Sobolev contractivity of gradient flow maximal functions

TL;DR

This paper demonstrates that certain maximal operators associated with gradient flows do not increase the Sobolev norm, extending energy dissipation properties to a broader class of operators and equations.

Contribution

It establishes Sobolev contractivity of gradient flow maximal functions in various PDE settings, including non-convolution solutions and parabolic/elliptic equations.

Findings

Vertical maximal function does not increase the $ abla W^{1,p}$ norm for $p > 2$.

Results extend to uniformly parabolic and elliptic equations with measurable coefficients.

Energy dissipation properties are preserved by these maximal operators.

Abstract

We prove that the energy dissipation property of gradient flows extends to the 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 $\dot{W}^{1,p}$ norm of $\dot{W}^{1,p}(\mathbb{R}^n) \cap L^{2}(\mathbb{R}^n)$ functions when $p > 2$. We also obtain analogous results in the setting of uniformly parabolic and elliptic equations with bounded, measurable, real and symmetric coefficients, where the solutions do not have a representation formula via a convolution.