Meyers inequality and strong stability for stable-like operators

Let alpha is an element of (0, 2), let epsilon(u, u) = integral(Rd) integral(Rd) (u(y) - u(x))2 A(x, y)/vertical bar x - y vertical bar(d+alpha) dydx be the Dirichlet form for a stable-like operator, let Gamma u(x) = (integral(Rd) (u(y) - u(x))2 A(x, y)/vertical bar x - y vertical bar(d+alpha) dy )(1/2) 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 me is the weak solution to Lu = then Gamma u is an element of 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. (C) 2013 Elsevier Inc. All rights reserved.