Fat sets and pointwise boundary estimates forp-harmonic functions in metric spaces
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We extend a result of John Lewis [L] by showing that if a doubling metric measure space supports a (1,q 0)-Poincaré inequality for some 1<q 0<p, then every uniformlyp-fat set is uniformlyq-fat for someq<p. This bootstrap result implies the Hardy inequality for Newtonian functions with zero boundary values for domains whose complements are uniformly fat. While proving this result, we also characterize positive Radon measures in the dual of the Newtonian space using the Wolff potential and obtain an estimate for the oscillation ofp-harmonic functions andp-energy minimizers near a boundary point.
KeywordsSobolev Inequality Radon Measure Harnack Inequality Hardy Inequality Strong Maximum Principle
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- [Bj] J. Björn,Boundary continuity for quasi-minimizers on metric spaces, Preprint (2000).Google Scholar
- [HaK] P. Hajłasz and P. Koskela,Sobolev met Poincaré, Mem. Amer. Math. Soc.145 (2000).Google Scholar
- [KaSh] S. Kallunki and N. Shanmugalingam,Modulus and continuous capacity, Ann. Acad. Sci. Fenn. Math. (to appear).Google Scholar
- [KL] J. Kinnunen and V. Latvala,Lebesgue points for Sobolev functions on metric measure spaces, Preprint (2000).Google Scholar
- [KiSh] J. Kinnunen and N. Shanmugalingam,Quasi-minimizers on metric spaces, Preprint (1999).Google Scholar
- [Mi] P. Mikkonen,On the Wolff potential and quasilinear elliptic equations involving measures, Ann. Acad. Sci. Fenn., Math. Dissertationes104 (1996).Google Scholar
- [Sh2] N. Shanmugalingam,Harmonic functions on metric spaces, Illinois J. Math. (to appear).Google Scholar