Matrix weighted Poincare inequalities and applications to degenerate elliptic systems

We prove Poincar\'e and Sobolev inequalities in matrix Aₚ 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 Aₚ weight. Such results parallel earlier results by Fabes, Kenig, and Serapioni for a single degenerate equation governed by a scalar Aₚ 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 Lᵖ spaces and go on to develop an entire matrix Ap, q theory.