Interior regularity of solutions of non-local equations in Sobolev and   Nikol'skii spaces
  We prove interior $H^{2s-\varepsilon}$ regularity for weak solutions of linear elliptic integro-differential equations close to the fractional $s$-Laplacian. The result is obtained via intermediate estimates in Nikol'skii spaces, which are in turn carried out by means of an appropriate modification of the classical translation method by Nirenberg. 
Matteo Cozzi
math.AP