Weighted norm inequalities for the Hardy maximal function

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

$$\int_J {{{[{f^ \ast }(x)]}^p}U(x)dx \leqq C\int_J {\vert f(x){\vert^p}U(x)dx,} }$$

where $1 < p < \infty$, J is a fixed interval, C is independent of f, and ${f^ \ast }$ is the Hardy maximal function,

$${f^ \ast }(x) = \mathop {\sup }\limits_{y \ne x;y \in J} \frac{1}{{y - x}}\int_x^y {\vert f(t)\vert dt.}$$

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

$$\left[ {\int_I {U(x)dx} } \right]{\left[ {\int_I {{{[U(x)]}^{ - 1/(p - 1)}}dx} } \right]^{p - 1}} \leqq K\vert I{\vert^p}$$

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.