Poincaré-type inequalities for the composite operator in \(L^{\mathcal{A}}\)-averaging domains (Q1724715)

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