Meyers inequality and strong stability for stable-like operators

Abstract

Let $\alpha \in (0,2)$, let $$\mathcal{E}(u,u)={\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}(u(y)-u(x){)}^2\frac{A(x,y)}{|x-y{|}^{d+\alpha }}dydx$$ be the Dirichlet form for a stable-like operator, let $$\Gamma u(x)={\int }_{{\mathbb{R}}^d}(u(y)-u(x){)}^2\frac{A(x,y)}{|x-y{|}^{d+\alpha }}dy,$$ 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.

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