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Transactions of the American Mathematical Society

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



Weighted norm inequalities for the Hardy maximal function

Author: Benjamin Muckenhoupt
Journal: Trans. Amer. Math. Soc. 165 (1972), 207-226
MSC: Primary 46E30; Secondary 26A86, 42A40
MathSciNet review: 0293384
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Abstract: The principal problem considered is the determination of all nonnegative functions, $ U(x)$, for which there is a constant, C, such that

$\displaystyle \int_J {{{[{f^ \ast }(x)]}^p}U(x)dx \leqq C\int_J {\vert f(x){\vert^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,

$\displaystyle {f^ \ast }(x) = \mathop {\sup }\limits_{y \ne x;y \in J} \frac{1}{{y - x}}\int_x^y {\vert f(t)\vert dt.} $

The main result is that $ U(x)$ is such a function if and only if

$\displaystyle \left[ {\int_I {U(x)dx} } \right]{\left[ {\int_I {{{[U(x)]}^{ - 1/(p - 1)}}dx} } \right]^{p - 1}} \leqq K\vert I{\vert^p}$

where I is any subinterval of J, $ \vert I\vert$ 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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Keywords: Hardy maximal function, mean summability, Fourier series, Gegenbauer series, weighted norm inequalities
Article copyright: © Copyright 1972 American Mathematical Society