Abstract
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.
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Björn, J., MacManus, P. & Shanmugalingam, N. Fat sets and pointwise boundary estimates forp-harmonic functions in metric spaces. J. Anal. Math. 85, 339–369 (2001). https://doi.org/10.1007/BF02788087
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DOI: https://doi.org/10.1007/BF02788087