Geodesic
Gradient estimates for elliptic systems in Carnot-Carathéodory spaces
Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 4, pp. 605-618

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Let $X=(X_1,X_2,\dots ,X_q)$ be a system of vector fields satisfying the Hör\-man\-der condition. We prove $L^{2,\lambda}_X$ local regularity for the gradient $Xu$ of a solution of the following strongly elliptic system $$ -X^{*}_{\alpha }(a^{\alpha \beta }_{ij}(x)X_{\beta } u^{j})= g_{i}-X^{*}_{\alpha } f^{\alpha }_{i}(x) \quad \forall i=1,2,\dots ,N, $$ where $a^{\alpha \beta }_{ij}(x)$ are bounded functions and belong to Vanishing Mean Oscillation space.
Classification : 35B65, 35D10, 35H20, 35J45, 35J50
Keywords: elliptic systems; Morrey space regularity; Carnot-Carathéodory metric 
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     author = {Di Fazio, Giuseppe and Fanciullo, Maria Stella},
     title = {Gradient estimates for elliptic systems in {Carnot-Carath\'eodory} spaces},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {605--618},
     publisher = {mathdoc},
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     number = {4},
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     zbl = {1090.35058},
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Di Fazio, Giuseppe; Fanciullo, Maria Stella. Gradient estimates for elliptic systems in Carnot-Carathéodory spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 4, pp. 605-618. http://geodesic.mathdoc.fr/item/CMUC_2002__43_4_a2/