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Sobolev contractivity of gradient flow maximal functions

Simon Bortz, Moritz Egert, Olli Saari

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.

Sobolev contractivity of gradient flow maximal functions

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 -parabolic extension does not increase the norm of functions when . 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.

Paper Structure

This paper contains 10 sections, 20 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Energy functionals
  3. The Euler--Lagrange equation
  4. A-subharmonic functions
  5. The A-parabolic equation
  6. Two Concrete Energies
  7. The gradient flow of $p$-energy
  8. $p$-energy of the maximal function
  9. Quadratic energies and related semigroups
  10. $A$-energy of the heat and Poisson maximal functions

Key Result

Theorem 1.2

Let $p > 2$, $n \geq 1$, $f \in L^{2}({\mathbb{R}^n}) \cap \dot{W}^{1,p}({\mathbb{R}^n})$ be non-negative and $S_t f$ the unique energy solution to the Cauchy problem Define $S^*f(x) := \sup_{t > 0} S_t f (x)$. Then $S^*f$ is weakly differentiable and satisfies

Theorems & Definitions (41)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.4: Elliptic comparison principle
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Proposition 2.9: Theorem 1.1 of Section III in DiBenedetto1993a
  • Lemma 2.10: Lemma 3.5 in Korte2010
  • Definition 3.2: Energy solution
  • ...and 31 more