Higher order elliptic equations in weighted Banach spaces

Abstract

We consider m-th order linear, uniformly elliptic equations \(\mathcal {L}u=f\) with non-smooth coefficients in Banach–Sobolev spaces \(W_{X_w}^m (\Omega )\) generated by weighted Banach Function Spaces (BFS) \(X_w (\Omega )\) on a bounded domain \(\Omega \subset {\mathbb R}^{n}\). Supposing boundedness of the Hardy–Littlewood Maximal operator and the Calderón–Zygmund singular integrals in \(X_w (\Omega )\) we obtain solvability in the small in \(W_{X_w}^m (\Omega )\) and establish interior Schauder type a priori estimates. These results will be used in order to obtain Fredholmness of the operator \(\mathcal {L}\) in \(X_w (\Omega )\).