Poincaré-Type Inequalities for the Composite Operator in L𝒜-Averaging Domains

Abstract

We first establish the local Poincaré inequality with L𝒜-averaging domains for the composition of the sharp maximal operator and potential operator, applied to the nonhomogenous A-harmonic equation. Then, according to the definition of L𝒜-averaging domains and relative properties, we demonstrate the global Poincaré inequality with L𝒜-averaging domains. Finally, we give some illustrations for these theorems.