Regularity of maximal functions on Hardy-Sobolev spaces

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 $1\mathrm{/}p<1+1\mathrm{/}d$. 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.