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Weighted norm inequalities for the Hardy maximal function. (English) Zbl 0236.26016

The principal problem considered is the determination of all nonnegative functions, $U\left(x\right)$, for which there is a constant, $C$, such that

${\int }_{J}{\left[{f}^{*}\left(x\right)\right]}^{p}U\left(x\right)\phantom{\rule{0.166667em}{0ex}}dx\le C{\int }_{J}{|f\left(x\right)|}^{p}U\left(x\right)\phantom{\rule{0.166667em}{0ex}}dx,$

where $1, $J$ is a fixed interval, $C$ is independent of $f$, and ${f}^{*}$ is the Hardy maximal function,

${f}^{*}\left(x\right)=\underset{y\ne x;\phantom{\rule{4pt}{0ex}}y\in J}{sup}\frac{1}{y-x}{\int }_{x}^{y}|f\left(t\right)|\phantom{\rule{0.166667em}{0ex}}dt·$

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

$\left[{\int }_{I}U\left(x\right)\phantom{\rule{0.166667em}{0ex}}dx\right]{\left[{\int }_{I}{\left[U\left(x\right)\right]}^{-1/\left(p-1\right)}\phantom{\rule{0.166667em}{0ex}}dx\right]}^{p-1}\le K{|I|}^{p}$

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.

##### MSC:
 42B25 Maximal functions, Littlewood-Paley theory 42B20 Singular and oscillatory integrals, several variables 26D15 Inequalities for sums, series and integrals of real functions 42A24 Summability and absolute summability of Fourier and trigonometric series