<p></p>
<p style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 6pt; padding-top: 3pt;"><span class="s170"> </span><span class="s171">(</span><span class="s112"></span></p>
<p></p>
<p class="s144" style="text-align: left; line-height: 6pt; text-indent: 0pt; padding-left: 6pt;">p <span class="s50">1</span>/p</p>
<p class="s44"><span class="s169">k</span><span class="s50">=1</span>|<span class="s34">a</span><span class="s146">k</span>| <span class="s145">)</span></p>
<p class="s34" style="text-align: left; line-height: 9pt; text-indent: 0pt; padding-left: 13pt;">, <span class="s145">1 </span><span class="s41">� </span>p <span class="s44"></span></p>
<p class="s44">∥<span class="s34">a</span>∥<span class="s169">ℓ</span><span class="s148">p</span> <span class="s145">:=</span></p>
<p class="s145" style="text-align: left; line-height: 8pt; text-indent: 0pt; padding-left: 22pt;">sup</p>
<p class="s144"><span class="s173"> </span>k<span class="s50">=1</span>,<span class="s50">2</span>,...</p>
<p class="s44">|<span class="s34">a</span><span class="s146">k</span>| <span class="s34">, p </span><span class="s145">= </span><span class="s34">. </span><span class="s174">.</span></p>
<p style="text-align: left; line-height: 17pt; text-indent: 0pt; padding-left: 21pt;">Let <span class="s34">N </span>be an Orlicz function, that is, an increasing convex continuous function on <span class="s145">[0</span><span class="s34">, </span><span class="s44"></span><span class="s145">) </span>such that <span class="s34">N </span><span class="s145">(0) =</span></p>
<p class="s144">k<span class="s50">=1</span></p>
<p></p>
<p class="s145" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 6pt;">= 0 <span class="p">and </span>lim<span class="s146">t</span><span class="s56"> </span><span class="s34">N </span>(<span class="s34">t</span>) = <span class="s44"></span><span class="p">. The </span><span class="s45">Orlicz sequence space </span><span class="s34">ℓ</span><span class="s146">N </span><span class="p">consists of all sequences </span><span class="s34">a </span>= (<span class="s34">a</span><span class="s146">k</span> )<span class="s55"></span></p>
<p>, for which</p>
<p style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 6pt;">the following (Luxemburg) norm</p>
<p></p>
<p><img src="Vestnik EN 2020_26_4/Image_085.png" alt="image" width="261" height="1" /></p>
<p class="s160" style="text-align: left; text-indent: 0pt; padding-left: 17pt; padding-top: 8pt;">2<span class="s161">See the next section.</span></p>
<p></p>
<p class="s44">∥<span class="s34">a</span>∥<span class="s146">ℓ</span><span class="s175">N</span> <span class="s145">:= inf</span></p>
<p>{</p>
<p class="s34" style="text-align: left; text-indent: 0pt; padding-left: 8pt; padding-top: 4pt;">u <span class="s145">0 :</span></p>
<p></p>
<p class="s56" style="text-align: left; line-height: 9pt; text-indent: 0pt; padding-left: 5pt;"></p>
<p style="text-align: left; line-height: 7pt; text-indent: 0pt; padding-left: 2pt;"></p>
<p></p>
<p class="s144">k<span class="s50">=1</span></p>
<p>}</p>
<p></p>
<p class="s71" style="text-align: center; line-height: 22pt; text-indent: 0pt; padding-top: 12pt;"><span class="s176">N</span> <span class="s177">(</span> |<span class="s178">a</span><span class="s179">k</span>| <span class="s177">)</span> <span class="s180">�</span> <span class="s145">1</span></p>
<p class="s34">u</p>
<p class="s40" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 5pt; padding-top: 2pt;">Вестник Самарского университета. Естественнонаучная серия. 2020. Том 26, № 4. С. 1524</p>
<p class="s47" style="text-align: left; line-height: 13pt; text-indent: 0pt; padding-left: 5pt;">Vestnik of Samara University. Natural Science Series. 2020, vol. 26, no. 4, pp. 1524 <span class="s164">17</span></p>
<p></p>
<p class="s34" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 5pt;"><span class="p">is finite. Without loss of generality, we will assume that </span>N <span class="s145">(1) = 1</span><span class="p">. In particular, if </span>N <span class="s145">(</span>t<span class="s145">) = </span>t<span class="s143">p</span><span class="p">, we get the</span></p>
<p class="s34" style="text-align: left; line-height: 16pt; text-indent: 0pt; padding-left: 5pt;">ℓ<span class="s143">p</span><span class="p">-space, </span><span class="s145">1 </span><span class="s41">� </span>p <span class="s44"></span><span class="p">.</span></p>
<p class="s144">N</p>
<p></p>
<p style="text-align: left; line-height: 11pt; text-indent: 0pt; padding-left: 20pt;">Recall that the Matuszewska-Orlicz indices (at zero) <span class="s34"></span><span class="s54">0</span></p>
<p class="s144" style="text-align: right; line-height: 3pt; text-indent: 0pt; padding-top: 5pt;">p</p>
<p class="s144">N</p>
<p></p>
<p style="text-align: left; line-height: 11pt; text-indent: 0pt; padding-left: 3pt;">and <span class="s34"></span><span class="s54">0</span></p>
<p style="text-align: left; line-height: 11pt; text-indent: 0pt; padding-left: 3pt;">of an Orlicz function <span class="s34">N </span>are defined by</p>
<p class="s144" style="text-align: center; line-height: 3pt; text-indent: 0pt; padding-top: 5pt;">p</p>
<p class="s181"><span class="s50">0</span></p>
<p></p>
<p class="s145"><span class="s153">N </span>:= sup <span class="s154">{</span><span class="s34">p </span>: sup</p>
<p class="s144">x,y<span class="s108">:s</span><span class="s50">1</span></p>
<p class="s178">N <span class="s182">(</span>x<span class="s182">)</span>y</p>
<p class="s34" style="text-align: left; text-indent: 0pt; padding-left: 3pt; padding-top: 1pt;">N <span class="s145">(</span>xy<span class="s145">)</span></p>
<p class="s144">N</p>
<p></p>
<p class="s34" style="text-align: left; line-height: 19pt; text-indent: 0pt; padding-left: 4pt;"> <span class="s44"></span><span class="s154">}</span>, <span class="s53">0</span></p>
<p class="s145" style="text-align: left; line-height: 15pt; text-indent: 0pt; padding-left: 2pt;">:= inf <span class="s154">{</span><span class="s34">p </span>: inf</p>
<p class="s144">x,y<span class="s108">:s</span><span class="s50">1</span></p>
<p class="s178">N <span class="s182">(</span>x<span class="s182">)</span>y</p>
<p class="s34" style="text-align: left; text-indent: 0pt; padding-left: 3pt; padding-top: 1pt;">N <span class="s145">(</span>xy<span class="s145">)</span></p>
<p class="s34" style="text-align: left; line-height: 17pt; text-indent: 0pt; padding-left: 4pt;"> <span class="s145">0</span><span class="s154">}</span>.</p>
<p class="s144">N</p>
<p></p>
<p style="text-align: left; text-indent: 0pt; padding-left: 5pt; padding-top: 3pt;">It can be easily checked that <span class="s145">1 </span><span class="s41">� </span><span class="s34"></span><span class="s54">0</span></p>
<p class="s144">N</p>
<p></p>
<p class="s41" style="text-align: left; text-indent: 0pt; padding-left: 2pt; padding-top: 3pt;">� <span class="s34"></span><span class="s54">0</span></p>
<p style="text-align: left; line-height: 18pt; text-indent: 0pt; padding-left: 2pt;"><span class="s41">� </span><span class="s44"></span>. It is well known also that an Orlicz sequence space <span class="s34">ℓ</span><span class="s146">N </span>is</p>
<p class="s144">N</p>
<p></p>
<p style="text-align: left; line-height: 11pt; text-indent: 0pt; padding-left: 5pt;">separable if and only if <span class="s34"></span><span class="s54">0</span></p>
<p style="text-align: left; line-height: 13pt; text-indent: -105pt; padding-left: 107pt;"><span class="s34"> </span><span class="s44"></span>, or equivalently, if the function <span class="s34">N </span>satisfies the <span class="s145">∆</span><span class="s52">2</span>-condition at zero, i.e.,</p>
<p class="s34" style="text-align: left; line-height: 9pt; text-indent: 0pt; padding-left: 107pt; padding-top: 3pt;">N <span class="s145">(2</span>u<span class="s145">)</span></p>
<p class="s145">lim sup</p>
<p class="s144">u<span class="s56"></span><span class="s50">0</span></p>
<p></p>
<p><img src="Vestnik EN 2020_26_4/Image_086.png" alt="image" width="37" height="1" /></p>
<p class="s34">N <span class="s145">(</span>u<span class="s145">)</span></p>
<p class="s34" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 1pt;"> <span class="s44"></span>.</p>
<p class="s144">k<span class="s50">=1</span></p>
<p></p>
<p style="text-align: left; line-height: 16pt; text-indent: 0pt; padding-left: 20pt;">The subset <span class="s34">h</span><span class="s146">N </span>of an Orlicz sequence space <span class="s34">ℓ</span><span class="s146">N </span>consists of all <span class="s145">(</span><span class="s34">a</span><span class="s146">k</span> <span class="s145">)</span><span class="s55"></span></p>
<p class="s44" style="text-align: left; line-height: 18pt; text-indent: 0pt; padding-left: 1pt;"> <span class="s34">ℓ</span><span class="s146">N </span><span class="p">such that</span></p>
<p class="s56" style="text-align: center; line-height: 7pt; text-indent: 0pt; padding-left: 6pt;"></p>
<p><img src="Vestnik EN 2020_26_4/Image_087.png" alt="image" width="22" height="1" /></p>
<p>( <span class="s181">a </span>) <span class="s183"></span></p>
<p></p>
<p class="s34" style="text-align: left; line-height: 15pt; text-indent: -37pt; padding-left: 211pt;"><span class="s155"></span> N <span class="s184">|</span> <span class="s151">k</span><span class="s184">|</span> <span class="p">for each </span>u <span class="s145">0</span>.</p>
<p class="s34" style="text-align: center; line-height: 7pt; text-indent: 0pt; padding-left: 6pt;">u</p>
<p class="s144" style="text-align: center; line-height: 8pt; text-indent: 0pt; padding-left: 6pt;">k<span class="s50">=1</span></p>
<p style="text-align: left; text-indent: 0pt; padding-left: 20pt; padding-top: 3pt;">One can easily check (see also [7, Proposition 4.a.2]) that <span class="s34">h</span><span class="s146">N </span>is a separable closed subspace of <span class="s34">ℓ</span><span class="s146">N </span>and</p>
<p class="s34" style="text-align: left; line-height: 6pt; text-indent: 0pt; padding-left: 5pt;"><span class="p">the canonical unit vectors </span>e<span class="s146">n</span> <span class="s145">= (</span>e<span class="s143">i</span> <span class="s145">) </span><span class="p">such that </span>e<span class="s143">n</span> <span class="s145">= 1 </span><span class="p">and </span>e<span class="s143">i</span></p>
<p class="s34" style="text-align: left; line-height: 6pt; text-indent: 0pt; padding-left: 3pt;"><span class="s145">= 0 </span><span class="p">if </span>i <span class="s44"≯</span><span class="s145">= </span>n<span class="p">, </span>n <span class="s145">= 1</span>, <span class="s145">2</span>, . . .<span class="p">, form a symmetric</span></p>
<p class="s144" style="text-align: left; line-height: 7pt; text-indent: 0pt; padding-left: 160pt;">n n n</p>
<p style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 5pt;">basis of the space</p>
<p class="s144">n<span class="s50">=1</span></p>
<p></p>
<p class="s34" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 2pt;">h<span class="s146">N</span> <span class="p">. Recall that a basis </span><span class="s44">{</span>x<span class="s146">n</span><span class="s44">}</span><span class="s55"></span></p>
<p style="text-align: left; line-height: 11pt; text-indent: 0pt; padding-left: 3pt;">of a Banach space <span class="s34">X </span>is said to be <span class="s45">symmetric </span>if there</p>
<p style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 5pt;">exists</p>
<p class="s34" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 2pt;">C <span class="s145">0 </span><span class="p">such that for any permutation </span> <span class="p">of the set of positive integers and all </span>a<span class="s146">n</span> <span class="s44"> </span><span class="s147">R </span><span class="p">we have</span></p>
<p class="s56" style="text-align: center; line-height: 7pt; text-indent: 0pt; padding-left: 153pt;"> </p>
<p class="s181">C<span class="s56"></span><span class="s50">1</span><span class="s150">1</span> <span class="s173"></span></p>
<p></p>
<p>1</p>
<p></p>
<p>1</p>
<p></p>
<p style="text-align: center; line-height: 7pt; text-indent: 0pt; padding-left: 145pt;">1 1 1 <span class="s177"></span></p>
<p>1</p>
<p></p>
<p class="s56"></p>
<p></p>
<p>1</p>
<p></p>
<p>1</p>
<p></p>
<p style="text-align: left; line-height: 14pt; text-indent: 0pt; padding-left: 32pt;">1 <span class="s185">1</span> <span class="s186"> </span>1</p>
<p>1</p>
<p class="s144">n<span class="s50">=1</span></p>
<p class="s144"><span class="s187">a</span>n<span class="s187">x</span>n<span class="s188">1</span><span class="s189">X </span><span class="s190">�</span> <span class="s188">1</span></p>
<p></p>
<p class="s144">n<span class="s50">=1</span></p>
<p class="s34">a<span class="s146">n</span>x<span class="s146"></span><span class="s50">(</span><span class="s144">n</span><span class="s50">)</span><span class="p">1</span><span class="s166">X</span> <span class="s41">� </span>C<span class="p">1</span></p>
<p></p>
<p class="s144">n<span class="s50">=1</span></p>
<p class="s34">a<span class="s146">n</span>x<span class="s146">n</span><span class="p">1</span><span class="s191">X</span> .</p>
<p class="s34" style="text-align: justify; line-height: 12pt; text-indent: 14pt; padding-left: 5pt; padding-top: 3pt;"><span class="p">Observe that the definition of an Orlicz sequence space </span>ℓ<span class="s146">N</span> <span class="p">is determined (up to equivalence of norms) by the behaviour of the function </span>N <span class="p">near zero. More precisely, the following conditions are equivalent: 1) </span>ℓ<span class="s146">N</span> <span class="s145">= </span>ℓ<span class="s146">M</span> <span class="p">(with equivalence of norms); 2) the canonical vector bases of the spaces </span>h<span class="s146">N </span><span class="p">и </span>h<span class="s146">M </span><span class="p">are equivalent; 3) there are constants </span>C <span class="s145">0</span><span class="p">, </span>c <span class="s145">0 </span><span class="p">and </span>t<span class="s52">0</span> <span class="s145">0 </span><span class="p">such that for all </span><span class="s145">0 </span><span class="s41">� </span>t <span class="s41">� </span>t<span class="s52">0 </span><span class="p">it holds</span></p>
