Explicit improvements for $\mathrm{L}^p$-estimates related to elliptic systems

TL;DR

This paper presents a straightforward method to establish explicit $L^p$-boundedness for heat semigroups related to elliptic systems, optimizing bounds at the endpoint and extending estimates to gradients.

Contribution

It introduces a simple Stein interpolation approach to derive explicit $L^p$ bounds for elliptic system semigroups, including gradient estimates, with results depending explicitly on ellipticity.

Findings

Explicit $p$ bounds in terms of ellipticity.

Optimal bounds at $p=\infty$.

Gradient estimates for $p>2$ depending on ellipticity.

Abstract

We give a simple argument to obtain $\mathrm{L}^p$-boundedness for heat semigroups associated to uniformly strongly elliptic systems on $\mathbb{R}^d$ by using Stein interpolation between Gaussian estimates and hypercontractivity. Our results give $p$ explicitly in terms of ellipticity. It is optimal at the endpoint $p=\infty$. We also obtain $\mathrm{L}^p$-estimates for the gradient of the semigroup, where $p>2$ depends on ellipticity but not on dimension.