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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.

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