Solyanik estimates and local Hölder continuity of halo functions of geometric maximal operators

Abstract

Let $\mathcal{B}$ be a homothecy invariant basis consisting of convex sets in ${\mathbb{R}}^n$, and define the associated geometric maximal operator $M_{\mathcal{B}}$ by $$M_{\mathcal{B}}f(x):=\underset{x\in R\in \mathcal{B}}{\sup }\frac{1}{|R|}{\int }_R|f|$$ and the halo function ${\varphi }_{\mathcal{B}}(\alpha )$ on $(1,\infty )$ by $${\varphi }_{\mathcal{B}}(\alpha ):=\underset{E\subset {\mathbb{R}}^n:0<|E|<\infty }{\sup }\frac{1}{|E|}|\{x\in {\mathbb{R}}^n:M_{\mathcal{B}}{\chi }_E(x)>1/\alpha \}|.$$ It is shown that if ${\varphi }_{\mathcal{B}}(\alpha )$ satisfies the Solyanik estimate ${\varphi }_{\mathcal{B}}(\alpha )-1\le C(1-\frac{1}{\alpha }{)}^p$ for $\alpha \in (1,\infty )$ sufficiently close to 1 then ${\varphi }_{\mathcal{B}}$ lies in the Hölder class $C^p(1,\infty )$. As a consequence we obtain that the halo functions associated with the Hardy-Littlewood maximal operator and the strong maximal operator on ${\mathbb{R}}^n$ lie in the Hölder class $C^{1/n}(1,\infty )$.

19 pages, 1 figure, minor typos corrected, incorporates referee's report, to appear in Adv. Math

References in corpus (3)

• Tauberian conditions, Muckenhoupt weights, and differentiation properties of weighted bases
• Variations on the Theme of Journe's Lemma
• Weighted Solyanik Estimates for the Hardy-Littlewood maximal operator and embedding of $A_{\infty }$ into $A_p$

Cited by in corpus (1)

• A note on local Hölder continuity of weighted Tauberian functions