Meyers inequality and strong stability for stable-like operators

TL;DR

This paper extends Meyers' inequality to stable-like operators, showing that weak solutions have gradients in L^p spaces, and applies this to establish stability bounds under perturbations of the operator.

Contribution

It proves a Meyers-type inequality for stable-like operators and derives explicit stability bounds for their semigroups and fundamental solutions.

Findings

Weak solutions have gradients in L^p for some p>2.

Established stability bounds for semigroups under operator perturbations.

Extended Meyers' inequality to non-divergence form stable-like operators.

Abstract

Let $\alpha\in (0,2)$, let $${\cal E}(u,u)=\int_{\Bbb R^d}\int_{\Bbb R^d} (u(y)-u(x))^2\frac{A(x,y)}{|x-y|^{d+\alpha}}\, dy\, dx$$ be the Dirichlet form for a stable-like operator, let $$\Gamma u(x)=\int_{\Bbb R^d} (u(y)-u(x))^2\frac{A(x,y)}{|x-y|^{d+\alpha}}\, dy,$$ let $L$ be the associated infinitesimal generator, and suppose $A(x,y)$ is jointly measurable, symmetric, bounded, and bounded below by a positive constant. We prove that if $u$ is the weak solution to $Lu=h$, then $\Gamma u\in L^p$ for some $p>2$. This is the analogue of an inequality of Meyers for solutions to divergence form elliptic equations. As an application, we prove strong stability results for stable-like operators. If $A$ is perturbed slightly, we give explicit bounds on how much the semigroup and fundamental solution are perturbed.