Poincaré-Type Inequalities for the Composite Operator in L 𝒜 -Averaging Domains

Abstract

We first establish the local Poincaré inequality with $L^{𝒜}$-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 $L^{𝒜}$-averaging domains and relative properties, we demonstrate the global Poincaré inequality with $L^{𝒜}$-averaging domains. Finally, we give some illustrations for these theorems.