Decomposition theorem and Riesz basis for axisymmetric potentials in the right half-plane

Abstract

Abstract. The Weinstein equation with complex coefficients is the equation governing generalized axisymmetric potentials (GASP) which can be written as Lm[u] = ∆u + (m/x) ∂xu = 0, where m ∈ C. We generalize results known for m ∈ R to m ∈ C. We give explicit expressions of fundamental solutions for Weinstein operators and their estimates near singularities, then we prove a Green's formula for GASP in the right half-plane H + for Re m < 1. We establish a new decomposition theorem for the GASP in any annular domains for m ∈ C, which is in fact a generalization of the Bôcher's decomposition theorem. In particular, using bipolar coordinates, we prove for annuli that a family of solutions for GASP equation in terms of associated Legendre functions of first and second kind is complete. For m ∈ C, we show that this family is even a Riesz basis in some non-concentric circular annulus.

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Solutions of Weinstein equations representable by Bessel Poisson integrals of BMO functions

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