Global regularity in Orlicz-Morrey spaces of solutions to nondivergence elliptic equations with VMO coefficients

Abstract

We show continuity in generalized Orlicz-Morrey spaces $M_{\Phi,\varphi}(\mathbb{R}^n)$ of sublinear integral operators generated by Calderon-Zygmund operator and their commutators with BMO functions. The obtained estimates are used to study global regularity of the solution of the Dirichlet problem for linear uniformly elliptic operator $\mathcal{L}=\sum_{i,j=1}^n a^{ij}(x)D_{ij}$ with discontinuous coefficients. We show that $\mathcal{L} u\in M_{\Phi,\varphi}$ implies the second-order derivatives belong to $M_{\Phi,\varphi}$.

Keywords

• Generalized Orlicz-Morrey spaces
• Calderon-Zygmund integrals
• commutators
• VMO
• elliptic equations
• Dirichlet problem