<p class="s34" style="text-align: center; text-indent: 0pt; padding-left: 70pt; padding-top: 1pt;">cN <span class="s145">(</span>C<span class="s123"></span><span class="s50">1</span>t<span class="s145">) </span><span class="s41">� </span>M <span class="s145">(</span>t<span class="s145">) </span><span class="s41">� </span>c<span class="s123"></span><span class="s50">1</span>N <span class="s145">(</span>Ct<span class="s145">)</span></p>
<p class="s34" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 5pt; padding-top: 6pt;"><span class="p">(see e.g. [7, Proposition 4.a.5] or [11, Theorem 3.4]). In particular, if </span>N <span class="p">is a degenerate Orlicz function, i. e., for some </span>t<span class="s52">0</span> <span class="s145">0 </span><span class="p">we have </span>N <span class="s145">(</span>t<span class="s145">) = 0 </span><span class="p">if </span><span class="s145">0 </span><span class="s41">� </span>t <span class="s41">� </span>t<span class="s52">0</span><span class="p">, then </span>ℓ<span class="s146">N</span> <span class="s145">= </span>l<span class="s159"> </span><span class="p">(with equivalence of norms).</span></p>
<p><img src="Vestnik EN 2020_26_4/Image_088.png" alt="image" width="6" height="1" /></p>
<p class="s50">2</p>
<p></p>
<p class="s34" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 20pt;"><span class="p">Given Orlicz function </span>N <span class="p">, we define the following subsets of the space </span>C<span class="s145">[0</span>, <span class="s53">1</span> <span class="s145">]</span><span class="p">:</span></p>
<p></p>
<p style="text-align: left; line-height: 1pt; text-indent: 0pt; padding-left: 172pt;"><img src="Vestnik EN 2020_26_4/Image_089.png" alt="image" width="130" height="1" /></p>
<p class="s61" style="text-align: left; line-height: 4pt; text-indent: 0pt; padding-left: 135pt;"><span class="s176">E</span>0 <span class="s188">{</span> <span class="s178">N </span><span class="s182">(</span><span class="s178">xy</span><span class="s182">) </span>0 <span class="s158"> </span>0</p>
<p></p>
<p style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 5pt;">and</p>
<p class="s144">N,a <span class="s192">:=</span></p>
<p class="s34" style="text-align: left; line-height: 11pt; text-indent: 33pt; padding-left: 5pt;"><span class="s145">: 0 </span> y a<span class="s154">}</span>, E<span class="s153">N</span> <span class="s145">:=</span></p>
<p class="s34" style="text-align: left; line-height: 9pt; text-indent: 0pt; padding-left: 5pt;">N <span class="s145">(</span>y<span class="s145">)</span></p>
<p></p>
<p class="s50">0<span class="s144">a</span>1</p>
<p class="s193">E<span class="s144">N,a</span></p>
<p><img src="Vestnik EN 2020_26_4/Image_090.png" alt="image" width="56" height="1" /></p>
<p class="s34" style="text-align: left; line-height: 2pt; text-indent: 0pt; padding-left: 169pt;"><span class="s153">N,a</span> <span class="s145">:= </span>convE<span class="s169">N,a</span>, C<span class="s153">N</span> <span class="s145">:= </span><span class="s155"></span></p>
<p class="s193" style="text-align: left; line-height: 0pt; text-indent: 0pt; padding-left: 6pt; padding-top: 2pt;">C<span class="s144">N,a</span>,</p>
<p class="s50"><span class="s194">C</span>0 0</p>
<p class="s50">0 0</p>
<p class="s50">0<span class="s144">a</span>1</p>
<p><img src="Vestnik EN 2020_26_4/Image_091.png" alt="image" width="6" height="1" /></p>
<p><img src="Vestnik EN 2020_26_4/Image_092.png" alt="image" width="6" height="1" /></p>
<p class="s50">2</p>
<p></p>
<p class="s50">2</p>
<p></p>
<p class="s34" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 5pt; padding-top: 2pt;"><span class="p">where </span><span class="s145">0 </span> a <span class="s145">1 </span><span class="p">and the closure is taken in the norm topology of </span>C<span class="s145">[0</span>, <span class="s53">1</span> <span class="s145">]</span>. <span class="p">All these sets are non-void norm compact subsets of the space </span>C<span class="s145">[0</span>, <span class="s53">1</span> <span class="s145">] </span><span class="p">[7, Lemma 4.a.6]. It is well known that they determine to a large extent</span></p>
<p style="text-align: left; line-height: 13pt; text-indent: 0pt; padding-left: 5pt;">the structure of disjoint sequences of Orlicz sequence spaces (see [7, 4.a] and [12]). Moreover, if <span class="s145">1 </span><span class="s41">� </span><span class="s34">p </span><span class="s44"></span>,</p>
<p class="s34" style="text-align: left; line-height: 6pt; text-indent: 0pt; padding-left: 5pt;"><span class="p">then </span>t<span class="s143">p</span> <span class="s44"> </span>C<span class="s54">0</span></p>
<p class="s34" style="text-align: left; line-height: 6pt; text-indent: 0pt; padding-left: 5pt;"><span class="p">if and only if </span>p <span class="s44"> </span><span class="s145">[</span><span class="s54">0</span> , <span class="s54">0</span> <span class="s145">] </span><span class="p">[7, Theorem 4.a.9].</span></p>
<p class="s144" style="text-align: left; line-height: 7pt; text-indent: 0pt; padding-left: 56pt;">N N N</p>
<p style="text-align: left; line-height: 4pt; text-indent: 0pt; padding-left: 1pt; padding-top: 3pt;"><span class="s34">N </span>satisfies the <span class="s145">∆</span><span class="s52">2</span>-condition at zero, the sets <span class="s34">E</span><span class="s54">0</span></p>
<p style="text-align: left; line-height: 4pt; text-indent: 0pt; padding-left: 7pt; padding-top: 3pt;">, <span class="s34">E</span><span class="s54">0</span> , <span class="s34">C</span><span class="s54">0</span></p>
<p style="text-align: left; line-height: 2pt; text-indent: 0pt; padding-left: 11pt; padding-top: 5pt;">and</p>
<p style="text-align: left; line-height: 10pt; text-indent: 14pt; padding-left: 5pt;">In the case when an Orlicz function</p>
<p class="s194" style="text-align: left; line-height: 4pt; text-indent: 0pt; padding-left: 5pt;">C<span class="s50">0</span></p>
<p class="s144" style="text-align: left; text-indent: 0pt; padding-left: 5pt; padding-top: 3pt;">N,a N</p>
<p class="s144" style="text-align: left; text-indent: 0pt; padding-left: 5pt; padding-top: 3pt;">N,a</p>
<p class="s34" style="text-align: left; line-height: 11pt; text-indent: 0pt; padding-left: 12pt;"><span class="s153">N </span><span class="p">can be considered as subsets of the space </span>C<span class="s145">[0</span>, <span class="s145">1] </span><span class="p">(see the remark after Lemma 4.a.6 in [7]).</span></p>
<p></p>
</li>
<li style="text-align: left; text-indent: -34pt; padding-left: 39pt; padding-top: 8pt;">
<h3>Orlicz function spaces</h3>
<p style="text-align: left; line-height: 12pt; text-indent: 14pt; padding-left: 5pt; padding-top: 6pt;">Let <span class="s34">N </span>be an Orlicz function such that <span class="s34">N </span><span class="s145">(1) = 1</span>. Denote by <span class="s34">L</span><span class="s146">N </span>the Orlicz space on <span class="s145">[0</span><span class="s34">, </span><span class="s145">1] </span>endowed with the Luxemburg norm</p>
<p class="s44" style="text-align: left; text-indent: 0pt; padding-left: 154pt; padding-top: 6pt;">∥<span class="s34">x</span>∥<span class="s146">L</span><span class="s175">N</span> <span class="s145">:= inf</span>{<span class="s34">u </span><span class="s145">0 :</span></p>
<p class="s50" style="text-align: left; line-height: 4pt; text-indent: 0pt; padding-left: 7pt;">1</p>
<p style="text-align: left; line-height: 18pt; text-indent: 0pt; padding-left: 2pt;"><span class="s150"></span> <span class="s176">N</span> <span class="s177">(</span> <span class="s71">|</span><span class="s178">x</span><span class="s182">(</span><span class="s178">t</span><span class="s182">)</span><span class="s71">| </span><span class="s177">)</span></p>
<p class="s34" style="text-align: left; line-height: 8pt; text-indent: 0pt; padding-left: 38pt;">u</p>
<p class="s50" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 3pt;">0</p>
<p class="s34">dt <span class="s41">� </span><span class="s145">1</span><span class="s44">}</span>.</p>
<p class="s34" style="text-align: left; line-height: 15pt; text-indent: 0pt; padding-left: 5pt;"><span class="p">In particular, if </span>N <span class="s145">(</span>t<span class="s145">) = </span>t<span class="s143">p</span><span class="p">, </span><span class="s145">1 </span><span class="s41">� </span>p <span class="s44"></span><span class="p">, we obtain the space </span>L<span class="s146">p</span> <span class="s145">= </span>L<span class="s146">p</span><span class="s145">[0</span>, <span class="s145">1] </span><span class="p">with the usual norm.</span></p>
<p class="s144">N</p>
<p></p>
<p style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 20pt;">The Matuszewska-Orlicz indices <span class="s34"></span><span class="s55"></span></p>
<p class="s144">N</p>
<p></p>
<p style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 4pt;">and <span class="s34"></span><span class="s55"></span></p>
<p style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 4pt;">(at infinity) of an Orlicz function <span class="s34">N </span>are defined by the</p>
<p>formulae</p>
<p class="s181"><span class="s56"></span></p>
<p></p>
<p class="s145" style="text-align: left; line-height: 17pt; text-indent: 0pt; padding-left: 11pt; padding-top: 7pt;"><span class="s153">N </span>= sup <span class="s154">{</span><span class="s34">p </span>: sup</p>
<p><img src="Vestnik EN 2020_26_4/Image_093.png" alt="image" width="44" height="1" /></p>
<p class="s34" style="text-align: left; text-indent: 0pt; padding-left: 4pt; padding-top: 8pt;">N <span class="s145">(</span>x<span class="s145">)</span>y<span class="s143">p</span></p>
<p class="s34"></p>
<p></p>
<p class="s56"></p>
<p></p>
<p class="s44"></p>
<p></p>
<p class="s154" style="text-align: left; line-height: 17pt; text-indent: 0pt; padding-left: 22pt; padding-top: 7pt;">}<span class="s34">, </span><span class="s153">N </span><span class="s145">= inf</span></p>
<p>{</p>
<p></p>
<p class="s34" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 43pt; padding-top: 8pt;">N <span class="s145">(</span>x<span class="s145">)</span>y<span class="s143">p</span></p>
<p class="s34" style="text-align: left; line-height: 7pt; text-indent: 0pt; padding-left: 6pt;">p <span class="s145">: inf</span></p>
<p class="s34" style="text-align: left; line-height: 17pt; text-indent: 0pt; padding-top: 7pt;"> <span class="s145">0</span><span class="s154">}</span>.</p>
<p class="s144">x,y<span class="s108">?:</span><span class="s50">1</span></p>
<p class="s34" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 3pt;">N <span class="s145">(</span>xy<span class="s145">)</span></p>
<p class="s144">x,y<span class="s108">?:</span><span class="s50">1</span></p>
<p class="s34" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 3pt;">N <span class="s145">(</span>xy<span class="s145">)</span></p>
<p class="s40" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 33pt; padding-top: 2pt;">Astashkin S.V. Symmetric finite representability of <span class="s16">ℓ</span><span class="s162">p </span>in Orlicz spaces</p>
<p><img src="Vestnik EN 2020_26_4/Image_094.png" alt="image" width="636" height="1" /></p>
<p class="s40" style="text-align: left; line-height: 13pt; text-indent: 0pt; padding-left: 5pt;"><span class="s164">18 </span>Асташкин С.В. Симметричная финитная представимость <span class="s16">ℓ</span><span class="s162">p </span>в пространствах Орлича</p>
<p></p>
<p class="s41" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 5pt;"><span class="p">Again </span><span class="s145">1 </span>� <span class="s34"></span><span class="s55"></span> � <span class="s34"></span><span class="s55"></span> � <span class="s44"></span><span class="p">. As in the case of sequence spaces, an Orlicz space </span><span class="s34">L</span><span class="s146">N </span><span class="p">is separable if and only</span></p>
<p class="s144" style="text-align: left; line-height: 3pt; text-indent: 0pt; padding-left: 60pt;">N N</p>
<p class="s194" style="text-align: left; line-height: 4pt; text-indent: 0pt; padding-left: 15pt;"><span class="s56"></span></p>
<p style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 5pt;">if <span class="s153">N </span><span class="s34"> </span><span class="s44"></span>, or equivalently, if the function <span class="s34">N </span>satisfies the <span class="s145">∆</span><span class="s52">2</span><span class="s45">-condition at infinity</span>, i.e.,</p>
<p class="s34" style="text-align: center; line-height: 9pt; text-indent: 0pt; padding-left: 10pt; padding-top: 3pt;">N <span class="s145">(2</span>u<span class="s145">)</span></p>
<p class="s145">lim sup</p>
<p class="s144">u<span class="s56"></span></p>
<p></p>
<p><img src="Vestnik EN 2020_26_4/Image_095.png" alt="image" width="37" height="1" /></p>
<p class="s34">N <span class="s145">(</span>u<span class="s145">)</span></p>
<p class="s34" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 1pt;"> <span class="s44"></span>.</p>
<p class="s34" style="text-align: left; line-height: 12pt; text-indent: 14pt; padding-left: 5pt; padding-top: 4pt;"><span class="p">In contrast to the sequence case, the definition of an Orlicz function space </span>L<span class="s146">N </span><span class="p">on </span><span class="s145">[0</span>, <span class="s145">1] </span><span class="p">is determined (up to equivalence of norms) by the behaviour of the function </span>N <span class="s145">(</span>t<span class="s145">) </span><span class="p">for large values of </span>t<span class="p">.</span></p>
<p><img src="Vestnik EN 2020_26_4/Image_096.png" alt="image" width="6" height="1" /></p>
<p class="s50">2</p>
<p></p>
<p class="s34" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 20pt;"><span class="p">For every Orlicz function </span>N <span class="p">we define the following subsets of the space </span>C<span class="s145">[0</span>, <span class="s53">1</span> <span class="s145">]</span><span class="p">:</span></p>
<p></p>
<p style="text-align: left; line-height: 1pt; text-indent: 0pt; padding-left: 147pt;"><img src="Vestnik EN 2020_26_4/Image_097.png" alt="image" width="109" height="1" /></p>
<p class="s181">E<span class="s56"></span></p>
<p></p>
