## Weighted norm inequalities for the Hardy maximal function

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- by Benjamin Muckenhoupt PDF
- 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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## Additional Information

- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**165**(1972), 207-226 - MSC: Primary 46E30; Secondary 26A86, 42A40
- DOI: https://doi.org/10.1090/S0002-9947-1972-0293384-6
- MathSciNet review: 0293384