Fat sets and pointwise boundary estimates forp-harmonic functions in metric spaces (original) (raw)

We extend a result of John Lewis [L] by showing that ifa doubling metric measure space supports a (1, q0)-Poincare inequality for some 1 < qo < p, then every uniformly p-fat set is uniformly q-fat for some q < 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 of pharmonic functions and p-energy minimizers near a boundary point. Sweden. We wish to thank both institutes for their support. The first author was also supported by grants from the Swedish Natural Science Research Council and the Knut and Alice Wallenberg Foundation. We also thank Juha Kinnunen for pointing out the reference [Mi] and for other useful discussions. We also wish to thank Andreas Wannebo for interesting discussions related to his work and Juha Heinonen for his encouragement.