Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0236.26016
Muckenhoupt, Benjamin
Weighted norm inequalities for the Hardy maximal function. (English)
[J] Trans. Am. Math. Soc. 165, 207-226 (1972). ISSN 0002-9947; ISSN 1088-6850/e

The principal problem considered is the determination of all nonnegative functions, $U(x)$, for which there is a constant, $C$, such that $$\int_J [f^*(x)]^p U(x)\,dx \leq C\int_J \vert f(x)\vert^p U(x)\,dx,$$ where $1 < p < \infty$, $J$ is a fixed interval, $C$ is independent of $f$, and $f^*$ is the Hardy maximal function, $$f^*(x) = \sup_{y \ne x;\ y \in J} \frac{1}{y - x}\int_x^y \vert f(t)\vert \,dt.$$ The main result is that $U(x)$ is such a function if and only if $$\left[\int_I U(x)\,dx\right]\left[\int_I [U(x)]^{-1/(p - 1)}\,dx\right]^{p-1} \leq K\vert I\vert^p$$ 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.

Display scanned Zentralblatt-MATH page with this review.
[Benjamin Muckenhoupt]

MSC 2000:: *42B25 Maximal functions
42B20 Singular integrals, several variables
26D15 Inequalities for sums, series and integrals of real functions
42A24 Summability of trigonometric series

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

Cited in: Zbl 1208.30050 Zbl 1101.26309 Zbl 1199.30125 Zbl 1067.62039 Zbl 0919.42002 Zbl 0885.31005 Zbl 0870.47021 Zbl 0802.47029 Zbl 0770.35014 Zbl 0731.30030 Zbl 0718.42018 Zbl 0704.49003 Zbl 0651.46016 Zbl 0677.42019 Zbl 0644.35036 Zbl 0612.28014 Zbl 0619.42014 Zbl 0596.49008 Zbl 0546.35035 Zbl 0521.42019 Zbl 0518.46045 Zbl 0473.42015 Zbl 0447.26008 Zbl 0429.60047 Zbl 0398.60050

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]