Properties Of Harmonic Measure Research Articles

Semi-Uniform Domains and the A∞ Property for Harmonic Measure

Abstract We study the properties of harmonic measure in semi-uniform domains. Aikawa and Hirata showed in [ 5] that, for John domains satisfying the capacity density condition (CDC), the doubling property for harmonic measure is equivalent to the domain being semi-uniform. Our 1st result removes the John condition by showing that any domain satisfying the CDC whose harmonic measure is doubling is semi-uniform. Next, we develop a substitute for some classical estimates on harmonic measure in nontangentially accessible domains that works in semi-uniform domains. We also show that semi-uniform domains with uniformly rectifiable boundary have big pieces of chord-arc subdomains. We cannot hope for big pieces of Lipschitz subdomains (as was shown for chord-arc domains by David and Jerison [ 19]) due to an example of Hrycak, which we review in the appendix. Finally, we combine these tools to study the $A_{\infty }$-property of harmonic measure. For a domain with Ahlfors–David regular boundary, it was shown by Hofmann and Martell that the $A_{\infty }$ property of harmonic measure implies uniform rectifiability of the boundary [ 25, 27]. Since $A_{\infty }$-weights are doubling, this also implies the domain is semi-uniform. Our final result shows that these two properties, semi-uniformity and uniformly rectifiable boundary, also imply the $A_{\infty }$ property for harmonic measure, thus classifying geometrically all domains for which this holds.

On an inequality for the universal spectrum of integral means

βf (t) = lim sup r→1 ln ∫ 2π 0 |f ′(reiθ)|t dθ | ln(1− r)| is the spectrum of the integral means of the conformal mapping of f . The universal spectrum B(t) is responsible for the boundary behavior of conformal mappings in the disk, in particular, for the metric properties of harmonic measure on Jordan curves as well as for the distortion of the Hausdorffmeasures of Borel sets on the circle under the conformal mappings of the disk (see, for example, [2]–[4]). It is well known that B(t) is a convex function of the variable t. Carleson and Makarov showed [5] that B(t) = |t| − 1 for t ≤ t0 for some t0 ≤ −2 (further, we assume that t0 takes the maximum possible value). In [6], Feng and MacGregor proved the equality B(t) = 3t− 1 for t ≥ 2 5 .

Metric properties of harmonic measures

Introduction Metric properties of harmonic measures, Green functions and equilibrium measures Sharpness Higher order smoothness Cantor-type sets Phargmen-Lindelof type theorems Markov and Bernstein type inequalities Fast decreasing polynomials Remez and Schur type inequalities Approximation on compact sets Appendix References List of symbols List of figures Index.

Harmonic measure and polynomial Julia sets

There is a natural conjecture that the universal bounds for the dimension spectrum of harmonic measure are the same for simply connected and for nonsimply connected domains in the plane. Because of the close relation to conformal mapping theory, the simply connected case is much better understood, and proving the above statement would give new results concerning the properties of harmonic measure in the general case. We establish the conjecture in the category of domains bounded by polynomial Julia sets. The idea is to consider the coefficients of the dynamical zeta function as subharmonic functions on a slice of Teichmüller's space of the polynomial and then to apply the maximum principle.

Properties of harmonic measures in the Dirichlet problem for nilpotent Lie groups of Heisenberg type

In groups of Heisenberg type we introduce a large class of domains, which we call ADP, admissible for the Dirichlet problem , and we prove that on the boundary of such domains, harmonic measure, ordinary surface measure, and the perimeter measure, are mutually absolutely continuous. We also establish the solvability of the Dirichlet problem when the boundary datum belongs to L p , 1 < p ≤ ∞, with respect to the ordinary surface measure. Here, the harmonic measure is that relative to a sub-Laplacian associated with a basis of the first layer of the Lie algebra. A domain is called ADP if it is a nontangentially accessible domain and it satisfies an intrinsic outer ball condition.