We first establish the local Poincaré inequality with -averaging domains for the composition of the sharp maximal operator and potential operator, applied to the nonhomogenous -harmonic equation. Then, according to the definition of -averaging domains and relative properties, we demonstrate the global Poincaré inequality with -averaging domains. Finally, we give some illustrations for these theorems.
Guannan Shi. Yuming Xing. Baiqing Sun. "Poincaré-Type Inequalities for the Composite Operator in -Averaging Domains." Abstr. Appl. Anal. 2014 (SI33) 1 - 7, 2014. https://doi.org/10.1155/2014/675464