A Note on Local Hölder Continuity of Weighted Tauberian Functions

Abstract

Let M and M _S respectively denote the Hardy-Littlewood maximal operator with respect to cubes and the strong maximal operator on $\mathbb{R}^{n}$, and let w be a nonnegative locally integrable function on $\mathbb{R}^{n}$. We define the associated Tauberian functions C _{HL, w} (α) and C _{S, w} (α) on (0, 1) by

$$\displaystyle{\mathsf{C}_{\mathsf{HL},w}(\alpha )\,:=\,\sup _{\begin{array}{c}E\subset \mathbb{R}^{n} \\ 0<w(E)<\infty \end{array}} \frac{1} {w(E)}w(\{x \in \mathbb{R}^{n}: \mathsf{M}\chi _{ E}(x)>\alpha \})}$$

and

$$\displaystyle{\mathsf{C}_{\mathsf{S},w}(\alpha )\,:=\,\sup _{\begin{array}{c}E\subset \mathbb{R}^{n} \\ 0<w(E)<\infty \end{array}} \frac{1} {w(E)}w(\{x \in \mathbb{R}^{n}: \mathsf{M}_{\mathsf{ S}}\chi _{E}(x)>\alpha \}).}$$

Utilizing weighted Solyanik estimates for M and M _S , we show that the function C _{HL, w} lies in the local Hölder class $C^{(c_{n}[w]_{A_{\infty }})^{-1} }(0,1)$ and C _{S, w} lies in the local Hölder class $C^{(c_{n}[w]_{A_{\infty }^{{\ast}}})^{-1} }(0,1)$, where the constant c _n > 1 depends only on the dimension n.

P. H. is partially supported by a grant from the Simons Foundation (#208831 to Paul Hagelstein).

I. P. is supported by IKERBASQUE.