We prove that maximal operators of convolution type associated to smooth kernels are bounded in the homogeneous Hardy–Sobolev spaces trueḢ1,pfalse(double-struckRdfalse) 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 trueḣ1,pfalse(double-struckRdfalse) in the same range of exponents.