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

DOI:
https://doi.org/10.1090/S0002-9947-1972-0293384-6

MathSciNet review:
0293384

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Abstract | References | Similar Articles | Additional Information

Abstract: The principal problem considered is the determination of all nonnegative functions, , for which there is a constant, *C*, such that

*J*is a fixed interval,

*C*is independent of

*f*, and is the Hardy maximal function,

*I*is any subinterval of

*J*, 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 or , 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

DOI:
https://doi.org/10.1090/S0002-9947-1972-0293384-6

Keywords:
Hardy maximal function,
mean summability,
Fourier series,
Gegenbauer series,
weighted norm inequalities

Article copyright:
© Copyright 1972
American Mathematical Society