Solvability in the small and interior Schauder type estimates for higher order elliptic equations in weighted Banach function spaces

We consider m-th order linear, uniformly elliptic equation L u=f with non-smooth coefficients in Banach-Sobolev space W_{X_{w} }^{m} (\Omega ) generated by weighted general Banach Function Space (BFS) X_{w} (\Omega) on a bounded domain \Omega \subset R^{n}. Supposing boundedness of the Hardy-Littlewood Maximal and Calderòn-Zygmund singular operators in X_{w} (\Omega) we obtain solvability in the small in W_{X_{w} }^{m} (\Omega ) and establish interior Schauder type a priori estimates for the corresponding elliptic operator. These results will be used in order to obtain Fredholmness of the operator under consideration in X_{w} (\Omega) with suitable weight. In addition, we analyze some examples of weighted BFS that verify our assumptions and in which the corresponding Schauder type estimates and Fredholmness of the operator hold true. This approach and the obtained results are new even for well studied spaces as the Morrey spaces, grand Lebesgue spaces, and Lebesgue spaces with variable exponents.