Оценки максимальных функций, иэмеряюших средние колебания

V. Kolyada; В. Коляда

doi:10.1007/bf02908442

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Analysis Mathematica

Volume/Issue:: Volume 25: Issue 1

Оценки максимальных функций, иэмеряюших средние колебания

Authors:

V. Kolyada

V. Kolyada

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and

В. Коляда

В. Коляда

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Pages:: 277–300

Online Publication Date:: 05 Jun 2008

Publication Date:: 01 Dec 1999

Article Category:: Research Article

DOI:: https://doi.org/10.1007/bf02908442

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Abstract

Letη be a nondecreasing function on (0, 1] such thatη(t)/t decreases andη(+0)=0. Letf ∈L(Iⁿ) (I≡[0,1]. Set

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\mathcal{N}}_\eta f(x) = \sup \frac{1}{{\left| Q \right|\eta (\left| Q \right|^{1/n} )}} \smallint _Q \left| {f(t) - f(x)} \right|dt,$$ \end{document}

, where the supremum is taken over all cubes containing the pointx. Forη=t^α (0<α≤1) this definition was given by A.Calderón. In the paper we prove estimates of the maximal functions

, along with some embedding theorems. In particular, we prove the following Sobolev type inequality: if

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$1 \leqslant p< q< \infty , \theta \equiv n(1/p - 1/q)< 1, and \eta (t) \leqslant t^\theta \sigma (t),$$ \end{document}

, then

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\parallel {\mathcal{N}}_\sigma {f} {\parallel_{q,p}} \leqslant c \parallel {\mathcal{N}}_\eta {f} {\parallel_p} .$$ \end{document}

. Furthermore, we obtain estimates of

in terms of theL^p-modulus of continuity off. We find sharp conditions for

to belong toL^p(Iⁿ) and the Orlicz classϕ(L), too.

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Analysis Mathematica

Print ISSN:: 0133-3852

Online ISSN:: 1588-273X

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Analysis Mathematica
Language	English
Size	B5
Year of Foundation	1975
Volumes per Year	1
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ISSN	0133-3852 (Print)
ISSN	1588-273X (Online)