Weighted Poincaré inequalities for solutions to $A$-harmonic equations

Abstract

We first prove a local $A_r$-weighted Poincaré inequality for solutions to $A$-harmonic equations of the form $d^{\star} A(x, d \omega) =B(x,d\omega)$. Then, as an application of this local result, we prove a global $A_r$-weighted Poincaré inequality for functions that are solutions to such equations in John domains.