\(L(\varphi ,\mu )\)-averaging domains and Poincaré inequalities with Orlicz norms. (English) Zbl 1202.46033

Ding, Shusen

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73, No. 1, 256-265 (2010).

Summary: We characterize \(L(\varphi ,\mu )\)-averaging domains using Whitney covers and the quasihyperbolic metric and study the invariance of \(L(\varphi ,\mu )\)-averaging domains under some mappings. As applications of the \(L(\varphi ,\mu )\)-averaging domains, we prove the Poincaré inequality with Orlicz norms for solutions of the non-homogeneous \(A\)-harmonic equation in \(L(\varphi ,\mu )\)-averaging domains.

MSC:

46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
28A25	Integration with respect to measures and other set functions
35J60	Nonlinear elliptic equations
26D10	Inequalities involving derivatives and differential and integral operators

Keywords:

Poincaré inequalities; Orlicz norms; \(A\)-harmonic equation; differential forms

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