## Density properties of harmonic measure.(English)Zbl 0842.31001

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.

### 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
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