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

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.


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
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