Homogeneous Herz spaces with variable exponents and regularity results

Abstract

In this paper we deal with the second order divergence form operators $\mathcal{L}$ with coefficients satisfying the vanishing mean oscillation property and we prove some regularity results for a solution to $\mathcal{L}u=\mathrm{div}\, f$, where $f$ belongs to homogeneous Herz spaces with variable exponents $\dot{K}_{p(\cdot)}^{\alpha,q(\cdot)}$.