Poincaré-Type inequalities for the composite operator in L -averaging domains

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