Harmonic measures, potentials, approximation and polynomials inequalities on general sets

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Research will be done in connection with metric properties of harmonic measures on infinitely connected domains, in connection with polynomial inequalities and orthogonal polynomials. The estimates for harmonic measures/Green functionswill be given via a local integral involving a density function associated withthe boundary of the domain. Such results directly lead to estimates for polynomials in the complex plane and to extensions of some classical polynomial inequalities for general compact sets. The results, in particular, will allow to find the order of the Markoff factors for Cantor-type sets, which in turnwill produce compact sets of zero measure for which the Markoff factors are of the minimal growth order O(n^2). Weighted polynomial inequalitieswith respect to general (say doubling) weights will also be studied together with applications in connection with approximation theory and orthogonal polynomials.Another part of the research is polynomial approximation and orthogonalpolynomials with varying weights (the weight varies with the degree), whichwill be studied with the help of logarithmic potentials in thepresence of an external field. The approximation problem is fundamentalin proving appropriate asymptotics for Christoffel functionsand orthogonal polynomials with varying weights, that inturn can be used to test the universality hypothesis in statistical-mechanical models of quantum physics. The proposed research uses different tools from classical analysis, but the main method is potential thoretic.Research will be done in mathematical analysis. The main effort will be to find new ways to estimate some classical quantities and to extend some well known results to more general situations. These will allow wider applicabilityof several classical tools and will provide some new insights into the theoretical aspects of some fundamental questions in approximation theoryand orthogonal polynomials.