In this paper we study potential-theoretic properties of the symmetric :-stable processes (0<:<2): establishing the boundary Harnack principle for ratios of :-harmonic functions on any open sets, identifying the Martin boundary with the Euclidean boundary for open sets with a certain interior fatness property, and extending earlier results on intrinsic ultracontractivity and the conditional gauge theorem to certain open sets.
Academic Press
Abstract. We consider decomposition spaces R 3 /G that are manifold factors and admit defining sequences consisting of cubes-with-handles of finite type. Metrics on R 3 /G constructed via modular embeddings of R 3 /G into a Euclidean space promote the controlled topology to a controlled geometry.The quasisymmetric parametrizability of the metric space R 3 /G×R m by R 3+m for any m ≥ 0 imposes quantitative topological constraints, in terms of the circulation and the growth of the cubes-with-handles, on the defining sequences for R 3 /G. We give a necessary condition and a sufficient condition for the existence of such a parametrization.The necessary condition answers negatively a question of Heinonen and Semmes on quasisymmetric parametrizability of spaces associated to the Bing double. The sufficient condition gives new examples of quasispheres in S 4 .
Abstract. Vol berg and Konyagin have proved that a compact metric space carries a nontrivial doubling measure if and only if it has finite uniform metric dimension. Their construction of doubling measures requires infinitely many adjustments. We give a simpler and more direct construction, and also prove that for any α > 0, the doubling measure may be chosen to have full measure on a set of Hausdorff dimension at most α.Let (X, ρ) be a compact metric space. Vol'berg and Konyagin proved in [VK] that (X, ρ) carries a nontrivial doubling measure µ (there exists Λ ≥ 1 so that µ(B(x, 2r)) ≤ Λµ(B(x, r)) for all x ∈ X and r > 0) if and only if (X, ρ) has finite uniform metric dimension (in each ball B(x, 2r), there exist at most N points with mutual distances at least r). Here B(x, r) = {y : ρ(x, y) < r}.Assume that (X, ρ) has finite uniform metric dimension. The construction of doubling measures in [VK] requires infinitely many adjustments which cannot be predicted in advance. In this note, we give a simpler and more direct construction, and prove that given any α > 0, there exists a doubling measure on X that has full measure on a set of Hausdorff dimension at most α. Also we observe that a doubling measure may be concentrated on a countable set even when X is a set on the real line of positive length. Some ideas have been adapted from [FKP], [VK] and [T].
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