On the Hardy-Littlewood maximal function and some applications

Abstract:

With a monotone family $F = \{ {S_\alpha }\} , {S_\alpha } \subset {{\textbf {R}}^n}$, we associate the Hardy-Littlewood maximal function ${M_F}f(x) = {\sup _\alpha }(1/\left | {{S_\alpha }} \right |)\int _{{S_\alpha } + x} {\left | f \right |}$. In general, ${M_F}$ is not weak type (1.1). However, if we replace in the denominator ${S_\alpha }$ by $S_F^ {\ast } = \{ x - y: x, y \in {S_\alpha }\}$, and denote the resulting maximal function by $M_F^ {\ast }$, then $M_F^ {\ast }$ is weak type (1, 1) with weak type constant 1.