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 $$\lim_{r\to 0}\frac{1}{[m(B_r(x))]^\alpha}\int_{B_r(x)} f(y) dy$$
  • $\begingroup$ I wouldn't call this a weighted H-L maximal function because the measure in the integral doesn't have a weight. I would call this a fractional H-L maximal function. This has been studied; see, for instance, "Regularity of the Fractional Maximal Function" by Kinnunen and Saksman. $\endgroup$ – Ryan Gibara Jun 29 '18 at 9:22
  • $\begingroup$ @Ryan Thank you for the reference. I'd like the $\alpha$ that appears in your paper as $0 \le \alpha < n$ to be negative though. $\endgroup$ – Dal Jun 29 '18 at 11:01
  • $\begingroup$ In the paper, the alpha is in the numerator, whereas yours is in the denominator. $\endgroup$ – Ryan Gibara Jun 29 '18 at 11:15
  • $\begingroup$ @Ryan Yes, and I want it in the denominator with positive exponent. $\endgroup$ – Dal Jun 29 '18 at 12:02
  • $\begingroup$ Ah, I see! I apologize. $\endgroup$ – Ryan Gibara Jun 29 '18 at 12:48

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