Regularity of maximal functions on Hardy–Sobolev spaces

Abstract

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