Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

Query:
Fill in the form 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

Highlights
Master Server