On some potential theoretic themes in function theory

Abstract

On the 60th anniversary of my friend Vladimir Maz’ya I am happy to remember a fortunate time long ago when we worked on some problems in function theory (1965-1972). Our themes were 1) L p -approximation by analytic and harmonic functions, 2) Uniqueness properties of analytic functions,. 3) The Cauchy problem for the Laplace equation,. 4) Non-linear potential theory. A characteristic feature of everything we did was the heavy use of potential theoretic methods and ideas. Even if a problem didn’t contain anything potential theoretic in its statement (as was the case with the first and to some extent with the second theme), then potential theory would emerge by itself in the solution. Working on 1) and 2) we were compelled to invent a “non-linear potential theory” (among the initial works on the subject are [MH1], [MH2], but this is another story, not to be discussed here; see [AH]). As to theme 3), the problem was potential theoretic from the outset, but it is related to and suggested by traditional themes of pure function theory (quasianalyticity, moment problem, and weighted polynomial approximation in the spirit of S. N. Bernstein) and so corresponds completely to the title of this article.KeywordsCauchy ProblemHarmonic FunctionSobolev SpaceEnglish TranslLaplace EquationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.