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We prove interior $H^{2s-\varepsilon}$ regularity for weak solutions oflinear elliptic integro-differential equations close to the fractional$s$-Laplacian. The result is obtained via intermediate estimates in Nikol'skiispaces, which are in turn carried out by means of an appropriate modificationof the classical translation method by Nirenberg.

Interior regularity of solutions of non-local equations in Sobolev and Nikol'skii spaces