Product formulas and convolutions for two-dimensional Laplace-Beltrami operators: beyond the trivial case

TL;DR

This paper develops a convolution framework for Laplace-Beltrami operators on two-dimensional manifolds with cone-like metrics, extending classical convolution theory to more complex geometric and differential operator settings.

Contribution

It introduces a new convolution structure for a class of Laplace-Beltrami operators on 2D manifolds, including explicit formulas for a specific operator on d7^+ d7, extending generalized convolution theory.

Findings

Established convolution semigroup representation for the Markovian semigroup.

Derived explicit convolution kernel and inversion formula using confluent hypergeometric functions.

Extended one-dimensional convolution theory to multiparameter eigenvalue problems.

Abstract

We introduce the notion of a family of convolution operators associated with a given elliptic partial differential operator. Such a convolution structure is shown to exist for a general class of Laplace-Beltrami operators on two-dimensional manifolds endowed with cone-like metrics. This structure gives rise to a convolution semigroup representation for the Markovian semigroup generated by the Laplace-Beltrami operator. In the particular case of the operator $\mathcal{L} = \partial_x^2 + {1 \over 2x} \partial_x + {1 \over x} \partial_\theta^2$ on $\mathbb{R}^+ \times \mathbb{T}$, we deduce the existence of a convolution structure for a two-dimensional integral transform whose kernel and inversion formula can be written in closed form in terms of confluent hypergeometric functions. The results of this paper can be interpreted as a natural extension of the theory of one-dimensional generalized convolutions to the framework of multiparameter eigenvalue problems.