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arXiv · 1601.00111

Matrix weighted Poincaré inequalities and applications to degenerate elliptic systems

Joshua Isralowitz·Kabe Moen
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Abstract

We prove Poincaré and Sobolev inequalities in matrix A${}_p$ weighted spaces. We then use these Poincaré 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 Hölder inequalities for weak solutions of the degenerate systems. As a means to prove the Poincaré 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.

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Joshua Isralowitz, Kabe Moen. 2019-09-16. Matrix weighted Poincaré inequalities and applications to degenerate elliptic systems. https://arxiv.org/abs/1601.00111

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