Estimates of rearrangements and imbedding theorems

V. I. Kolyada

Sbornik. Mathematics, Tome 64 (1989) no. 1, pp. 1-21

Résumé

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 $$ \omega_p(f;\delta)=\sup_{|h|\leqslant\delta}\biggl(\int_{I^N}|f(x)-f(x+h)|^p\,dx\biggr)^{1/p}. $$ The following estimate is established for the nonincreasing rearrangement of a function $f\in L^p(I^N)$ ($p,N\geqslant1$; $\Delta A_n=A_{n+1}-A_n$): \begin{equation} \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}). \end{equation} 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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