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 Mat. Sb. (N.S.), 1988, Volume 136(178), Number 1(5), Pages 3–23 (Mi msb1725)

Estimates of rearrangements and imbedding theorems

Abstract: The modulus of continuity of a function $f\in L^p(I^N)$ ($1\leqslant p<\infty$, $I=[0,1]$), 1-periodic in each variable is defined by
$$\omega_p(f;\delta)=\sup_{|h|\leqslant\delta}(\int_{I^N}|f(x)-f(x+h)|^p dx)^{1/p}.$$
The following estimate is established for the nonincreasing rearrangement of a function $f\in L^p(I^N)$ ($p,N\geqslant1$; $\Delta A_n=A_{n+1}-A_n$):
$$\sum^\infty_{n=s}2^{-nN}(\Delta f^*(2^{-nN}))^p +2^{-sp}\sum_{n=1}^s2^{n(p-N)}(\Delta f^*(2^{-nN}))^p\leqslant c\omega_p^p(f;2^{-s}).$$
Also, analytic functions of Hardy class $H^p$ in the unit disk are considered. It is proved that the inequality (1) ($N=1$) holds for the rearrangements of their boundary values also when $0<p<1$ (this is false for real functions of class $L^p$).
Inequality (1) is used to find necessary and sufficient conditions for the space $H^\omega_{p,N}$ ($1\leqslant p<N$) of functions with a given majorant of the $L^p$-modulus of continuity to be imbedded in the Orlicz classes $\varphi(L)$, where $\varphi$ satisfies the $\Delta_2$-condition and $\varphi(t)t^{-p}\uparrow$ on $(0,\infty)$. For $p\geqslant N$ the solution of this problem follows from estimates obtained earlier by the author (RZh.Mat., 1975, 8B 62).
An analogous result is established for classes of functions in the Hardy space $H^p$ ($0<p<1$).
The imbeddings with limiting exponent (Sobolev and Hardy–Littlewood theorems) are limiting cases of the results in this article.
Bibliography: 27 titles.

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English version:
Mathematics of the USSR-Sbornik, 1989, 64:1, 1–21

Bibliographic databases:

UDC: 517.5
MSC: Primary 46E35, 46E30; Secondary 26A15, 26A16, 30D55

Citation: V. I. Kolyada, “Estimates of rearrangements and imbedding theorems”, Mat. Sb. (N.S.), 136(178):1(5) (1988), 3–23; Math. USSR-Sb., 64:1 (1989), 1–21

Citation in format AMSBIB
\Bibitem{Kol88} \by V.~I.~Kolyada \paper Estimates of rearrangements and imbedding theorems \jour Mat. Sb. (N.S.) \yr 1988 \vol 136(178) \issue 1(5) \pages 3--23 \mathnet{http://mi.mathnet.ru/msb1725} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=945897} \zmath{https://zbmath.org/?q=an:0693.46030} \transl \jour Math. USSR-Sb. \yr 1989 \vol 64 \issue 1 \pages 1--21 \crossref{https://doi.org/10.1070/SM1989v064n01ABEH003291} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. I. Kolyada, “Rearrangements of functions and embedding theorems”, Russian Math. Surveys, 44:5 (1989), 73–117
2. A. M. Stokolos, “Differentiation of integrals by bases without the density property”, Sb. Math., 187:7 (1996), 1061–1085
3. Lazaro, FJP, “A note on extreme cases of Sobolev embeddings”, Journal of Mathematical Analysis and Applications, 320:2 (2006), 973
4. Amiran Gogatishvili, Luboš Pick, Jan Schneider, “Characterization of a rearrangement-invariant hull of a Besov space via interpolation”, Rev Mat Complut, 2011
5. Ioku N., “Sharp Sobolev Inequalities in Lorentz Spaces for a Mean Oscillation”, J. Funct. Anal., 266:5 (2014), 2944–2958
6. O. V. Besov, “Embeddings of Sobolev spaces in the case of the limit exponent”, Dokl. Math, 91:3 (2015), 277
7. O. V. Besov, “Embedding of Sobolev Space in the Case of the Limit Exponent”, Math. Notes, 98:4 (2015), 550–560
8. E. D. Kosov, “Klassy Besova na konechnomernykh i beskonechnomernykh prostranstvakh”, Matem. sb., 210:5 (2019), 41–71
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