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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 675464, 7 pages
http://dx.doi.org/10.1155/2014/675464
Research Article

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

1Department of Math, Harbin Institute of Technology, Harbin 150001, China
2School of Management, Harbin Institute of Technology, Harbin 150001, China

Received 12 November 2013; Accepted 10 January 2014; Published 25 February 2014

Academic Editor: Shusen Ding

Copyright © 2014 Guannan Shi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.