<p class="s144" style="text-align: right; line-height: 7pt; text-indent: 0pt; padding-top: 4pt;">N,A <span class="s192">:=</span></p>
<p class="s178" style="text-align: left; line-height: 14pt; text-indent: -9pt; padding-left: 10pt;"><span class="s154">{</span> N <span class="s182">(</span>xy<span class="s182">) </span><span class="s145">: </span><span class="s34">y A</span><span class="s154">}</span><span class="s34">, E</span><span class="s123"></span> <span class="s145">=</span></p>
<p class="s34" style="text-align: left; line-height: 2pt; text-indent: 0pt; padding-left: 10pt;">N <span class="s145">(</span>y<span class="s145">)</span></p>
<p style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 3pt;"> <span class="s195">E</span><span class="s196"></span></p>
<p><img src="Vestnik EN 2020_26_4/Image_098.png" alt="image" width="50" height="1" /></p>
<p class="s34" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 4pt;">, C<span class="s123"></span> <span class="s145">:= </span>convE<span class="s55"></span>,</p>
<p class="s144">N</p>
<p class="s144">A<span class="s50">0</span></p>
<p class="s144" style="text-align: left; line-height: 5pt; text-indent: 0pt; padding-left: 7pt;">N,A N N</p>
<p><img src="Vestnik EN 2020_26_4/Image_099.png" alt="image" width="6" height="1" /></p>
<p><img src="Vestnik EN 2020_26_4/Image_100.png" alt="image" width="6" height="1" /></p>
<p class="s50">2</p>
<p></p>
<p class="s50">2</p>
<p></p>
<p class="s34" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 5pt; padding-top: 5pt;"><span class="p">where the closure is taken in the norm topology of </span>C<span class="s145">[0</span>, <span class="s53">1</span> <span class="s145">]</span>. <span class="p">Again all these sets are non-void norm compact subsets of the space </span>C<span class="s145">[0</span>, <span class="s53">1</span> <span class="s145">] </span><span class="p">and they determine largely the structure of disjoint sequences in Orlicz function</span></p>
<p class="s34" style="text-align: left; line-height: 7pt; text-indent: 0pt; padding-left: 5pt;"><span class="p">spaces (see [12, Propositions 3 and 4]). Moreover, if </span><span class="s145">1 </span><span class="s41">� </span>p <span class="s44"></span><span class="p">, then </span>t<span class="s143">p</span> <span class="s44"> </span>C<span class="s55"></span></p>
<p class="s34" style="text-align: left; line-height: 7pt; text-indent: 0pt; padding-left: 3pt;"><span class="p">if and only if </span>p <span class="s44"> </span><span class="s145">[</span><span class="s55"></span>, <span class="s55"></span><span class="s145">]</span></p>
<p style="text-align: left; text-indent: 0pt; padding-left: 5pt; padding-top: 4pt;">[12].</p>
<p class="s144" style="text-align: left; line-height: 7pt; text-indent: 0pt; padding-left: 5pt;">N N N</p>
<p class="s144">N,A</p>
<p></p>
<p style="text-align: left; line-height: 15pt; text-indent: 0pt; padding-left: 20pt;">Finally, if an Orlicz function <span class="s34">N </span>satisfies the <span class="s145">∆</span><span class="s52">2</span>-condition at infinity, the sets <span class="s34">E</span><span class="s55"></span></p>
<p class="s144">N</p>
<p></p>
<p>, <span class="s34">E</span><span class="s55"></span></p>
<p class="s144">N</p>
<p></p>
<p style="text-align: left; line-height: 13pt; text-indent: 0pt; padding-left: 3pt;">and <span class="s34">C</span><span class="s55"></span></p>
<p>can</p>
<p class="s34" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 5pt;"><span class="p">be considered as subsets of the space </span>C<span class="s145">[0</span>, <span class="s145">1]</span><span class="p">.</span></p>
<p></p>
</li>
</ol>
</li>
<li style="text-align: left; text-indent: -28pt; padding-left: 33pt; padding-top: 6pt;">
<h2>Symmetric finite representability of <span class="s197">ℓ</span><span class="s198">p</span> in Orlicz sequence spaces</h2>
<h4 style="text-align: left; text-indent: 0pt; padding-left: 20pt; padding-top: 9pt;">Theorem 1</h4>
<p></p>
<p class="s34" style="text-align: left; line-height: 46%; text-indent: 14pt; padding-left: 5pt;"><span class="p">Let </span>M <span class="p">be an Orlicz function satisfying the </span><span class="s145">∆</span><span class="s52">2</span><span class="p">-condition at zero. Then </span>ℓ<span class="s143">p </span><span class="p">is symmetrically finitely representable in the Orlicz sequence space </span>ℓ<span class="s146">M </span><span class="p">if and only if </span>p <span class="s44"> </span><span class="s145">[</span><span class="s54">0 </span>, <span class="s54">0</span> <span class="s145">]</span><span class="p">, i.e., </span><span class="s44">F </span><span class="s145">(</span>ℓ<span class="s146">M</span> <span class="s145">) = [</span><span class="s54">0 </span>, <span class="s54">0</span> <span class="s145">]</span><span class="p">.</span></p>
<h4 style="text-align: left; text-indent: 0pt; padding-left: 20pt; padding-top: 6pt;">Proof.</h4>
<p class="s144" style="text-align: left; line-height: 2pt; text-indent: 0pt; padding-left: 20pt;">M M M M</p>
<p class="s145" style="text-align: left; line-height: 13pt; text-indent: 0pt; padding-left: 20pt;"><span class="p">As was observed in 1, we always have </span><span class="s44">F </span>(<span class="s34">ℓ</span><span class="s146">M</span> ) <span class="s44"> </span>[<span class="s34"></span><span class="s54">0</span> <span class="s34">, </span><span class="s54">0</span> ]<span class="p">. Therefore, it suffices to prove only the opposite</span></p>
<p class="s144" style="text-align: center; line-height: 6pt; text-indent: 0pt; padding-left: 27pt;">M M</p>
<p class="s144">M</p>
<p></p>
<p class="s34" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 5pt;"><span class="p">embedding. In other words, we need to show that for every </span>p <span class="s44"> </span><span class="s145">[</span><span class="s54">0</span> ,</p>
<p class="s144">M</p>
<p></p>
<p class="s34"><span class="s54">0</span> <span class="s145">]</span><span class="p">, </span>m <span class="s44"> </span><span class="s147">N </span><span class="p">and each </span> <span class="s145">0 </span><span class="p">there exists</span></p>
<p class="s144">k<span class="s50">=1</span></p>
<p></p>
<p class="s34" style="text-align: left; line-height: 13pt; text-indent: 0pt; padding-left: 5pt;"><span class="p">an element </span>x<span class="s52">0</span> <span class="s44"> </span>ℓ<span class="s146">M </span><span class="p">such that for its disjoint copies </span>x<span class="s146">k</span> <span class="p">, </span>k <span class="s145">= 1</span>, <span class="s145">2</span>, . . . , m<span class="p">, and for every </span>c <span class="s145">= (</span>c<span class="s146">k</span> <span class="s145">)</span><span class="s143">m</span></p>
<p class="s44" style="text-align: left; line-height: 13pt; text-indent: 0pt; padding-left: 1pt;"> <span class="s147">R</span><span class="s143">n</span> <span class="p">we</span></p>
<p style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 5pt;">have</p>
<p style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 5pt; padding-top: 1pt;">1 <span class="s153">m </span>1</p>
<p class="s34" style="text-align: left; line-height: 7pt; text-indent: 0pt; padding-left: 150pt;"><span class="s145">(1 + </span><span class="s145">)</span><span class="s123"></span><span class="s50">1</span><span class="s44">∥</span>c<span class="s44">∥</span><span class="s146">p </span><span class="s41">� </span><span class="s154">1</span> <span class="s155"></span> c<span class="s146">k</span> x<span class="s146">k</span> <span class="s154">1</span></p>
<p class="s34" style="text-align: left; line-height: 7pt; text-indent: 0pt; padding-left: 12pt;"><span class="s41">� </span><span class="s145">(1 + </span><span class="s145">)</span><span class="s44">∥</span>c<span class="s44">∥</span><span class="s146">p</span>. <span class="p">(1)</span></p>
<p>1</p>
<p class="s154">1 <span class="s144">k</span><span class="s50">=1</span></p>
<p style="text-align: left; line-height: 1pt; text-indent: 0pt; padding-left: 19pt;">1</p>
<p style="text-align: left; line-height: 15pt; text-indent: 0pt; padding-left: 19pt;">1<span class="s166">ℓ</span><span class="s199">M</span></p>
<p class="s144">M</p>
<p></p>
<p class="s34" style="text-align: left; line-height: 15pt; text-indent: 0pt; padding-left: 20pt;"><span class="p">According to the proof of Theorem 4.a.9 in [7] and a comment followed this proof on p. 144, </span>t<span class="s143">p</span> <span class="s44"> </span>C<span class="s54">0</span></p>
<p style="text-align: left; line-height: 11pt; text-indent: 0pt; padding-left: 5pt;">(see also 2.1). Since</p>
<p class="s144">M</p>
<p></p>
<p style="text-align: left; line-height: 11pt; text-indent: 0pt; padding-left: 2pt;"><span class="s34">M </span>satisfies the <span class="s145">∆</span><span class="s52">2</span>-condition at zero, the set <span class="s34">C</span><span class="s54">0</span></p>
<p style="text-align: left; line-height: 11pt; text-indent: 0pt; padding-left: 3pt;">may be considered as a subset of the</p>
<p class="s144">M</p>
<p></p>
<p class="s34" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 5pt;"><span class="p">space </span>C<span class="s145">[0</span>, <span class="s145">1] </span><span class="p">(see the remark after Lemma 4.a.6 in [7] or again 2.1). Therefore, since </span>C<span class="s54">0</span></p>
<p class="s145" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 2pt;">:= <span class="s150"></span></p>
<p class="s50" style="text-align: left; line-height: 9pt; text-indent: 0pt; padding-top: 4pt;">0<span class="s144">a</span>1</p>
<p class="s34">C</p>
<p></p>
<p class="s50">0</p>
<p></p>
<p class="s144">M,a<span class="s158">,</span></p>
<p class="s144">M,<span class="s50">2</span><span class="s200"></span><span class="s172">n</span></p>
<p></p>
<p class="s34" style="text-align: left; line-height: 13pt; text-indent: -14pt; padding-left: 20pt;"><span class="p">we conclude that </span>t<span class="s143">p</span> <span class="s44"> </span>C<span class="s54">0</span></p>
<p style="text-align: left; line-height: 11pt; text-indent: 0pt; padding-left: 20pt;">Note that the mapping</p>
<p style="text-align: left; line-height: 13pt; text-indent: 0pt; padding-left: 3pt;">for each <span class="s34">n </span><span class="s44"> </span><span class="s147">N</span>.</p>
<p class="s34" style="text-align: left; text-indent: 0pt; padding-left: 47pt; padding-top: 5pt;"> <span class="s44">1 </span>M<span class="s146"></span><span class="s145">(</span>t<span class="s145">) := </span>M <span class="s145">(</span>t<span class="s145">)</span>/M <span class="s145">(</span><span class="s145">) </span><span class="p">(2)</span></p>
<p class="s144">M,<span class="s50">2</span><span class="s200"></span><span class="s172">n</span></p>
<p></p>
<p class="s145" style="text-align: left; line-height: 15pt; text-indent: 0pt; padding-left: 5pt;"><span class="p">is continuous from </span><span class="s34">I</span><span class="s146">n</span> := (0<span class="s34">, </span>2<span class="s55"></span><span class="s144">n</span>] <span class="p">into the subset </span><span class="s34">E</span><span class="s54">0</span></p>
<p class="s34" style="text-align: left; text-indent: 0pt; padding-left: 3pt; padding-top: 2pt;"><span class="p">of </span>C<span class="s145">[0</span>, <span class="s145">1]</span><span class="p">. Indeed, as it is well known (see e.g. [9,</span></p>
<p>Theorem 1.1]),</p>
<p class="s34" style="text-align: left; text-indent: 0pt; padding-left: 5pt; padding-top: 19pt;">M <span class="s145">(</span>t<span class="s145">) =</span></p>
<p style="text-align: center; line-height: 13pt; text-indent: 0pt; padding-top: 6pt;"> <span class="s153">t</span></p>
<p class="s34" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 15pt;"><span class="s145">(</span>s<span class="s145">) </span>ds, <span class="p">(3)</span></p>
<p class="s50">0</p>
<p style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 5pt;">where <span class="s34"> </span>is a nondecreasing right-continuous function.</p>
<p class="s34" style="text-align: left; line-height: 13pt; text-indent: 0pt; padding-left: 20pt;"><span class="p">Therefore, for arbitrary </span><span class="s52">2</span> <span class="s52">1</span> <span class="s145">0 </span><span class="p">and all </span><span class="s145">0 </span><span class="s41">� </span>t <span class="s41">� </span><span class="s145">1 </span><span class="p">we have</span></p>
<p class="s178" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 206pt;">M <span class="s182">(</span><span class="s103">1</span><span class="s182">)</span>M <span class="s182">(</span><span class="s103">2</span>t<span class="s182">) </span><span class="s71"> </span>M <span class="s182">(</span><span class="s103">2</span><span class="s182">)</span>M <span class="s182">(</span><span class="s103">1</span>t<span class="s182">)</span><span class="s71">|</span></p>
<p class="s145" style="text-align: left; line-height: 16pt; text-indent: 0pt; padding-left: 97pt;"><span class="s44">|</span><span class="s34">M</span><span class="s146"></span><span class="s201">2</span> (<span class="s34">t</span>) <span class="s44"> </span><span class="s34">M</span><span class="s146"></span><span class="s201">1</span> (<span class="s34">t</span>)<span class="s44">| </span>= <span class="s184">|</span></p>
<p class="s145" style="text-align: right; line-height: 10pt; text-indent: 0pt; padding-top: 4pt;">1</p>
<p class="s34" style="text-align: left; line-height: 14pt; text-indent: 0pt; padding-left: 22pt; padding-top: 8pt;">M <span class="s145">(</span><span class="s52">1</span><span class="s145">)</span>M <span class="s145">(</span><span class="s52">2</span><span class="s145">)</span></p>
<p class="s34" style="text-align: left; line-height: 8pt; text-indent: 0pt; padding-left: 11pt;"><span class="s145">(</span>M <span class="s145">(</span><span class="s52">2</span>t<span class="s145">) </span><span class="s44"> </span>M <span class="s145">(</span><span class="s52">1</span>t<span class="s145">) + </span>M <span class="s145">(</span><span class="s52">2</span><span class="s145">) </span><span class="s44"> </span>M <span class="s145">(</span><span class="s52">1</span><span class="s145">))</span></p>
