Regularity in Orlicz spaces for nondivergence degenerate elliptic equations in carnot groups

Abstract

In this paper, we consider nondivergence degenerate elliptic equations of the kind $$\begin{aligned} \underset{i,j=1}{\overset{q}{\sum }}a_{ij}(\xi ) X_{i} X_{j}u=f,\quad \text{ in } \Omega \end{aligned}$$ where \(\{X_{1},\ldots ,X_{q}\} \) is the basis of the space of horizontal vector fields in a homogeneous Carnot group \(\mathbb G \,{=}\,(\mathbb R ^{n};\circ ) \), the coefficients \(a_{ij}(\xi )\) are real valued bounded measurable functions defined in \(\Omega \subset \mathbb G \), satisfying the uniform ellipticity. We establish the regularity in Orlicz spaces for the solutions if the coefficients \(a_{ij}(\xi ) \) belong to \(VMO_{loc}\).