Poisson integrals for standard weighted Laplacians in the unit disc. (English) Zbl 1272.31002

The article deals with solutions to the weighted Dirichlet boundary value problem \(\Delta_{\alpha}u=0\) on the unit disc in \(\mathbb C\) with distributional boundary value conditions. Here \(\Delta_{\alpha}\) is the so-called weighted laplacian, corresponding to the (standard) weight function in this theory, namely \(w_{\alpha}(z)=(1-|z|^2)^{\alpha}\). It seems that the main achievement of the authors is to calculate explicitly the Poisson kernel, which enables one to solve the so-stated Dirichlet problem. It turns out that the kernel is of particularly nice form, namely \[ P_{\alpha}(z)=\frac{(1-|z|^2)^{\alpha+1}}{(1-z)(1-\bar z)^{\alpha+1}}. \] Among others the authors show some regularity results for the solutions of the Dirichlet problem as well as get the expansions of the solutions in power series. The authors use deep methods from functional analysis, in particular from the theory of homogeneous Banach spaces and their relative completions. Some functional-analytical results seem to be new and of interest to specialists in functional analysis, although the authors develop the machinery to deal with particular problems. Some results of Fatou type are also included.


31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
35J25 Boundary value problems for second-order elliptic equations
Full Text: DOI Euclid


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