<p><img src="Vestnik EN 2020_26_4/Image_101.png" alt="image" width="39" height="1" /></p>
<p class="s34" style="text-align: left; line-height: 18pt; text-indent: -30pt; padding-left: 215pt;"><span class="s66">� </span>M <span class="s145">(</span><span class="s52">2</span><span class="s145">)</span></p>
<p class="s145">1</p>
<p><img src="Vestnik EN 2020_26_4/Image_102.png" alt="image" width="39" height="1" /></p>
<p class="s34"><span class="s66">� </span>M <span class="s145">(</span><span class="s52">2</span><span class="s145">)</span></p>
<p style="text-align: left; line-height: 20pt; text-indent: -12pt; padding-left: 12pt; padding-top: 6pt;">( <span class="s158"></span> <span class="s144"></span><span class="s202">2</span> <span class="s144">t </span><span class="s203"></span><span class="s149">1 </span><span class="s203">t</span></p>
<p></p>
<p class="s34"><span class="s145">(</span>s<span class="s145">) </span>ds <span class="s145">+</span></p>
<p class="s149" style="text-align: left; line-height: 20pt; text-indent: -5pt; padding-left: 5pt; padding-top: 6pt;"><span class="p"> </span><span class="s153"></span><span class="s204">2</span> <span class="s203"></span>1</p>
<p></p>
<p class="s34"><span class="s145">(</span>s<span class="s145">) </span>ds<span class="s165">)</span></p>
<p><img src="Vestnik EN 2020_26_4/Image_103.png" alt="image" width="39" height="1" /></p>
<p class="s145"><span class="s180">� </span>2<span class="s34"></span>(<span class="s34"></span><span class="s52">2</span>) <span class="s205">(</span><span class="s34"></span></p>
<p class="s34">M <span class="s145">(</span><span class="s52">2</span><span class="s145">) </span><span class="s97">2</span></p>
<p class="s34"><span class="s44"> </span><span class="s52">1</span><span class="s145">)</span>.</p>
<p style="text-align: left; line-height: 18pt; text-indent: 0pt; padding-left: 5pt;">Thus, mapping (2) may be extended uniquely to a map <span class="s34"> </span><span class="s44">1 </span><span class="s34">M</span><span class="s146"> </span>from the Stone-C<span class="s158">˘</span> ech compactification <span class="s34">I</span><span class="s146">n</span></p>
<p class="s136"><span class="s153">M,</span><span class="s50">2</span><span class="s172">n</span><span class="s149">1</span></p>
<p></p>
<p style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 5pt;">of <span class="s34">I</span><span class="s146">n </span>onto the set <span class="s34">E</span><span class="s54">0</span></p>
<p class="s144">M,<span class="s50">2</span><span class="s200"></span><span class="s172">n</span></p>
<p></p>
<p class="s34"><span class="p">. Since </span>t<span class="s143">p</span> <span class="s44"> </span>C<span class="s54">0</span></p>
<p class="s144">M,<span class="s50">2</span><span class="s200"></span><span class="s172">n</span></p>
<p></p>
<p style="text-align: left; line-height: 11pt; text-indent: 0pt; padding-left: 4pt;">and the extreme points of <span class="s34">C</span><span class="s54">0</span></p>
<p style="text-align: left; line-height: 11pt; text-indent: 0pt; padding-left: 4pt;">are contained in the</p>
<p class="s40" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 5pt; padding-top: 2pt;">Вестник Самарского университета. Естественнонаучная серия. 2020. Том 26, № 4. С. 1524</p>
<p class="s47" style="text-align: left; line-height: 13pt; text-indent: 0pt; padding-left: 5pt;">Vestnik of Samara University. Natural Science Series. 2020, vol. 26, no. 4, pp. 1524 <span class="s164">19</span></p>
<p></p>
<p class="s144">M,<span class="s50">2</span><span class="s200"></span><span class="s172">n</span></p>
<p></p>
<p style="text-align: left; text-indent: 0pt; padding-left: 5pt; padding-top: 2pt;">compact set <span class="s34">E</span><span class="s54">0</span></p>
<p>, by the Krein-Milman theorem (see e.g. [13, Theorem 3.28]), there exists a probability</p>
<p>measure <span class="s34"></span><span class="s146">n </span>on the set <span class="s34">I</span><span class="s146">n</span> such that</p>
<p></p>
<p class="s34">t<span class="s157">p</span> <span class="s145">=</span></p>
<p></p>
<p>Let us show that</p>
<p></p>
<p></p>
<p></p>
<p class="s203"><span class="s144">I</span><span class="s172">n</span></p>
<p></p>
<p class="s34">M<span class="s146"></span> <span class="s145">(</span>t<span class="s145">) </span>d<span class="s146">n</span><span class="s145">(</span><span class="s145">)</span>, <span class="s145">0 </span><span class="s41">� </span>t <span class="s41">� </span><span class="s145">1</span>. <span class="p">(4)</span></p>
<p>for some probability measure <span class="s34"></span><span class="s146">n </span>on <span class="s34">I</span><span class="s146">n </span>we have</p>
<p class="s92" style="text-align: center; line-height: 3pt; text-indent: 0pt; padding-left: 6pt; padding-top: 5pt;">2<span class="s136"></span><span class="s172">n</span></p>
<p class="s186" style="text-align: left; line-height: 2pt; text-indent: 0pt; padding-left: 152pt;"><span class="p"></span></p>
<p class="s144">p</p>
<p></p>
<p class="s144">n</p>
<p></p>
<p style="text-align: left; line-height: 6pt; text-indent: 0pt; padding-left: 152pt;"></p>
<p></p>
<p></p>
<p class="s34">t <span class="s44"></span></p>
<p class="s50">0</p>
<p class="s34" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 14pt;">M<span class="s146"></span><span class="s145">(</span>t<span class="s145">) </span>d<span class="s146">n</span><span class="s145">(</span><span class="s145">) </span> <span class="s145">2</span><span class="s123"></span></p>
<p class="s34" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 3pt;">, <span class="s145">0 </span><span class="s41">� </span>t <span class="s41">� </span><span class="s145">1</span>. <span class="p">(5)</span></p>
<p style="text-align: left; line-height: 18pt; text-indent: 0pt; padding-left: 20pt;">First, the fact that the set <span class="s147">Q</span><span class="s146">n</span> <span class="s145">:= </span><span class="s147">Q </span><span class="s44"> </span><span class="s34">I</span><span class="s146">n </span>(<span class="s147">Q </span>is the set of rationals) is dense in <span class="s34">I</span><span class="s146">n</span> implies that the set</p>
<p class="s144">k<span class="s50">=1</span></p>
<p></p>
<p class="s34" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 5pt;"><span class="s44">{</span>M<span class="s146">r</span> , r <span class="s44"> </span><span class="s147">Q</span><span class="s146">n</span><span class="s44">} </span><span class="p">is dense in the subset </span><span class="s44">{</span>M<span class="s146"></span> , <span class="s44"> </span>I<span class="s146">n</span><span class="s44">} </span><span class="p">of </span>C<span class="s145">[0</span>, <span class="s145">1]</span><span class="p">. Consequently, putting </span><span class="s147">Q</span><span class="s146">n</span> <span class="s145">= </span><span class="s44">{</span>r<span class="s146">k</span><span class="s44">}</span><span class="s55"></span></p>
<p style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 3pt;">and</p>
<p class="s144">n</p>
<p></p>
<p class="s34" style="text-align: left; line-height: 19pt; text-indent: 0pt; padding-left: 114pt;">E<span class="s146">k</span> <span class="s145">:= </span><span class="s44">{</span> <span class="s44"> </span>I<span class="s146">n</span> <span class="s145">: </span><span class="s44">|</span>M<span class="s146"></span> <span class="s145">(</span>t<span class="s145">) </span><span class="s44"> </span>M<span class="s146">r</span><span class="s175">k</span> <span class="s145">(</span>t<span class="s145">)</span><span class="s44">| </span> <span class="s145">2</span><span class="s123"></span></p>
<p class="s41" style="text-align: left; line-height: 19pt; text-indent: 0pt; padding-left: 4pt;"><span class="p">for all </span><span class="s145">0 </span>� <span class="s34">t </span>� <span class="s145">1</span><span class="s44">}</span><span class="s34">, </span><span class="p">(6)</span></p>
<p class="s144">k<span class="s50">=1</span></p>
<p></p>
<p style="text-align: left; line-height: 19pt; text-indent: 0pt; padding-left: 5pt;">we have <span class="s34">I</span><span class="s146">n</span> <span class="s145">= </span><span class="s150"></span><span class="s99"></span></p>
<p class="s144">k<span class="s50">=1</span></p>
<p></p>
<p class="s34">E<span class="s146">k</span> <span class="p">. Now, if </span>F<span class="s146">m</span> <span class="s145">:= </span>E<span class="s146">m</span> <span class="s44">\ </span><span class="s145">(</span><span class="s150"></span><span class="s151">m</span><span class="s56"></span><span class="s50">1 </span>E<span class="s146">k</span> <span class="s145">)</span><span class="p">, </span>m <span class="s145">= 1</span>, <span class="s145">2</span>, . . .<span class="p">, then </span>F<span class="s146">m </span><span class="p">are pairwise disjoint and</span></p>
<p class="s56"></p>
<p></p>
<p class="s34" style="text-align: left; line-height: 13pt; text-indent: 0pt; padding-left: 5pt;">I<span class="s146">n</span> <span class="s145">= </span><span class="s44"></span><span class="s153">m</span><span class="s50">=1</span>F<span class="s146">m</span><span class="p">. Define the measure </span><span class="s146">n </span><span class="p">on </span><span class="p">-algebra of Borel subsets </span>U <span class="p">of the interval </span>I<span class="s146">n </span><span class="p">by</span></p>
<p class="s34"><span class="s146">n</span><span class="s145">(</span>U <span class="s145">) := </span><span class="s155"></span></p>
<p class="s144"><span class="s56">{</span>k<span class="s50">: </span>r<span class="s148">k</span> <span class="s56"></span>U<span class="s56">}</span></p>
<p style="text-align: left; text-indent: 0pt; padding-left: 5pt; padding-top: 5pt;">where <span class="s34"></span><span class="s146">n </span>is the probability measure from (4). Since</p>
<p class="s34"><span class="s146">n</span><span class="s145">(</span>F<span class="s146">k</span> <span class="s145">)</span>,</p>
<p class="s56" style="text-align: center; line-height: 9pt; text-indent: 0pt; padding-left: 70pt; padding-top: 1pt;"></p>
<p class="s34" style="text-align: center; line-height: 17pt; text-indent: 0pt; padding-left: 70pt;"><span class="s146">n</span><span class="s145">(</span>I<span class="s146">n</span><span class="s145">) = </span><span class="s155"></span> <span class="s146">n</span><span class="s145">(</span>F<span class="s146">k</span> <span class="s145">) = </span><span class="s146">n</span><span class="s145">(</span>I<span class="s146">n</span><span class="s145">) = 1</span>,</p>
<p class="s144" style="text-align: center; text-indent: 0pt; padding-left: 70pt; padding-top: 1pt;">k<span class="s50">=1</span></p>
<p style="text-align: left; text-indent: 0pt; padding-left: 5pt; padding-top: 3pt;">then <span class="s34"></span><span class="s146">n </span>is a probability measure on <span class="s34">I</span><span class="s146">n</span>. Moreover, by (4) and (6), for all <span class="s145">0 </span><span class="s41">� </span><span class="s34">t </span><span class="s41">� </span><span class="s145">1</span></p>
<p></p>
<p><span class="s191">p</span></p>
<p></p>
<p></p>
<p><span class="s34">t </span><span class="s44"></span></p>
<p style="text-align: left; text-indent: -5pt; padding-left: 5pt; padding-top: 2pt;"> <span class="s61">2</span><span class="s136"></span><span class="s172">n</span></p>
<p class="s50" style="text-align: left; text-indent: 0pt; padding-left: 5pt; padding-top: 9pt;">0</p>
<p style="text-align: right; line-height: 11pt; text-indent: 0pt; padding-top: 4pt;"></p>
<p class="s145"><span class="s34">M</span><span class="s146"></span>(<span class="s34">t</span>) <span class="s34">d</span><span class="s146">n</span>(<span class="s34"></span>) =</p>
<p></p>
<p></p>
<p></p>
<p class="s203"><span class="s144">I</span><span class="s172">n</span></p>
<p class="s34">M<span class="s146"></span> <span class="s145">(</span>t<span class="s145">) </span>d<span class="s146">n</span><span class="s145">(</span><span class="s145">) </span><span class="s44"></span></p>
<p class="s172">n</p>
<p></p>
<p> <span class="s61">2</span><span class="s136"></span></p>
<p></p>
<p class="s50">0</p>
<p style="text-align: left; line-height: 11pt; text-indent: 0pt; padding-left: 57pt; padding-top: 4pt;"></p>
<p class="s34">M<span class="s146"></span><span class="s145">(</span>t<span class="s145">) </span>d<span class="s146">n</span><span class="s145">(</span><span class="s145">)</span></p>
<p style="text-align: left; line-height: 6pt; text-indent: 0pt; padding-left: 57pt;"></p>
<p class="s56"></p>
<p></p>
<p style="text-align: center; line-height: 10pt; text-indent: 0pt; padding-left: 6pt;"><span class="s206">� </span><span class="s170"></span> <span class="s207">M </span></p>
<p></p>
<p class="s144">k<span class="s50">=1 </span><span class="s143">F</span><span class="s208">k</span></p>
<p class="s145" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 10pt;"><span class="s146"></span> (<span class="s34">t</span>) <span class="s34">d</span><span class="s146">n</span>(<span class="s34"></span>) <span class="s44"></span></p>
<p class="s56" style="text-align: left; text-indent: 0pt; padding-left: 5pt; padding-top: 5pt;">{<span class="s144">r</span><span class="s148">k</span> }</p>
<p class="s34">M<span class="s146"></span><span class="s145">(</span>t<span class="s145">) </span>d<span class="s146">n</span><span class="s145">(</span><span class="s145">)</span></p>
<p></p>
<p class="s56"></p>
<p></p>
<p style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 183pt;"><span class="s206">� </span><span class="s170"></span> <span class="s209">n </span></p>
<p></p>
<p class="s144">k<span class="s50">=1</span></p>
<p class="s56" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 10pt; padding-top: 5pt;">{<span class="s144">r</span><span class="s148">k</span> }</p>
<p class="s56" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 5pt;"></p>
<p class="s145">(<span class="s34">M</span><span class="s146"></span>(<span class="s34">t</span>) + 2<span class="s123"></span></p>
<p class="s145" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 3pt;">) <span class="s34">d</span><span class="s146">n</span>(<span class="s34"></span>) <span class="s44"></span></p>
