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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, $$U(x)$$, for which there is a constant, $$C$$, such that
$\int_J [f^*(x)]^p U(x)\,dx \leq C\int_J | f(x)|^p U(x)\,dx,$ where $$1 < p < \infty$$, $$J$$ is a fixed interval, $$C$$ is independent of $$f$$, and $$f^*$$ is the Hardy maximal function,
$f^*(x) = \sup_{y \neq x;\;y \in J} \frac{1}{y - x}\int_x^y | f(t)| \,dt.$ The main result is that $$U(x)$$ is such a function if and only if
$\left[\int_I U(x)\,dx\right]\left[\int_I [U(x)]^{-1/(p - 1)}\,dx\right]^{p-1} \leq K| I|^p$ 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 (Calderón-Zygmund, etc.) 26D15 Inequalities for sums, series and integrals 42A24 Summability and absolute summability of Fourier and trigonometric series
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References:
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