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We prove that maximal operators of convolution type associated to smooth kernels are bounded in the homogeneous Hardy–Sobolev spacesH  ̇    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 spacesh  ̇    1  ,  p(    R   d   )in the same range of exponents.

Regularity of maximal functions on Hardy–Sobolev spaces