<p class="s56" style="text-align: left; text-indent: 0pt; padding-left: 5pt; padding-top: 5pt;">{<span class="s144">r</span><span class="s148">k</span> }</p>
<p class="s34">M<span class="s146"></span><span class="s145">(</span>t<span class="s145">) </span>d<span class="s146">n</span><span class="s145">(</span><span class="s145">)</span></p>
<p></p>
<p></p>
<p style="text-align: left; text-indent: 0pt; padding-left: 5pt; padding-top: 7pt;">and inequality (5) is proved.</p>
<p class="s145" style="text-align: left; line-height: 16pt; text-indent: 0pt; padding-left: 5pt;"><span class="s41">� </span>2<span class="s123"></span><span class="s144">n </span><span class="s155"></span> <span class="s34"></span><span class="s146">n</span>(<span class="s44">{</span><span class="s34">r</span><span class="s146">k</span><span class="s44">}</span>) = 2<span class="s123"></span><span class="s144">n</span><span class="s34"></span><span class="s146">n</span>(<span class="s34">I</span><span class="s146">n</span>) = 2<span class="s123"></span><span class="s144">n</span><span class="s56"></span><span class="s50">1</span><span class="s34">,</span></p>
<p class="s144" style="text-align: left; line-height: 11pt; text-indent: 0pt; padding-left: 41pt;">k<span class="s50">=1</span></p>
<p class="s34"><span class="p">Next, for any </span>s <span class="s44"> </span><span class="s145">(0</span>, <span class="s145">1) </span><span class="p">and </span>n, j <span class="s44"> </span><span class="s147">N </span><span class="p">we set</span></p>
<p></p>
<p class="s34">a<span class="s146">j,n</span> <span class="s145">:=</span></p>
<p></p>
<p class="s136"><span class="p"> </span><span class="s153">s</span><span class="s172">j</span><span class="s149">1 </span><span class="s61">2</span><span class="s172">n</span></p>
<p></p>
<p class="s178">d<span class="s179">n</span><span class="s182">(</span><span class="s182">)</span></p>
<p></p>
<p class="s34">. <span class="p">(7)</span></p>
<p></p>
<p>Then, by inequality (5), we have</p>
<p class="s56" style="text-align: center; line-height: 12pt; text-indent: 0pt; padding-top: 1pt;"></p>
<p class="s172" style="text-align: left; text-indent: 0pt; padding-left: 16pt; padding-top: 2pt;"><span class="s153">s</span>j <span class="s61">2</span><span class="s136"></span>n</p>
<p></p>
<p class="s56" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 5pt;"></p>
<p class="s34" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 5pt;">M <span class="s145">(</span><span class="s145">)</span></p>
<p class="s145" style="text-align: left; line-height: 16pt; text-indent: 0pt; padding-left: 69pt;"><span class="s155"></span>[<span class="s34">a</span><span class="s146">j,n</span>]<span class="s34">M </span>(<span class="s34">s</span><span class="s157">j</span> 2<span class="s123"></span><span class="s144">n</span><span class="s34">t</span>) <span class="s44"> </span>2<span class="s123"></span><span class="s144">n </span><span class="s34"> t</span><span class="s157">p </span><span class="s34"> </span><span class="s155"></span>[<span class="s34">a</span><span class="s146">j,n</span>]<span class="s34">M </span>(<span class="s34">s</span><span class="s157">j</span><span class="s56"></span><span class="s50">1</span>2<span class="s123"></span><span class="s144">n</span><span class="s34">t</span>) + <span class="s34">M </span>(<span class="s34">t</span>)2<span class="s123"></span><span class="s144">n</span><span class="s34">/</span>(1 <span class="s44"> </span><span class="s34">s</span>) + 2<span class="s123"></span><span class="s144">n</span><span class="s34">,</span></p>
<p class="s144">j<span class="s50">=1</span></p>
<p class="s144" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 69pt;">j<span class="s50">=1</span></p>
<p style="text-align: left; text-indent: 0pt; padding-left: 5pt; padding-top: 5pt;">where by <span class="s145">[</span><span class="s34">z</span><span class="s145">] </span>we denote the integer part of a real number <span class="s34">z</span>. Choosing now <span class="s34">k</span><span class="s146">n </span>such that</p>
<p></p>
<p class="s34"><span class="p">as </span>M <span class="s145">(</span>t<span class="s145">) </span><span class="s41">� </span>M <span class="s145">(1) = 1</span><span class="p">, we get</span></p>
<p class="s56" style="text-align: center; line-height: 9pt; text-indent: 0pt; padding-left: 5pt; padding-top: 1pt;"></p>
<p style="text-align: center; line-height: 7pt; text-indent: 0pt; padding-left: 5pt;"></p>
<p></p>
<p class="s144">j<span class="s50">=</span>k<span class="s148">n</span> <span class="s50">+1</span></p>
<p class="s145">[<span class="s34">a</span><span class="s146">j,n</span>]<span class="s34">M </span>(<span class="s34">s</span><span class="s157">j</span><span class="s56"></span><span class="s50">1</span>2<span class="s123"></span><span class="s144">n</span>) <span class="s34"> </span>2<span class="s123"></span><span class="s144">n</span><span class="s34">,</span></p>
<p></p>
<p>where</p>
<p class="s34" style="text-align: left; text-indent: 0pt; padding-left: 5pt; padding-top: 1pt;">F<span class="s146">n</span><span class="s145">(</span>st<span class="s145">) </span><span class="s44"> </span><span class="s145">2</span><span class="s123"></span><span class="s144">n</span><span class="s50">+1 </span> t<span class="s157">p</span> F<span class="s146">n</span><span class="s145">(</span>t<span class="s145">) + 2</span><span class="s123"></span><span class="s144">n</span>/<span class="s145">(1 </span><span class="s44"> </span>s<span class="s145">) + 2</span><span class="s123"></span><span class="s144">n</span><span class="s50">+1</span>, <span class="s145">0 </span><span class="s41">� </span>t <span class="s41">� </span><span class="s145">1</span>, <span class="p">(8)</span></p>
<p></p>
<p class="s144" style="text-align: left; line-height: 4pt; text-indent: 0pt; padding-left: 118pt;">k<span class="s148">n</span></p>
<p class="s145" style="text-align: left; line-height: 18pt; text-indent: 0pt; padding-left: 77pt;"><span class="s34">F</span><span class="s146">n</span>(<span class="s34">t</span>) := <span class="s155"></span>[<span class="s34">a</span><span class="s146">j,n</span>]<span class="s34">M </span>(<span class="s34">s</span><span class="s157">j</span><span class="s56"></span><span class="s50">1</span>2<span class="s123"></span><span class="s144">n</span><span class="s34">t</span>)<span class="s34">. </span><span class="p">(9)</span></p>
<p class="s144" style="text-align: left; text-indent: 0pt; padding-left: 116pt; padding-top: 1pt;">j<span class="s50">=1</span></p>
<p class="s40" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 33pt; padding-top: 2pt;">Astashkin S.V. Symmetric finite representability of <span class="s16">ℓ</span><span class="s162">p </span>in Orlicz spaces</p>
<p><img src="Vestnik EN 2020_26_4/Image_104.png" alt="image" width="636" height="1" /></p>
<p class="s40" style="text-align: left; line-height: 13pt; text-indent: 0pt; padding-left: 5pt;"><span class="s164">20 </span>Асташкин С.В. Симметричная финитная представимость <span class="s16">ℓ</span><span class="s162">p </span>в пространствах Орлича</p>
<p></p>
<p style="text-align: left; text-indent: 0pt; padding-left: 5pt; padding-top: 2pt;">Since the right derivative <span class="s34"> </span>of <span class="s34">M </span>(see (3)) is a nondecreasing function and <span class="s145">0 </span><span class="s34"> s </span><span class="s145">1</span>, from (7) it follows that</p>
<p class="s34" style="text-align: left; text-indent: 0pt; padding-left: 98pt; padding-top: 8pt;">F<span class="s146">n</span><span class="s145">(</span>t<span class="s145">) </span><span class="s44"> </span>F<span class="s146">n</span><span class="s145">(</span>st<span class="s145">) </span><span class="s41">�</span></p>
<p class="s144" style="text-align: left; line-height: 4pt; text-indent: 0pt; padding-left: 10pt; padding-top: 4pt;">k<span class="s148">n</span></p>
<p class="s34" style="text-align: left; line-height: 19pt; text-indent: 0pt; padding-left: 7pt;"><span class="s155"></span> a<span class="s146">j,n</span><span class="s145">(</span>M <span class="s145">(</span>s<span class="s157">j</span><span class="s56"></span><span class="s50">1</span><span class="s145">2</span><span class="s123"></span><span class="s144">n</span>t<span class="s145">) </span><span class="s44"> </span>M <span class="s145">(</span>s<span class="s157">j</span> <span class="s145">2</span><span class="s123"></span><span class="s144">n</span>t<span class="s145">))</span></p>
<p class="s144" style="text-align: left; line-height: 11pt; text-indent: 0pt; padding-left: 8pt;">j<span class="s50">=1</span></p>
<p class="s144" style="text-align: right; line-height: 3pt; text-indent: 0pt; padding-top: 1pt;">k</p>
<p><img src="Vestnik EN 2020_26_4/Image_105.png" alt="image" width="152" height="1" /></p>
<p class="s172">n</p>
<p></p>
<p class="s210">� <span class="p"> </span><span class="s211">2</span><span class="s62"></span></p>
<p class="s34"><span class="s143">n</span>s<span class="s143">j</span><span class="s56"></span><span class="s50">1</span><span class="s145">(1 </span><span class="s44"> </span>s<span class="s145">)</span><span class="s145">(</span>s<span class="s143">j</span><span class="s56"></span><span class="s50">1</span><span class="s145">2</span><span class="s55"></span></p>
<p class="s144">n<span class="s212">)</span> <span class="s213"></span> s</p>
<p class="s136"><span class="s172">j</span><span class="s149">1 </span><span class="s92">2</span><span class="s172">n</span></p>
<p></p>
<p class="s34" style="text-align: left; line-height: 8pt; text-indent: 0pt; padding-left: 2pt;">d<span class="s146">n</span></p>
<p></p>
<p class="s145">(<span class="s34"></span>)<span class="s34">.</span></p>
<p></p>
<p style="text-align: left; text-indent: 0pt; padding-left: 5pt; padding-top: 8pt;">Furthermore, the estimate</p>
<p class="s144" style="text-align: left; text-indent: 0pt; padding-left: 5pt; padding-top: 5pt;">j<span class="s50">=1</span></p>
<p></p>
<p class="s158"> <span class="s50">2</span><span class="s144">x</span></p>
<p class="s145" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 5pt;"><span class="s34">M </span>(<span class="s34">s</span><span class="s143">j</span> 2<span class="s55"></span><span class="s144">n</span>)</p>
<p class="s172" style="text-align: left; text-indent: 0pt; padding-left: 5pt; padding-top: 2pt;"><span class="s153">s</span>j <span class="s61">2</span><span class="s136"></span>n</p>
<p class="s34">F <span class="s145">(2</span>x<span class="s145">) </span><span class="s41">;;:</span></p>
<p class="s144">x</p>
<p class="s34" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 8pt;"><span class="s145">(</span>s<span class="s145">) </span>ds <span class="s41">;;: </span>x<span class="s145">(</span>x<span class="s145">)</span>, <span class="s145">0 </span><span class="s41">� </span>x <span class="s41">� </span><span class="s145">1</span>,</p>
<p>combined with the hypothesis that <span class="s34">M </span>satisfies the <span class="s145">∆</span><span class="s52">2</span>-condition at zero, shows that</p>
<p class="s34" style="text-align: center; line-height: 9pt; text-indent: 0pt; padding-left: 30pt; padding-top: 2pt;">x<span class="s145">(</span>x<span class="s145">)</span></p>
<p></p>
<p>Hence,</p>
<p><img src="Vestnik EN 2020_26_4/Image_106.png" alt="image" width="33" height="1" /></p>
<p class="s34" style="text-align: left; line-height: 7pt; text-indent: 0pt; padding-left: 5pt;">K <span class="s145">:= sup</span></p>
<p class="s50" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 30pt;">0<span class="s144">x</span><span class="s108">:s</span>1 <span class="s187">M</span> <span class="s145">(</span><span class="s34">x</span><span class="s145">)</span></p>
<p class="s34" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 1pt;"> <span class="s44"></span>.</p>
<p class="s136"><span class="s172">j</span><span class="s149">1 </span><span class="s172">n</span></p>
<p></p>
<p class="s144" style="text-align: left; line-height: 2pt; text-indent: 0pt; padding-left: 234pt;"><span class="s156">k</span><span class="s214">n</span> <span class="s194">M</span> <span class="s145">(</span><span class="s34">s</span>j<span class="s56"></span><span class="s50">1</span><span class="s212">2</span><span class="s56"></span>n<span class="s212">)</span> <span class="s213"></span> s <span class="s50">2</span></p>
<p class="s101" style="text-align: left; line-height: 4pt; text-indent: 134pt; padding-left: 115pt;"></p>
<p class="s146">n <span class="s44"> </span>n <span class="s44"></span></p>
<p></p>
<p class="s34" style="text-align: left; line-height: 11pt; text-indent: 0pt; padding-left: 115pt;">F <span class="s145">(</span>t<span class="s145">) </span>F <span class="s145">(</span>st<span class="s145">) </span><span class="s41">� </span>K<span class="s145">(1 </span>s<span class="s145">) </span><span class="s155"></span></p>
<p class="s145"><span class="s34">M </span>(<span class="s34">s</span><span class="s143">j</span> 2<span class="s55"></span><span class="s144">n</span>)</p>
<p></p>
<p class="s172" style="text-align: left; line-height: 5pt; text-indent: 0pt; padding-left: 4pt;"><span class="s153">s</span>j <span class="s61">2</span><span class="s136"></span>n</p>
<p class="s34" style="text-align: left; text-indent: 0pt; padding-left: 15pt; padding-top: 7pt;">d<span class="s146">n</span><span class="s145">(</span><span class="s145">)</span>.</p>
<p class="s144" style="text-align: left; line-height: 8pt; text-indent: 0pt; padding-left: 232pt;">j<span class="s50">=1</span></p>
<p class="s144">M</p>
<p></p>
<p style="text-align: left; text-indent: 0pt; padding-left: 5pt; padding-top: 1pt;">Moreover, one can readily check that the upper Matuszewska-Orlicz index <span class="s34"></span><span class="s54">0</span></p>
<p></p>
<p>is finite (see also 2.1) and,</p>
<p class="s34" style="text-align: left; line-height: 14pt; text-indent: 0pt; padding-left: 5pt;"><span class="p">by its definition, for each </span>q <span class="s146">M </span><span class="p">there is a constant </span>c<span class="s52">0</span> <span class="s145">0 </span><span class="p">such that</span></p>
