Giuseppe Di Fazio, Maria Stella Fanciullo
Gradient estimates for elliptic systems in Carnot-Carath\'eodory spaces

Comment.Math.Univ.Carolinae 43,4 (2002) 605-618.

Abstract:Let $X=(X_1,X_2,...,X_q)$ be a system of vector fields satisfying the H\"ormander condition. We prove $L^{2,\lambda }_X$ local regularity for the gradient $Xu$ of a solution of the following strongly elliptic system $$ -X^{*}_{\alpha }(a^{\alpha \beta }_{ij}(x)X_{\beta } u^{j})= g_{i}-X^{*}_{\alpha } f^{\alpha }_{i}(x) \hskip 1em\relax \forall i=1,2,...,N, $$ where $a^{\alpha \beta }_{ij}(x)$ are bounded functions and belong to Vanishing Mean Oscillation space.

Keywords: elliptic systems, Morrey space regularity, Carnot-Carath\'eodory metric
AMS Subject Classification: 35J50