Modified Morrey regularity for elliptic equations

Aim of this paper is to prove regularity results, in some modified local generalized Morrey spaces, for the second derivatives of solutions of a nondivergence elliptic second order equation of the form (Formula presented.) where the coefficients aij,i,j=1,..,n, belong to the central Sarason class CVMO and f is assumed to be in the class of modified local generalized Morrey spaces LM~{x0}p,φ. The relevant key of the proof is to use an explicit representation formula for the first derivatives of solutions of the elliptic equation in nondivergence form, in terms of singular integral operators and commutators with Calderón–Zygmund kernels. By applying the representation formula and by using some Morrey-type estimates for each operator in the aforementioned formula, a regularity result is established.