Sobolev contractivity of gradient flow maximal functions

Abstract We prove that the energy dissipation property of gradient flows extends to semigroup maximal operators in various settings. In particular, we show that the vertical maximal function relative to the p -parabolic extension does not increase the p -norm of the gradient when p > 2 {p>2} . We also obtain analogous results in the setting of uniformly parabolic and elliptic equations with bounded, measurable, real and symmetric coefficients. These are the first regularity results for vertical maximal functions without convolution structure.