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A new weight class and Poincaré inequalities with the Radon measure. (English) Zbl 1279.26038

Summary: We first introduce and study a new family of weights, the \(A(\alpha, \beta, \gamma, E)\)-class which contains the well-known \(A_r(E)\)-weight as a proper subset. Then, as applications of the \(A(\alpha,\beta, \gamma;E)\)-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.

MSC:

26D10 Inequalities involving derivatives and differential and integral operators
35J60 Nonlinear elliptic equations
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
58A10 Differential forms in global analysis
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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