Meyers inequality and strong stability for stable-like operators (Q393067)

In this paper, the authors study a large class of integro-differential operators, called stable-like operators. These operators bear the same relationship to the fractional Laplacian as divergence form operators do to the Laplacian. They appear in many mathematical models where discontinuities can occur.NEWLINENEWLINEAn inequality of \textit{N. Meyers} [Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 17, 189--206 (1963; Zbl 0127.31904)] says that, if \(u\) is a weak solution of an elliptic equation in divergence form, then \(\nabla u\) is not only locally in \(L^2\), but it is locally in \(L^p\) for some \(p>2\). In the present paper, the authors prove the validity of this result for stable-like operators. The proof, as the one given by Meyers, is based on a Caccioppoli inequality, which, in this case, has a non-local nature.NEWLINENEWLINEIn the case of stable-like operators, the situation is more difficult to treat with respect to the divergence form case and, for this, the proof needs some localization arguments and the use of Sobolev-Besov embedding theorems.NEWLINENEWLINEAs an application, the authors prove strong stability results for stable-like operators.