Null sets for doubling and dyadic doubling measures.
Characterizations of snowflake metric spaces.
Nonlinear harmonic measures on trees.
Doubling measures with different bases
Boundary limits of Green's potentials along curves II. Lipschitz domains
Boundary limits of Green's potentials along curves
Harmonic measures for symmetric stable processes
Let D be an open set in ℝⁿ (n ≥ 2) and ω(·,D) be the harmonic measure on with respect to the symmetric α-stable process (0 < α < 2) killed upon leaving D. We study inequalities on volumes or capacities which imply that a set S on ∂D has zero harmonic measure and others which imply that S has positive harmonic measure. In general, it is the relative sizes of the sets S and that determine whether ω(S,D) is zero or positive.
Growth and asymptotic sets of subharmonic functions (II)
We study the relation between the growth of a subharmonic function in the half space R and the size of its asymptotic set. In particular, we prove that for any n ≥ 1 and 0 < α ≤ n, there exists a subharmonic function u in the R satisfying the growth condition of order α : u(x) ≤ x for 0 < x < 1, such that the Hausdorff dimension of the asymptotic set ∪A(λ) is exactly n-α. Here A(λ) is the set of boundary points at which f tends...
Comparisons of kernel functions boundary Harnack principle and relative Fatou theorem on Lipschitz domains
On a Lipschitz domain in , three theorems on harmonic functions are proved. The first (boundary Harnack principle) compares two positive harmonic functions at interior points near an open subset of the boundary where both functions vanish. The second extends some familiar geometric facts about the Poisson kernel on a sphere to the Poisson kernel on . The third theorem, on non-tangential limits of quotient of two positive harmonic functions in , generalizes Doob’s relative Fatou theorem on a...
Singularity of parabolic measures
Parabolic measure on domains of class Lip
Two problems on doubling measures.
Doubling measures appear in relation to quasiconformal mappings of the unit disk of the complex plane onto itself. Each such map determines a homeomorphism of the unit circle on itself, and the problem arises, which mappings f can occur as boundary mappings?
Quasiconformal dimensions of self-similar fractals.
The Sierpinski gasket and other self-similar fractal subsets of R, d ≥ 2, can be mapped by quasiconformal self-maps of R onto sets of Hausdorff dimension arbitrarily close to one. In R we construct explicit mappings. In R, d ≥ 3, the results follow from general theorems on the equivalence of invariant sets for iterated function systems under quasisymmetric maps and global quasiconformal maps. More specifically, we present geometric conditions ensuring that (i) isomorphic systems have quasisymmetrically...
Smooth quasiregular maps with branching in
According to a theorem of Martio, Rickman and Väisälä, all nonconstant C-smooth quasiregular maps in , ≥3, are local homeomorphisms. Bonk and Heinonen proved that the order of smoothness is sharp in . We prove that the order of smoothness is sharp in . For each ≥5 we construct a C-smooth quasiregular map in with nonempty branch set.
-harmonic measure is not additive on null sets
When and the -harmonic measure on the boundary of the half plane is not additive on null sets. In fact, there are finitely many sets , ,..., in , of -harmonic measure zero, such that .
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