M-harmonic reproducing kernels on the ball

Miroslav Englis; El-Hassan Youssfi

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arXiv · 2208.07358

M-harmonic reproducing kernels on the ball

Miroslav Englis·El-Hassan Youssfi

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Abstract

Using the machinery of unitary spherical harmonics due to Koornwinder, Folland and other authors, we~obtain expansions for the Szeg\"o and the weighted Bergman kernels of $M$-harmonic functions, i.e.~functions annihilated by the invariant Laplacian on the unit ball of the complex $n$-space. This yields, among others, an explicit formula for the $M$-harmonic Szeg\"o kernel in terms of multivariable as well as single-variable hypergeometric functions, and also shows that most likely there is no explicit (``closed'') formula for the corresponding weighted Bergman kernels.

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Miroslav Englis, El-Hassan Youssfi. 2022-08-15. M-harmonic reproducing kernels on the ball. https://arxiv.org/abs/2208.07358

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