Interior Schauder-type estimates for m − th order elliptic operators inrearrangement-invariant Sobolev spaces

Abstract

: In this study, we investigate the m -th order elliptic operators on n -dimensional bounded domain Ω ⊂ R n with discontinuous coefficients in the rearrangement-invariant Sobolev space W mX (Ω) . In general, the considered rearrangement-invariant spaces are not separable, so the use of classical methods in these spaces requires substantial modification of classical methods and a lot of preparation, concerning correctness of substitution operator, problems related to the extension operator in such spaces, etc. For this purpose, the corresponding separable subspaces of these spaces, in which the set of compact supported infinitely differentiable functions is dense, are introduced based on the shift operator. We establish interior Schauder-type estimates in the above subspaces. Note that Lebesgue spaces L p (Ω) , grand-Lebesgue spaces, Marcinkiewicz spaces, weak-type L wp spaces, etc. are also covered by such spaces.