# Properties of a weighted maximal function $\sup_{r>0}\frac{1}{[m(B_r(x))]^\alpha}\int_{B_r(x)} f(y) dy$

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 $$\lim_{r\to 0}\frac{1}{[m(B_r(x))]^\alpha}\int_{B_r(x)} f(y) dy$$
• 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. – Ryan Gibara Jun 29 '18 at 9:22
• @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. – Dal Jun 29 '18 at 11:01
• In the paper, the alpha is in the numerator, whereas yours is in the denominator. – Ryan Gibara Jun 29 '18 at 11:15
• @Ryan Yes, and I want it in the denominator with positive exponent. – Dal Jun 29 '18 at 12:02
• Ah, I see! I apologize. – Ryan Gibara Jun 29 '18 at 12:48