Title: Sobolev contractivity of the gradient flow maximal function

Abstract:  In 2013, Carneiro and Svaiter showed that the heat flow maximal function is contractive in $\dot{W}^{1,2}(\mathbb{R}^n)$ for $W^{1,2}(\mathbb{R}^n)$ functions. In other words, if $K_t$ is the heat kernel then $u_*(x) = \sup_{t > 0} (K_t \ast |f|)(x)$ for some $f \in W^{1,2}(\mathbb{R}^n)$ then $\|\nabla u_*\|_{L^2(\mathbb{R}^n)} \le \|\nabla f\|_{L^2(\mathbb{R}^n)}$. Equivalently, one can define $u_*$ by solving the Cauchy (initial value) problem for the heat equation, $\partial_t u – \Delta u = 0$, in the upper half-space with data $|f|$ and take $u_*$ to be the vertical maximal function $u_*(x) = \sup_{t > 0} u(x,t)$. Moreover, one may view the heat equation as the `gradient system’ for the energy $\mathcal{E}_2(g) := \tfrac{1}{2}\int_{\mathbb{R}^n} |\nabla g|^2 \, dx$.

We show that if $p > 2$ the vertical maximal function for solutions to the gradient system for the p-energy $\mathcal{E}_p(g) := \tfrac{1}{p} \int_{\mathbb{R}^n} |\nabla g|^p \, dx$ is a contraction on $\dot{W}^{1,p}(\mathbb{R}^n)$ for positive data in $L^2(\mathbb{R}^n) \cap \dot{W}^{1,p}(\mathbb{R}^n)$. Equivalently, one can (and we do!) work with the vertical maximal function for solutions to $\partial_t u – \div(|\nabla u|^{p-2}\nabla u) = 0$ in the upper half-space. The most interesting aspect of our result (aside from the numerous technicalities and in view of Carneiro and Svaiter’s work) is that our equation is non-linear and hence the solutions cannot be given in terms of a kernel representation. This is joint work with M. Egert and O. Saari.