Estimates in Morrey spaces and Hölder continuity for weak solutions to degenerate elliptic systems (Q431143)

The authors consider the family of smooth vector fields in a bounded domain \(\Omega \subset {\mathbb R}^n\) \[ X_j = \sum_{k=1}^n b_{jk}(x) \frac{\partial}{\partial x_k} \, , \quad j = 1 \, \dots q \leqslant n, \] where the \(b_{jk}\) satisfy the Hörmander condition of step \(s\). For the degenerate elliptic system \[ - X_{\alpha}^* (a_{ij}^{\alpha \beta} (x) X_{\beta} u^j) = g_i - X_{\alpha}^* f_i^{\alpha} (x) \tag{*} \] where \(\alpha, \beta = 1, \dots , q\), \(i, j = 1, \dots N\), \(X_j^{\ast} = - X_j + c_j\) is the transposed vector field of \(X_j\) with \[ c_j = - \sum_{k=1}^n \frac{\partial b_{jk}}{\partial x_k} \, , \] with suitable assumptions in the coefficients \(a_{ij}\) and with \(g_i, f_i^{\alpha}\) belonging to the Morrey space \(L^{p.\lambda}_X (\Omega)\) with \(p \geqslant 2\) and \(\lambda \in (Q - \lambda, Q)\) where \(Q\) is the homogeneous dimension relative to \(\Omega\) they prove a higher sommability result. Precisely, they prove that if \(u\) belonging to the Sobolev space \(S^1_X (\Omega, {\mathbb R}^N)\) is a solution to the system (*) then there is \(p > 2\) such that \[ X u \in L^{p, \lambda}_{X, loc} (\Omega , {\mathbb R}^{qN}) \, . \] As a consequence they also prove that \(u\) is \(\alpha\)-Hölder continuous with \(\alpha = 1 - \lambda/p\).