Regularity of maximal functions on Hardy-Sobolev spaces

TL;DR

This paper establishes the boundedness of convolution-type maximal operators on Hardy-Sobolev spaces within a sharp range of exponents, extending understanding of regularity properties in harmonic analysis.

Contribution

It proves the boundedness of maximal operators on Hardy-Sobolev spaces for a sharp exponent range, including local variants, advancing the theory of function space regularity.

Findings

Maximal operators are bounded on Hardy-Sobolev spaces for 1/p < 1+1/d

Results are sharp with respect to the exponent range

Similar boundedness results hold for local Hardy-Sobolev spaces

Abstract

We prove that maximal operators of convolution type associated to smooth kernels are bounded in the homogeneous Hardy-Sobolev spaces $\dot{H}^{1,p}(\mathbb{R}^d)$ when $1/p < 1+1/d$. This range of exponents is sharp. As a by-product of the proof, we obtain similar results for the local Hardy-Sobolev spaces $\dot{h}^{1,p}(\mathbb{R}^d)$ in the same range of exponents.