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\(L(\varphi ,\mu )\)-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.

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
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[1] Gehring, F.W.; Osgood, B.G., Uniform domains and the quasihyperbolic metric, J. anal. math., 36, 50-74, (1979) · Zbl 0449.30012
[2] Vaisala, J., Domains and maps, (), 119-131 · Zbl 0762.30009
[3] Staples, S.G., \(L^p\)-averaging domains and the Poincaré inequality, Ann. acad. sci. fenn, ser. AI math., 14, 103-127, (1989) · Zbl 0706.26010
[4] Ding, S.; Nolder, C.A., \(L^s(\mu)\)-averaging domains, J. math. anal. appl., 283, 85-99, (2003) · Zbl 1027.30053
[5] Bao, G.; Ding, S.; Ling, Y., \(L(\varphi, \mu)\)-averaging domains and their geometric characterizations, Chinese ann. math., 21A, 5, 613-622, (2000) · Zbl 0983.30008
[6] Liu, B.; Ding, S., The monotonic property of the \(L^s(\mu)\)-averaging domains, J. math. anal. appl., 237, 730-739, (1999) · Zbl 0946.26005
[7] Ding, S.; Liu, B., Whitney covers and quasi-isometric mappings of \(L^s(\mu)\)-averaging domains, J. inequal. appl., 6, 435-449, (2001) · Zbl 1017.30031
[8] Staples, S.G., Averaging domains: from Euclidean spaces to homogeneous spaces, (), 1041-1048 · Zbl 1129.43005
[9] Nolder, C.A., Hardy – littlewood theorems for \(A\)-harmonic tensors, Illinois J. math., 43, 613-631, (1999) · Zbl 0957.35046
[10] Stein, E.M., Harmonic analysis, (1993), Princeton University Press Princeton
[11] Wang, Y.; Wu, C., Sobolev imbedding theorems and Poincaré inequalities for green’s operator on solutions of the nonhomogeneous \(A\)-harmonic equation, Comput. math. appl., 47, 1545-1554, (2004) · Zbl 1155.31303
[12] Xing, Y., Weighted integral inequalities for solutions of the \(A\)-harmonic equation, J. math. anal. appl., 279, 350-363, (2003) · Zbl 1021.31004
[13] Xing, Y., Weighted Poincaré-type estimates for conjugate \(A\)-harmonic tensors, J. inequal. appl., 1, 1-6, (2005) · Zbl 1087.31009
[14] Ding, S., Two-weight Caccioppoli inequalities for solutions of nonhomogeneous \(A\)-harmonic equations on Riemannian manifolds, Proc. amer. math. soc., 132, 2367-2375, (2004) · Zbl 1127.35021
[15] Agarwal, R.P.; O’Regan, D.; Shakhmurov, V., Separable anisotropic differential operators in weighted abstract spaces and applications, J. math. anal. appl., 338, 970-983, (2008) · Zbl 1138.47037
[16] Agarwal, R.P.; Diagana, T.; Hernández, E.M., Weighted pseudo almost periodic solutions to some partial neutral functional differential equations, J. nonlinear convex anal., 8, 397-415, (2007) · Zbl 1155.35104
[17] Buckley, S.M.; Koskela, P., Orlicz – hardy inequalities, Illinois J. math., 48, 787-802, (2004) · Zbl 1070.46018
[18] Y. Xing, S. Ding, A new weight class and Poincaré inequalities with the Radon measures, preprint.
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