Matrix weighted Poincar\'e inequalities and applications to degenerate elliptic systems

We prove Poincar'e and Sobolev inequalities in matrix A${}_p$ weighted spaces. We then use these Poincar'e inequalities to prove existence and regularity results for degenerate systems of elliptic equations whose degeneracy is governed by a matrix A${}_p$ weight. Such results parallel earlier results by Fabes, Kenig, and Serapioni for a single degenerate equation governed by a scalar A${}_p$ weight. In addition, we prove Cacciopoli and reverse H"older inequalities for weak solutions of the degenerate systems. As a means to prove the Poincar'e inequalities we prove that the Riesz potential and fractional maximal function operators are bounded on matrix weighted $L^p$ spaces and go on to develop an entire matrix A${}_{p,q}$ theory.