<p class="s34" style="text-align: center; line-height: 19pt; text-indent: 0pt; padding-left: 70pt;">M <span class="s145">(</span>s<span class="s157">j</span> <span class="s145">2</span><span class="s123"></span><span class="s144">n</span><span class="s145">) </span><span class="s41">;;: </span>c<span class="s52">0</span>M <span class="s145">(</span>s<span class="s157">j</span><span class="s56"></span><span class="s50">1</span><span class="s145">2</span><span class="s123"></span><span class="s144">n</span><span class="s145">)</span>s<span class="s157">q</span> .</p>
<p style="text-align: left; text-indent: 0pt; padding-left: 5pt; padding-top: 4pt;">As a result, since <span class="s34"></span><span class="s146">n </span>is a probability measure, we conclude</p>
<p class="s172" style="text-align: center; line-height: 12pt; text-indent: 0pt; padding-left: 35pt; padding-top: 2pt;"><span class="s146">k</span><span class="s175">n</span> <span class="p"> </span><span class="s153">s</span>j<span class="s136"></span><span class="s149">1 </span><span class="s61">2</span><span class="s136"></span>n</p>
<p class="s50">0</p>
<p></p>
<p class="s34">F<span class="s146">n</span><span class="s145">(</span>t<span class="s145">) </span><span class="s44"> </span>F<span class="s146">n</span><span class="s145">(</span>st<span class="s145">) </span><span class="s41">� </span>K<span class="s145">(1 </span><span class="s44"> </span>s<span class="s145">)</span>s<span class="s123"></span><span class="s144">q </span>c<span class="s123"></span><span class="s50">1 </span><span class="s155"></span></p>
<p class="s144">j<span class="s50">=1</span></p>
<p></p>
<p class="s172"><span class="s153">s</span>j <span class="s61">2</span><span class="s136"></span>n</p>
<p class="s50">0</p>
<p></p>
<p class="s34" style="text-align: left; line-height: 16pt; text-indent: 0pt; padding-left: 15pt;">d<span class="s146">n</span><span class="s145">(</span><span class="s145">) </span><span class="s41">� </span>K<span class="s145">(1 </span><span class="s44"> </span>s<span class="s145">)</span>s<span class="s123"></span><span class="s144">q </span>c<span class="s123"></span><span class="s50">1</span>. <span class="p">(10)</span></p>
<p class="s34" style="text-align: left; line-height: 19pt; text-indent: 0pt; padding-left: 20pt;"><span class="p">Let </span>m <span class="s44"> </span><span class="s147">N </span><span class="p">and </span> <span class="s145">0 </span><span class="p">be arbitrary. Choose and fix </span>s <span class="s44"> </span><span class="s145">(0</span>, <span class="s145">1) </span><span class="p">so that</span></p>
<p class="s50">0</p>
<p></p>
<p class="s34" style="text-align: left; line-height: 19pt; text-indent: 0pt; padding-left: 190pt;">K<span class="s145">(1 </span><span class="s44"> </span>s<span class="s145">)</span>s<span class="s123"></span><span class="s144">q </span>c<span class="s123"></span><span class="s50">1 </span> /<span class="s145">(2</span>m<span class="s145">)</span>. <span class="p">(11)</span></p>
<p style="text-align: left; text-indent: 0pt; padding-left: 5pt; padding-top: 4pt;">Then, from (8) and (10) it follows</p>
<p class="s34" style="text-align: right; line-height: 9pt; text-indent: 0pt; padding-top: 2pt;"></p>
<p></p>
<p class="s144" style="text-align: left; line-height: 4pt; text-indent: 0pt; padding-left: 5pt;">n<span class="s50">+1</span></p>
<p></p>
<p class="s144" style="text-align: left; line-height: 7pt; text-indent: 0pt; padding-left: 5pt; padding-top: 7pt;"><span class="s56"></span>n<span class="s50">+1 </span>p</p>
<p><img src="Vestnik EN 2020_26_4/Image_107.png" alt="image" width="19" height="1" /></p>
<p class="s34" style="text-align: left; line-height: 15pt; text-indent: 0pt; padding-left: 126pt;">F<span class="s146">n</span><span class="s145">(</span>t<span class="s145">) </span><span class="s44"> </span><span class="s171">2</span>m <span class="s44"> </span><span class="s145">2</span><span class="s123"></span></p>
<p style="text-align: left; line-height: 18pt; text-indent: 0pt; padding-left: 5pt;">Now, taking <span class="s34">n </span><span class="s44"> </span><span class="s147">N </span>satisfying the inequality</p>
<p class="s145"><span class="s34"> F</span><span class="s146">n</span>(<span class="s34">st</span>) <span class="s44"> </span>2</p>
<p></p>
<p class="s212">2<span class="s56"></span><span class="s144">n</span></p>
<p class="s34" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 5pt;"> t , <span class="s145">0 </span><span class="s41">� </span>t <span class="s41">� </span><span class="s145">1</span>. <span class="p">(12)</span></p>
<p></p>
<p>from (8) and (12), we obtain</p>
<p class="s145" style="text-align: left; line-height: 24pt; text-indent: 5pt; padding-left: 5pt; padding-top: 1pt;">1 <span class="s44"> </span><span class="s34">s </span><span class="s178"> </span><span class="s153">p</span></p>
<p><img src="Vestnik EN 2020_26_4/Image_108.png" alt="image" width="30" height="1" /></p>
<p><img src="Vestnik EN 2020_26_4/Image_109.png" alt="image" width="19" height="1" /></p>
<p class="s215">+<span class="s145"> 2</span><span class="s56"></span><span class="s144">n</span><span class="s50">+1 </span><span class="s181"> </span><span class="s193"></span></p>
<p class="s145" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 50pt;">2<span class="s34">m</span></p>
<p></p>
<p class="s178"></p>
<p class="s34">, <span class="p">(13)</span></p>
<p class="s34" style="text-align: left; line-height: 10pt; text-indent: 153pt; padding-left: 5pt;">F<span class="s146">n</span><span class="s145">(</span>t<span class="s145">) </span><span class="s44"> </span><span class="s216">m</span> t</p>
<p class="s34" style="text-align: left; line-height: 17pt; text-indent: 0pt; padding-left: 5pt;"><span class="p">Therefore, for any </span>c<span class="s146">i</span> <span class="s44"> </span><span class="s145">[0</span>, <span class="s145">1]</span><span class="p">, </span>i <span class="s145">= 1</span>, <span class="s145">2</span>, . . . , m<span class="p">,</span></p>
<p class="s34" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 5pt;"> F<span class="s146">n</span><span class="s145">(</span>t<span class="s145">) + </span><span class="s216">m</span>, <span class="s145">0 </span><span class="s41">� </span>t <span class="s41">� </span><span class="s145">1</span>. <span class="p">(14)</span></p>
<p></p>
<p class="s144">k<span class="s50">=1</span></p>
<p></p>
<p class="s34"><span class="p">whence for all </span>c <span class="s145">= (</span>c<span class="s146">k</span> <span class="s145">)</span><span class="s143">n</span></p>
<p class="s144" style="text-align: center; line-height: 6pt; text-indent: 0pt; padding-left: 45pt; padding-top: 3pt;">m</p>
<p> <span class="s195">c</span><span class="s217">p</span></p>
<p></p>
<p class="s169" style="text-align: left; line-height: 17pt; text-indent: 19pt; padding-left: 46pt;">i <span class="s44"> </span><span class="s34"> </span></p>
<p class="s144" style="text-align: center; line-height: 10pt; text-indent: 0pt; padding-left: 45pt;">i<span class="s50">=1</span></p>
<p style="text-align: left; line-height: 19pt; text-indent: 0pt; padding-left: 1pt;"><span class="s44"> </span><span class="s147">R</span><span class="s143">n</span>, <span class="s34">c</span><span class="s146">k</span> <span class="s41">;;: </span><span class="s145">0</span>,</p>
<p class="s144" style="text-align: left; line-height: 4pt; text-indent: 0pt; padding-left: 4pt; padding-top: 3pt;">m</p>
<p></p>
<p></p>
<p class="s144">i<span class="s50">=1</span></p>
<p></p>
<p class="s34">F<span class="s146">n</span><span class="s145">(</span>c<span class="s146">i</span><span class="s145">) </span></p>
<p class="s144" style="text-align: left; line-height: 4pt; text-indent: 0pt; padding-left: 4pt; padding-top: 3pt;">m</p>
<p></p>
<p></p>
<p class="s144">i<span class="s50">=1</span></p>
<p class="s144">i</p>
<p></p>
<p class="s34">c<span class="s151">p</span> <span class="s145">+ </span>,</p>
<p class="s144" style="text-align: right; line-height: 4pt; text-indent: 0pt; padding-top: 3pt;">m</p>
<p class="s34"><span class="s145">1 </span><span class="s44"> </span> <span class="s155"></span> F<span class="s146">n</span></p>
<p>( <span class="s216">c</span><span class="s209">i</span></p>
<p style="text-align: left; line-height: 11pt; text-indent: 0pt; padding-left: 5pt;">)</p>
<p><img src="Vestnik EN 2020_26_4/Image_110.png" alt="image" width="26" height="1" /></p>
<p class="s34" style="text-align: left; line-height: 9pt; text-indent: 0pt; padding-left: 15pt; padding-top: 1pt;"> <span class="s145">1 + </span>.</p>
<p class="s144">i<span class="s50">=1</span></p>
<p class="s44" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 22pt;">∥<span class="s34">c</span>∥<span class="s146">p</span></p>
<p style="text-align: left; text-indent: 0pt; padding-left: 5pt; padding-top: 3pt;">Moreover, since <span class="s34">F</span><span class="s146">n</span> is a convex function, from the latter inequality it follows that</p>
<p class="s144" style="text-align: right; line-height: 4pt; text-indent: 0pt; padding-top: 4pt;">m</p>
<p> <span class="s195">F</span><span class="s218">n</span></p>
<p>( <span class="s216">c</span><span class="s209">i </span>)</p>
<p><img src="Vestnik EN 2020_26_4/Image_111.png" alt="image" width="65" height="1" /></p>
<p class="s41" style="text-align: left; line-height: 6pt; text-indent: 0pt; padding-left: 68pt;">� <span class="s145">1</span></p>
<p></p>
<p style="text-align: left; text-indent: 0pt; padding-left: 5pt; padding-top: 6pt;">and</p>
<p class="s144" style="text-align: left; text-indent: -1pt; padding-left: 8pt; padding-top: 4pt;">i<span class="s50">=1</span></p>
<p></p>
<p class="s144" style="text-align: left; line-height: 4pt; text-indent: 0pt; padding-left: 8pt;">m</p>
<p style="text-align: left; line-height: 18pt; text-indent: 0pt; padding-left: 5pt;"> <span class="s195">F</span><span class="s218">n</span></p>
<p class="s144" style="text-align: left; text-indent: 0pt; padding-left: 5pt; padding-top: 1pt;">i<span class="s50">=1</span></p>
<p class="s145">(1 + <span class="s34"></span>)<span class="s44">∥</span><span class="s34">c</span><span class="s44">∥</span><span class="s146">p</span></p>
<p style="text-align: center; line-height: 18pt; text-indent: 0pt; padding-top: 10pt;">( <span class="s216">c</span><span class="s209">i </span>)</p>
<p><img src="Vestnik EN 2020_26_4/Image_112.png" alt="image" width="65" height="1" /></p>
<p class="s44" style="text-align: left; line-height: 17pt; text-indent: 0pt; padding-left: 8pt;"><span class="s145">(1 </span> <span class="s34"></span><span class="s145">)</span>∥<span class="s34">c</span>∥<span class="s146">p</span></p>
<p></p>
<p class="s34"> <span class="s145">1</span>.</p>
<p class="s40" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 5pt; padding-top: 2pt;">Вестник Самарского университета. Естественнонаучная серия. 2020. Том 26, № 4. С. 1524</p>
<p class="s47" style="text-align: left; line-height: 13pt; text-indent: 0pt; padding-left: 5pt;">Vestnik of Samara University. Natural Science Series. 2020, vol. 26, no. 4, pp. 1524 <span class="s164">21</span></p>
<p></p>
<p class="s144">k<span class="s50">=1</span></p>
<p></p>
<p class="s34" style="text-align: left; line-height: 18pt; text-indent: 0pt; padding-left: 5pt;"><span class="p">Therefore, by the definition of the norm in an Orlicz sequence space, for every </span>m <span class="s44"> </span><span class="s147">N </span><span class="p">and all </span>c <span class="s145">= (</span>c<span class="s146">k</span> <span class="s145">)</span><span class="s143">n</span></p>
<p class="s44" style="text-align: left; line-height: 18pt; text-indent: 0pt; padding-left: 1pt;"> <span class="s147">R</span><span class="s143">n</span></p>
<p style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 5pt;">we have</p>
<p class="s144" style="text-align: right; line-height: 3pt; text-indent: 0pt; padding-top: 5pt;">m</p>
<p>1</p>
<p class="s44"><span class="s145">(1 </span> <span class="s34"></span><span class="s145">)</span>∥<span class="s34">c</span>∥<span class="s146">p</span> <span class="s41">� </span><span class="s155"></span></p>
<p></p>
<p>1</p>
<p>1</p>
<p class="s144">i<span class="s50">=1</span></p>
<p>1</p>
<p></p>
<p class="s34" style="text-align: left; line-height: 18pt; text-indent: 0pt; padding-top: 3pt;">c<span class="s146">i</span>e<span class="s146">i</span><span class="s165">1</span></p>
<p style="text-align: left; line-height: 13pt; text-indent: 0pt; padding-left: 15pt;">1<span class="s166">ℓ</span><span class="s199">F</span><span class="s219">n</span></p>
<p class="s34" style="text-align: left; text-indent: 0pt; padding-left: 2pt; padding-top: 9pt;"><span class="s41">� </span><span class="s145">(1 + </span><span class="s145">)</span><span class="s44">∥</span>c<span class="s44">∥</span><span class="s146">p</span>, <span class="p">(15)</span></p>
<p class="s34" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 5pt;"><span class="p">where </span>e<span class="s146">i</span><span class="p">, </span>i <span class="s145">= 1</span>, <span class="s145">2</span>, . . .<span class="p">, are the canonical unit vectors in </span>ℓ<span class="s146">F</span><span class="s175">n</span> <span class="p">.</span></p>
<p class="s34" style="text-align: left; line-height: 15pt; text-indent: 0pt; padding-left: 20pt;"><span class="p">Given </span>m <span class="s44"> </span><span class="s147">N </span><span class="p">and </span> <span class="s145">0</span><span class="p">, select </span>s <span class="p">and </span>n <span class="p">to satisfy (11) and (13). For any </span>i <span class="s145">= 1</span>, <span class="s145">2</span>, . . . , m <span class="p">and </span>j <span class="s145">= 1</span>, <span class="s145">2</span>, . . . , k<span class="s146">n</span></p>
<p class="s144">j,n</p>
<p></p>
<p style="text-align: left; line-height: 11pt; text-indent: 0pt; padding-left: 5pt;">denote by <span class="s34">A</span><span class="s143">i</span></p>
<p class="s144">j,n</p>
<p></p>
<p style="text-align: left; line-height: 11pt; text-indent: 0pt; padding-left: 3pt;">pairwise disjoint subsets of positive integers such that <span class="s145">card </span><span class="s34">A</span><span class="s143">i</span></p>
