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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 formfor 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 associatedinfinitesimal generator, and suppose $A(x,y)$ is jointly measurable, symmetric,bounded, and bounded below by a positive constant. We prove that if $u$ is theweak solution to $Lu=h$, then $\Gamma u\in L^p$ for some $p>2$. This is theanalogue of an inequality of Meyers for solutions to divergence form ellipticequations. As an application, we prove strong stability results for stable-likeoperators. If $A$ is perturbed slightly, we give explicit bounds on how muchthe semigroup and fundamental solution are perturbed.

Meyers inequality and strong stability for stable-like operators