Higher order elliptic equations in weighted Banach spaces

Abstract

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