<p class="s203" style="text-align: center; line-height: 8pt; text-indent: 0pt; padding-top: 7pt;">k<span class="s172">n</span></p>
<p class="s145" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 1pt;">= [<span class="s34">a</span><span class="s146">j,n</span>]<span class="p">. Then, the vectors</span></p>
<p class="s144" style="text-align: left; line-height: 14pt; text-indent: 0pt; padding-left: 152pt;"><span class="s181">u</span><span class="s189">i</span> <span class="s215">:=</span><span class="s145"> 2</span><span class="s56"></span>n <span class="s173"></span> <span class="s181">s</span>j<span class="s56"></span><span class="s50">1 </span><span class="s173"></span></p>
<p class="s34" style="text-align: left; text-indent: 0pt; padding-left: 5pt; padding-top: 1pt;">e<span class="s146">k</span> , i <span class="s145">= 1</span>, <span class="s145">2</span>, . . . , m,</p>
<p class="s144">j<span class="s50">=1</span></p>
<p class="s144">k<span class="s56"></span>A</p>
<p></p>
<p class="s172" style="text-align: left; line-height: 79%; text-indent: 0pt; padding-left: 36pt; padding-top: 3pt;">i j,n</p>
<p style="text-align: left; text-indent: 0pt; padding-left: 5pt; padding-top: 5pt;">are copies of an element from <span class="s34">l</span><span class="s146">m</span>. Moreover, by formula (9), we have</p>
<p class="s144" style="text-align: right; line-height: 3pt; text-indent: 0pt; padding-top: 3pt;">m</p>
<p>1</p>
<p></p>
<p></p>
<p>1</p>
<p>1</p>
<p class="s144">i<span class="s50">=1</span></p>
<p style="text-align: left; line-height: 9pt; text-indent: 0pt; padding-left: 16pt; padding-top: 1pt;">1</p>
<p class="s34">c<span class="s146">i</span>u<span class="s146">i</span><span class="s173">1</span></p>
<p style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 16pt;">1<span class="s166">ℓ</span><span class="s199">M</span></p>
<p class="s144" style="text-align: right; line-height: 3pt; text-indent: 0pt; padding-top: 3pt;">m</p>
<p style="text-align: center; line-height: 4pt; text-indent: 0pt; padding-left: 10pt;">1</p>
<p></p>
<p></p>
<p class="s220" style="text-align: left; line-height: 9pt; text-indent: 0pt; padding-left: 2pt;">= <span class="p">1</span></p>
<p style="text-align: center; line-height: 6pt; text-indent: 0pt; padding-left: 10pt;">1</p>
<p class="s144">i<span class="s50">=1</span></p>
<p style="text-align: left; line-height: 9pt; text-indent: 0pt; padding-left: 15pt; padding-top: 1pt;">1</p>
<p class="s34">c<span class="s146">i</span>e<span class="s146">i</span><span class="s173">1</span></p>
<p style="text-align: left; line-height: 13pt; text-indent: 0pt; padding-left: 15pt;">1<span class="s166">ℓ</span><span class="s199">F</span><span class="s219">n</span></p>
<p style="text-align: left; line-height: 16pt; text-indent: 0pt; padding-left: 5pt;">for all <span class="s34">c</span><span class="s146">i</span> <span class="s44"> </span><span class="s147">R</span>. Combining this with (15), we get (1), which completes the proof.</p>
<p></p>
</li>
<li>
<h2>Symmetric finite representability of <span class="s197">ℓ</span><span class="s198">p</span> in Orlicz function spaces</h2>
</li>
</ol>
<h4 style="text-align: left; text-indent: 0pt; padding-left: 20pt; padding-top: 8pt;">Theorem 2</h4>
<p></p>
<p class="s34" style="text-align: left; line-height: 46%; text-indent: 14pt; padding-left: 5pt;"><span class="p">Let </span>M <span class="p">be an Orlicz function satisfying </span><span class="s145">∆</span><span class="s52">2</span><span class="p">-condition at infinity. Then </span>ℓ<span class="s143">p </span><span class="p">is symmetrically finitely representable in the Orlicz function space </span>L<span class="s146">M </span><span class="p">if and only if </span>p <span class="s44"> </span><span class="s145">[</span><span class="s55"></span> , <span class="s55"></span><span class="s145">]</span><span class="p">, i.e., </span><span class="s44">F </span><span class="s145">(</span>L<span class="s146">M</span> <span class="s145">) = [</span><span class="s55"></span> , <span class="s55"></span><span class="s145">]</span><span class="p">.</span></p>
<h4 style="text-align: left; text-indent: 0pt; padding-left: 20pt; padding-top: 6pt;">Proof.</h4>
<p class="s144" style="text-align: left; line-height: 2pt; text-indent: 0pt; padding-left: 20pt;">M M M M</p>
<p class="s145" style="text-align: left; line-height: 13pt; text-indent: 0pt; padding-left: 20pt;"><span class="p">As in the sequence case, we need only to prove the embedding </span>[<span class="s34"></span><span class="s55"></span> <span class="s34">, </span><span class="s55"></span>] <span class="s44"> F </span>(<span class="s34">L</span><span class="s146">M</span> )<span class="p">. More precisely, we</span></p>
<p class="s144" style="text-align: left; line-height: 7pt; text-indent: 0pt; padding-left: 325pt;">M M</p>
<p class="s144">M</p>
<p></p>
<p class="s144">M</p>
<p></p>
<p class="s34" style="text-align: left; line-height: 11pt; text-indent: 0pt; padding-left: 5pt;"><span class="p">have to check that for every </span>p <span class="s44"> </span><span class="s145">[</span><span class="s55"></span> , <span class="s55"></span></p>
<p><span class="s145">]</span>, <span class="s34">m </span><span class="s44"> </span><span class="s147">N </span>and each <span class="s34"> </span><span class="s145">0 </span>there exist equimeasurable and disjointly</p>
<p style="text-align: left; line-height: 11pt; text-indent: 0pt; padding-left: 5pt;">supported functions</p>
<p class="s144">k<span class="s50">=1</span></p>
<p></p>
<p class="s34" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 3pt;">u<span class="s146">k</span> <span class="p">, </span>k <span class="s145">= 1</span>, <span class="s145">2</span>, . . . , m<span class="p">, satisfying for all </span>c <span class="s145">= (</span>c<span class="s146">k</span> <span class="s145">)</span><span class="s143">m</span></p>
<p class="s44" style="text-align: left; line-height: 13pt; text-indent: 0pt; padding-left: 1pt;"> <span class="s147">R</span><span class="s143">m</span> <span class="p">the inequality:</span></p>
<p style="text-align: center; line-height: 10pt; text-indent: 0pt; padding-left: 70pt;">1 <span class="s153">m </span>1</p>
<p class="s34" style="text-align: left; line-height: 7pt; text-indent: 0pt; padding-left: 150pt;"><span class="s145">(1 + </span><span class="s145">)</span><span class="s123"></span><span class="s50">1</span><span class="s44">∥</span>c<span class="s44">∥</span><span class="s146">p</span> <span class="s41">� </span><span class="s154">1</span> <span class="s155"></span> c<span class="s146">k</span> u<span class="s146">k</span> <span class="s154">1</span></p>
<p class="s145" style="text-align: left; line-height: 7pt; text-indent: 0pt; padding-left: 14pt;"><span class="s41">� </span>(1 + <span class="s34"></span>)<span class="s44">∥</span><span class="s34">c</span><span class="s44">∥</span><span class="s146">p</span> <span class="p">(16)</span></p>
<p>1</p>
<p class="s154">1 <span class="s144">k</span><span class="s50">=1</span></p>
<p style="text-align: left; line-height: 1pt; text-indent: 0pt; padding-left: 19pt;">1</p>
<p style="text-align: left; line-height: 15pt; text-indent: 0pt; padding-left: 19pt;">1<span class="s166">L</span><span class="s199">M</span></p>
<p class="s144">M</p>
<p></p>
<p class="s34" style="text-align: left; line-height: 60%; text-indent: 14pt; padding-left: 5pt; padding-top: 2pt;"><span class="p">First, </span>t<span class="s143">p</span> <span class="s44"> </span>C<span class="s55"></span> <span class="s44"> </span>C<span class="s145">[0</span>, <span class="s145">1] </span><span class="p">and then the same reasoning as in the proof of Theorem 1 shows that and that for every </span>n <span class="s44"> </span><span class="s147">N </span><span class="p">there is a probabilistic measure </span><span class="s146">n </span><span class="p">on </span><span class="s145">[2</span><span class="s143">n</span>, <span class="s44"></span><span class="s145">) </span><span class="p">such that for all </span>t <span class="s44"> </span><span class="s145">[0</span>, <span class="s145">1]</span></p>
<p style="text-align: right; line-height: 3pt; text-indent: 0pt; padding-top: 8pt;"><span class="s191">p</span></p>
<p></p>
<p></p>
<p><span class="s34">t </span><span class="s44"></span></p>
<p class="s178"><span class="s150"></span> <span class="s123"> </span>M <span class="s182">(</span>t<span class="s182">)</span></p>
<p></p>
<p></p>
<p class="s34" style="text-align: center; line-height: 12pt; text-indent: 0pt; padding-top: 5pt;">d<span class="s146">n</span><span class="s145">(</span><span class="s145">) </span> <span class="s145">2</span><span class="s123"></span><span class="s144">n</span>.</p>
<p></p>
<p style="text-align: left; text-indent: 0pt; padding-left: 5pt; padding-top: 7pt;">For any <span class="s34">s </span><span class="s145">1 </span>and <span class="s34">n, j </span><span class="s44"> </span><span class="s147">N </span>we define</p>
<p class="s34" style="text-align: left; line-height: 13pt; text-indent: 0pt; padding-left: 5pt;"><span class="s61">2</span><span class="s172">n </span>M <span class="s145">(</span><span class="s145">)</span></p>
<p class="s172">j n</p>
<p></p>
<p class="s158" style="text-align: left; text-indent: 0pt; padding-left: 27pt; padding-top: 7pt;"> <span class="s144">s </span><span class="s50">2</span></p>
<p class="s178" style="text-align: left; line-height: 9pt; text-indent: 0pt; padding-left: 5pt; padding-top: 26pt;">d<span class="s179">n</span><span class="s182">(</span><span class="s182">)</span></p>
<p style="text-align: left; line-height: 11pt; text-indent: 0pt; padding-left: 5pt; padding-top: 18pt;">Then, by the preceding inequality,</p>
<p class="s56"></p>
<p class="s34" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 5pt;">a<span class="s146">j,n</span> <span class="s145">:=</span></p>
<p class="s172" style="text-align: left; text-indent: 0pt; padding-left: 5pt; padding-top: 8pt;"><span class="s153">s</span>j<span class="s136"></span><span class="s149">1 </span><span class="s61">2</span>n</p>
<p class="s34" style="text-align: left; line-height: 7pt; text-indent: 0pt; padding-left: 33pt;">.</p>
<p class="s34" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 5pt;">M <span class="s145">(</span><span class="s145">)</span></p>
<p></p>
<p class="s56" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 9pt;"></p>
<p class="s34" style="text-align: left; line-height: 16pt; text-indent: 0pt; padding-left: 122pt;"><span class="s155"></span> a<span class="s146">j,n</span>M <span class="s145">(</span>s<span class="s157">j</span><span class="s56"></span><span class="s50">1</span><span class="s145">2</span><span class="s157">n</span>t<span class="s145">) </span><span class="s44"> </span><span class="s145">2</span><span class="s123"></span><span class="s144">n </span> t<span class="s157">p</span> <span class="s155"></span> a<span class="s146">j,n</span>M <span class="s145">(</span>s<span class="s157">j</span> <span class="s145">2</span><span class="s157">n</span>t<span class="s145">) + 2</span><span class="s123"></span><span class="s144">n</span>.</p>
<p class="s144">j<span class="s50">=1</span></p>
<p style="text-align: left; text-indent: 0pt; padding-left: 5pt; padding-top: 2pt;">Next, as <span class="s34">M </span>satisfies the <span class="s145">∆</span><span class="s52">2</span>-condition at infinity, we have</p>
<p class="s144">j<span class="s50">=1</span></p>
<p class="s145" style="text-align: center; line-height: 15pt; text-indent: 0pt; padding-left: 70pt;"><span class="s34">M </span>(<span class="s34">s</span><span class="s157">j</span> 2<span class="s157">n</span><span class="s34">t</span>) <span class="s41">� </span>(1 + 2<span class="s123"></span><span class="s144">n</span>)<span class="s34">M </span>(<span class="s34">s</span><span class="s157">j</span><span class="s56"></span><span class="s50">1</span>2<span class="s157">n</span><span class="s34">t</span>)</p>
<p class="s34" style="text-align: left; line-height: 19pt; text-indent: 0pt; padding-left: 5pt;"><span class="p">for all </span>j <span class="s44"> </span><span class="s147">N </span><span class="p">and </span>t <span class="s44"> </span><span class="s145">[0</span>, <span class="s145">1] </span><span class="p">whenever </span>s <span class="p">is sufficiently close to 1. Fixing such a </span>s<span class="p">, we get</span></p>
<p class="s56" style="text-align: left; line-height: 9pt; text-indent: 0pt; padding-left: 100pt;"> </p>
<p class="s34" style="text-align: left; line-height: 16pt; text-indent: 0pt; padding-left: 97pt;"><span class="s155"></span> a<span class="s146">j,n</span>M <span class="s145">(</span>s<span class="s157">j</span><span class="s56"></span><span class="s50">1</span><span class="s145">2</span><span class="s157">n</span>t<span class="s145">) </span><span class="s44"> </span><span class="s145">2</span><span class="s123"></span><span class="s144">n </span> t<span class="s157">p</span> <span class="s155"></span><span class="s145">(1 + 2</span><span class="s123"></span><span class="s144">n</span><span class="s145">)</span>a<span class="s146">j,n</span>M <span class="s145">(</span>s<span class="s157">j</span><span class="s56"></span><span class="s50">1</span><span class="s145">2</span><span class="s157">n</span>t<span class="s145">) + 2</span><span class="s123"></span><span class="s144">n</span>.</p>
<p class="s144" style="text-align: center; line-height: 10pt; text-indent: 0pt; padding-left: 3pt;">j<span class="s50">=1</span></p>
<p style="text-align: center; line-height: 11pt; text-indent: 0pt; padding-left: 5pt; padding-top: 2pt;">Combining this inequality with the estimate</p>
<p class="s56"></p>
<p class="s144" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 5pt;">j<span class="s50">=1</span></p>
<p></p>
<p style="text-align: left; line-height: 11pt; text-indent: 0pt; padding-left: 5pt;">we deduce</p>
