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Estimates of rearrangements and imbedding theorems
V. I. Kolyada
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
$$
\omega_p(f;\delta)=\sup_{|h|\leqslant\delta}\biggl(\int_{I^N}|f(x)-f(x+h)|^p\,dx\biggr)^{1/p}.
$$
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.
UDC:
517.5
MSC: Primary
46E35,
46E30; Secondary
26A15,
26A16,
30D55 Received: 04.09.1987