Higher order elliptic equations in weighted Banach spaces

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