# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## Weighted norm inequalities for the Hardy maximal functionHTML articles powered by AMS MathViewer

by Benjamin Muckenhoupt
Trans. Amer. Math. Soc. 165 (1972), 207-226 Request permission

## Abstract:

The principal problem considered is the determination of all nonnegative functions, $U(x)$, for which there is a constant, C, such that $\int _J {{{[{f^ \ast }(x)]}^p}U(x)dx \leqq C\int _J {|f(x){|^p}U(x)dx,} }$ where $1 < p < \infty$, J is a fixed interval, C is independent of f, and ${f^ \ast }$ is the Hardy maximal function, ${f^ \ast }(x) = \sup \limits _{y \ne x;y \in J} \frac {1}{{y - x}}\int _x^y {|f(t)|dt.}$ The main result is that $U(x)$ is such a function if and only if $\left [ {\int _I {U(x)dx} } \right ]{\left [ {\int _I {{{[U(x)]}^{ - 1/(p - 1)}}dx} } \right ]^{p - 1}} \leqq K|I{|^p}$ where I is any subinterval of J, $|I|$ denotes the length of I and K is a constant independent of I. Various related problems are also considered. These include weak type results, the problem when there are different weight functions on the two sides of the inequality, the case when $p = 1$ or $p = \infty$, a weighted definition of the maximal function, and the result in higher dimensions. Applications of the results to mean summability of Fourier and Gegenbauer series are also given.
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