Meyers inequality and strong stability for stable-like operators

Let α∈(0,2)α∈(0,2), letE(u,u)=∫Rd∫Rd(u(y)−u(x))2A(x,y)|x−y|d+αdydx be the Dirichlet form for a stable-like operator, letΓu(x)=(∫Rd(u(y)−u(x))2A(x,y)|x−y|d+αdy)1/2, let L   be the associated infinitesimal generator, and suppose A(x,y)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=hLu=h, then Γu∈LpΓu∈Lp for some p>2p>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.