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We first introduce and study a new family of weights, the A(α, β, γ E)-class which contains the well-known Ar(E)-weight as a proper subset. Then, as applications of the A(α, β, γ E)-class, we prove the local and global Poincaré inequalities with the Radon measure for the solutions of the non-homogeneous A-harmonic equation which belongs to a kind of the nonlinear partial differential equations. 2000 Mathematics Subject Classification: Primary 26D10; Secondary 35J60; 31B05; 58A10; 46E35. © 2012 Xing.

A new weight class and Poincaré inequalities with the Radon measure