On weights which admit harmonic Bergman kernel and minimal solutions of Laplace’s equation - Żynda, Tomasz Łukasz (Annales Mathematicae Silesianae, 2022)
In this paper we consider spaces of weight square-integrable and harmonic functions L2H(Ω,μ). Weights μ for which there exists reproducing kernel of L2H(Ω,μ) are named ’admissible weights’ and such kernels are named ’harmonic Bergman kernels’. We prove that if only weight of integration is integrable in some negative power, then it is admissible. Next we construct a weight μ on the unit circle which is non-admissible and using Bell-Ligocka theorem we show that such weights exist for a large class of domains in ℝ2. Later we conclude from the classical result of reproducing kernel Hilbert spaces theory that if the set f∈L2H(Ω,μ)| f(z) = c for admissible weight μ is non-empty, then there is exactly one element with minimal norm. Such an element in this paper is called ’a minimal (z,c)-solution in weight μ of Laplace’s equation on Ω’ and upper estimates for it are given.