Where can I find a reference on the properties of the following "weighted Hardy-Littlewood maximal function"?
$$Mf(x) = \sup_{r>0}\frac{1}{[m(B_r(x))]^\alpha}\int_{B_r(x)} f(y) dy, $$ where $m(B_r(x))$ is the Lebesgue's measure of the ball of radius $r$ and centre $x$ and $\alpha > 0$.
In particular, I'll be interested
- in $L^p$ estimates of $Mf$,
- in the relationship between $Mf(x)$ and Holder continuity of $f$,
- and in the relationship between $Mf(x)$ and $$\lim_{r\to 0}\frac{1}{[m(B_r(x))]^\alpha}\int_{B_r(x)} f(y) dy$$