<p class="s34" style="text-align: left; line-height: 15pt; text-indent: 0pt; padding-left: 5pt;"><span class="s145">2</span><span class="s123"></span><span class="s144">n </span><span class="s155"></span> a<span class="s146">j,n</span>M <span class="s145">(</span>s<span class="s157">j</span><span class="s56"></span><span class="s50">1</span><span class="s145">2</span><span class="s157">n</span>t<span class="s145">) </span> <span class="s145">2</span><span class="s123"></span><span class="s50">2</span><span class="s144">n </span><span class="s145">+ 2</span><span class="s123"></span><span class="s144">n</span>t<span class="s157">p</span> <span class="s145">2</span><span class="s123"></span><span class="s144">n</span><span class="s50">+1</span>, <span class="s145">0 </span><span class="s41">� </span>t <span class="s41">� </span><span class="s145">1</span>,</p>
<p class="s144" style="text-align: left; text-indent: 0pt; padding-left: 23pt; padding-top: 1pt;">j<span class="s50">=1</span></p>
<p class="s56" style="text-align: left; line-height: 4pt; text-indent: 0pt; padding-left: 115pt;"> </p>
<p class="s34" style="text-align: left; line-height: 16pt; text-indent: 0pt; padding-left: 112pt;"><span class="s155"></span> a<span class="s146">j,n</span>M <span class="s145">(</span>s<span class="s157">j</span><span class="s56"></span><span class="s50">1</span><span class="s145">2</span><span class="s157">n</span>t<span class="s145">) </span><span class="s44"> </span><span class="s145">2</span><span class="s123"></span><span class="s144">n </span> t<span class="s157">p</span> <span class="s155"></span> a<span class="s146">j,n</span>M <span class="s145">(</span>s<span class="s157">j</span><span class="s56"></span><span class="s50">1</span><span class="s145">2</span><span class="s157">n</span>t<span class="s145">) + 2</span><span class="s123"></span><span class="s144">n</span><span class="s50">+2</span>. <span class="p">(17)</span></p>
<p class="s144">j<span class="s50">=1</span></p>
<p class="s144" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 112pt;">j<span class="s50">=1</span></p>
<p class="s40" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 33pt; padding-top: 2pt;">Astashkin S.V. Symmetric finite representability of <span class="s16">ℓ</span><span class="s162">p </span>in Orlicz spaces</p>
<p><img src="Vestnik EN 2020_26_4/Image_113.png" alt="image" width="636" height="1" /></p>
<p class="s40" style="text-align: left; line-height: 13pt; text-indent: 0pt; padding-left: 5pt;"><span class="s164">22 </span>Асташкин С.В. Симметричная финитная представимость <span class="s16">ℓ</span><span class="s162">p </span>в пространствах Орлича</p>
<p></p>
<p class="s34"><span class="p">On the other hand, since </span>M <span class="s145">(</span>u<span class="s145">) </span><span class="s41">;;: </span>u <span class="p">for all </span>u <span class="s41">;;: </span><span class="s145">1</span><span class="p">, we have</span></p>
<p class="s107" style="text-align: right; line-height: 9pt; text-indent: 0pt; padding-top: 4pt;"><span class="s192">2 </span><span class="s56"></span><span class="s144">n</span></p>
<p></p>
<p class="s56" style="text-align: left; line-height: 7pt; text-indent: 0pt; padding-left: 2pt;"><span class="s144">j</span><span class="s50">+1</span></p>
<p></p>
<p>which implies that</p>
<p class="s34" style="text-align: center; line-height: 17pt; text-indent: 0pt; padding-left: 4pt;">a<span class="s146">j,n</span> <span class="s41">� </span><span class="s216">M</span> <span class="s145">(2</span><span class="s217">n</span><span class="s216">s</span><span class="s217">j</span><span class="s56"></span><span class="s50">1 </span><span class="s41">� </span><span class="s145">2 </span>s ,</p>
<p class="s56" style="text-align: center; line-height: 12pt; text-indent: 0pt; padding-left: 4pt; padding-top: 10pt;"> <span class="s221">s</span></p>
<p class="s144" style="text-align: left; line-height: 16pt; text-indent: 0pt; padding-left: 159pt;"><span class="s173"></span> <span class="s181">a</span><span class="s189">j,n</span> <span class="s222">�</span> <span class="s145">2</span><span class="s56"></span>n <span class="s173"></span> <span class="s181">s</span><span class="s56"></span>j<span class="s50">+1 </span><span class="s215">=</span><span class="s145"> 2</span><span class="s56"></span>n<span class="s50">+1 </span><span class="s223"></span></p>
<p class="s34" style="text-align: left; line-height: 6pt; text-indent: 0pt; padding-left: 24pt; padding-top: 2pt;">.</p>
<p><img src="Vestnik EN 2020_26_4/Image_114.png" alt="image" width="30" height="1" /></p>
<p class="s34" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 1pt;">s <span class="s44"> </span><span class="s145">1</span></p>
<p class="s144">j<span class="s50">=1</span></p>
<p class="s144" style="text-align: left; line-height: 8pt; text-indent: 0pt; padding-left: 48pt;">j<span class="s50">=1</span></p>
<p style="text-align: left; line-height: 16pt; text-indent: 0pt; padding-left: 5pt;">Let <span class="s34">m </span><span class="s44"> </span><span class="s147">N </span>and <span class="s34"> </span><span class="s145">0 </span>be arbitrary. Fix <span class="s34">n </span>so that</p>
<p class="s212" style="text-align: right; line-height: 14pt; text-indent: 0pt; padding-top: 2pt;">2<span class="s56"></span><span class="s144">n</span><span class="s50">+1</span><span class="s194">s</span> <span class="s145">1</span></p>
<p class="s34"></p>
<p><img src="Vestnik EN 2020_26_4/Image_115.png" alt="image" width="42" height="1" /></p>
<p><img src="Vestnik EN 2020_26_4/Image_116.png" alt="image" width="12" height="1" /></p>
<p class="s34">s <span class="s44"> </span><span class="s145">1 </span>m</p>
<p style="text-align: left; text-indent: 0pt; padding-left: 7pt; padding-top: 8pt;">and <span class="s145">2</span><span class="s123"></span><span class="s144">n</span><span class="s50">+2</span><span class="s34">m . </span>(18)</p>
<p class="s144">j</p>
<p></p>
<p class="s34" style="text-align: left; line-height: 14pt; text-indent: 0pt; padding-left: 5pt;"><span class="p">The first of the inequalities (18) allows us to take pairwise disjoint sets </span>E<span class="s143">i</span> <span class="s44"> </span><span class="s145">[0</span>, <span class="s145">1]</span><span class="p">, </span>j <span class="s44"> </span><span class="s147">N</span><span class="p">, </span>i <span class="s145">= 1</span>, <span class="s145">2</span>, . . . , m<span class="p">,</span></p>
<p class="s144">j</p>
<p></p>
<p class="s34" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 5pt;"><span class="p">with </span>m<span class="s145">(</span>E<span class="s143">i</span> <span class="s145">) = </span>a<span class="s146">j,n</span><span class="p">. Then, the functions</span></p>
<p></p>
<p class="s56" style="text-align: center; line-height: 6pt; text-indent: 0pt; padding-left: 6pt;"></p>
<p class="s144">E</p>
<p></p>
<p class="s34" style="text-align: center; line-height: 16pt; text-indent: 0pt; padding-left: 6pt;">u<span class="s146">i</span> <span class="s145">:= </span><span class="s155"></span> <span class="s145">2</span><span class="s157">n</span>s<span class="s157">j</span><span class="s56"></span><span class="s50">1</span> <span class="s172">i</span></p>
<p class="s172" style="text-align: center; line-height: 4pt; text-indent: 0pt; padding-left: 86pt;">j</p>
<p class="s144" style="text-align: center; line-height: 7pt; text-indent: 0pt; padding-left: 70pt;">j<span class="s50">=1</span></p>
<p class="s145" style="text-align: left; line-height: 18pt; text-indent: 0pt; padding-left: 5pt;"><span class="p">are equimeasurable and disjointly supported on </span>[0<span class="s34">, </span>1]<span class="p">. Moreover, for all </span><span class="s34">c</span><span class="s146">i</span> <span class="s44"> </span><span class="s147">R</span></p>
<p style="text-align: right; line-height: 12pt; text-indent: 0pt; padding-top: 2pt;"><span class="s158"></span> <span class="s50">1 </span>( <span class="s203">m</span></p>
<p class="s224" style="text-align: left; line-height: 12pt; text-indent: 0pt; padding-left: 31pt; padding-top: 2pt;">) <span class="s144">m </span><span class="s56"></span></p>
<p class="s34">M <span class="s155"></span> c<span class="s146">i</span>u<span class="s146">i</span><span class="s145">(</span>t<span class="s145">)</span></p>
<p class="s34" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-left: 5pt;">dt <span class="s145">= </span><span class="s155"></span><span class="p"> </span>M <span class="s145">(2</span><span class="s157">n</span>s<span class="s157">j</span><span class="s56"></span><span class="s50">1</span><span class="s44">|</span>c<span class="s146">i</span><span class="s44">|</span><span class="s145">)</span>a<span class="s146">j,n</span>.</p>
<p></p>
<p class="s50">0</p>
<p></p>
<p class="s144">i<span class="s50">=1</span></p>
<p class="s144" style="text-align: left; text-indent: 0pt; padding-left: 62pt; padding-top: 5pt;">i<span class="s50">=1 </span>j<span class="s50">=1</span></p>
<p style="text-align: left; text-indent: 0pt; padding-left: 5pt; padding-top: 2pt;">Therefore, by (17) and the second inequality in (18), we get</p>
<p></p>
<table cellspacing="0">
<tbody>
<tr>
<td colspan="2">
<p class="s225" style="text-align: left; line-height: 2pt; text-indent: 0pt; padding-left: 5pt; padding-top: 3pt;">m</p>
<p class="s226" style="text-align: left; line-height: 6pt; text-indent: 0pt; padding-left: 2pt;"> <span class="s227"></span></p>
<p class="s228" style="text-align: center; line-height: 12pt; text-indent: 0pt; padding-left: 9pt;">|<span class="s229">c</span><span class="s230">i</span>|<span class="s231">p</span> <span class="s229"> </span></p>
</td>
<td></td>
<td>
<p class="s232" style="text-align: left; line-height: 10pt; text-indent: 0pt; padding-top: 3pt;">1</p>
<p class="s229" style="text-align: left; line-height: 9pt; text-indent: 0pt; padding-left: 6pt;">M</p>
</td>
<td>
<p class="s225" style="text-align: center; line-height: 3pt; text-indent: 0pt; padding-right: 2pt; padding-top: 3pt;">m</p>
<p class="s227">( <span class="s233"></span></p>
<p class="s229" style="text-align: left; line-height: 9pt; text-indent: 0pt; padding-left: 6pt;">c<span class="s230">i</span>u</p>
</td>
<td>
<p class="s227" style="text-align: center; line-height: 11pt; text-indent: 0pt; padding-left: 2pt; padding-top: 2pt;">)</p>
<p class="s235" style="text-align: center; line-height: 10pt; text-indent: 0pt; padding-right: 1pt;"><span class="s230">i</span>(<span class="s229">t</span>) <span class="s229">dt</span></p>
</td>
<td>
<p class="s225" style="text-align: left; line-height: 4pt; text-indent: 0pt; padding-left: 15pt; padding-top: 3pt;">m</p>
<p class="s236"> <span class="s227"></span></p>
</td>
</tr>
<tr>
<td>
<p class="s225" style="text-align: left; text-indent: 0pt; padding-left: 2pt; padding-top: 3pt;">i<span class="s232">=1</span></p>
</td>
<td></td>
<td></td>
<td></td>
<td>
<p class="s227" style="text-align: left; line-height: 2pt; text-indent: 0pt; padding-left: 6pt;"></p>
<p class="s225" style="text-align: left; text-indent: 0pt; padding-left: 12pt; padding-top: 1pt;">i<span class="s232">=1</span></p>
</td>
<td>
<p class="s227" style="text-align: center; line-height: 2pt; text-indent: 0pt; padding-right: 2pt;"></p>
</td>
<td>
<p class="s225" style="text-align: left; text-indent: 0pt; padding-left: 12pt; padding-top: 3pt;">i<span class="s232">=1</span></p>
</td>
</tr>
</tbody>
</table>
<p></p>
<p class="s44" style="text-align: left; line-height: 17pt; text-indent: 0pt; padding-left: 327pt;">|<span class="s34">c</span><span class="s146">i</span>|<span class="s157">p</span> <span class="s145">+ </span><span class="s34">.</span></p>
<p class="s50" style="text-align: center; line-height: 9pt; text-indent: 0pt; padding-left: 70pt;">0</p>
<p style="text-align: left; text-indent: 0pt; padding-left: 5pt; padding-top: 3pt;">Repeating further the arguments from the end of the proof of Theorem 1, we come to (16) and so complete the proof.</p>[[1] Tsirel’son B.S. Not every Banach space contains an imbedding ofl p or c0. Functional Analysis and Its Applications, 1974, vol. 8, no. 2, pp. 138–141. DOI: https://doi.org/10.1007/BF01078599. (English; Russian original)][[2] Krivine J.L. Sous-espaces de dimension finie des espaces de Banach r´eticul´es. Annals of Mathematics, 1976, vol. 104, no. 2, pp. 1–29. Available at: https://www.irif.fr/krivine/articles/Espaces_reticules.pdf.][[3] Rosenthal H.P. On a theorem of J.L. Krivine concerning block finite representability of ℓp in general Banach spaces. Journal of Functional Analysis, 1978, vol. 28, pp. 197–225. DOI: http://dx.doi.org/10.1016/0022-1236(78)90086-1.][[4] Albiac F., Kalton N.J. Topics in Banach Space Theory. Graduate Texts in Mathematics 233. New York: Springer-Verlag, 2006. 373 p. DOI: http://dx.doi.org/10.1007/0-387-28142-8.][[5] Krein S.G., Petunin Yu.I., Semenov E.M. Interpolation of linear operators. Moscow: Nauka, 1978, 400 p. Available at: https://elibrary.ru/item.asp?id=21722209; https://booksee.org/book/577975. (In Russ.)][[6] Lindenstrauss J., Tzafriri L. Classical Banach Spaces, II. Function Spaces. Berlin, Heidelberg, New York: Springer-Verlag, 1979, 243 p. Available at: https://1lib.education/book/2307307/8b833b?dsource=recommend.][[7] Lindenstrauss J., Tzafriri L. Classical Banach Spaces, I. Sequence Spaces. Berlin–New York: Springer-Verlag, 1977. 190 p. Available at: https://1lib.education/book/2264754/01841c?dsource=recommend.][[8] Astashkin S.V. On the finite representability of ℓp-spaces in rearrangement invariant spaces. St. Petersburg Math. J., 2012, vol. 23, no. 2, pp. 257–273. DOI: http://doi.org/10.1090/S1061-0022-2012-01196-9. (English; Russian original)][[9] Krasnoselskii M.A., Rutickii Ya.B. Convex functions and Orlicz spaces. Moscow: Gos. izd. fiz.-mat. lit., 1958, 271 p. Available at: https://1lib.education/book/2078048/983381?id=2078048&secret=983381. (In Russ.)][[10] Rao M.M., Ren Z.D. Theory of Orlicz spaces. Monographs and Textbooks in Pure and Applied Mathematics, vol. 146. New York: Marcel Dekker Inc., 1991. 445 p.][[11] Maligranda L. Orlicz Spaces and Interpolation. Seminars in Mathematics 5. Campinas: University of Campinas, 1989. 206 p.][[12] Lindenstrauss Y., Tzafriri L. On Orlicz sequence spaces. III. Israel Journal of Mathematics, 1973, vol. 14, pp. 368–389. DOI: https://doi.org/10.1007/BF02771656.][[13] Rudin W. Functional Analysis. Moscow: Mir, 1975, 443 p. Available at: https://www.nehudlit.ru/books/funktsionalnyy-analiz.html.]