Density properties of harmonic measure. (English) Zbl 0842.31001

Jones, P. W.; Makarov, Nikolai G.

Ann. Math. (2) 142, No. 3, 427-455 (1995).

Some metric properties of harmonic measure are studied in the paper. The basis of all considerations is the known formula \(f(x)= \int_{\partial \Omega} u dw_z\) which solves the Dirichlet problem \(\Delta f=0\) on \(\Omega \subset \mathbb{R}^n\), \(f=u\) on \(\partial \Omega\) and \(w_z\) a Borel probability measure on \(\partial \Omega\). Some previous papers, connected with the present one, quoted in the bibliography, study the metric behaviour of \(w\), proving the independence of \(w\) from \(\Omega\). The authors of the paper analyse the behaviour of \(w\) on subsets of dimension almost \(n\) or subsets of positive Lebesgue measure. They improve the already obtained results in the domain, finding the order of decay of some harmonic measures, prove a density theorem, study the singularity of the boundary distortion, the Beurling and Marcinkiewicz integrals and Bourgain’s dichotomy.

Reviewer: V.Ionescu (Bucureşti)

MSC:

31A15	Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
28A78	Hausdorff and packing measures
28A25	Integration with respect to measures and other set functions

Keywords:

boundary distorsion; Beurling integrals; Marcinkiewicz integrals; Bourgain dichotomy