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

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 operatorwe 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.

Closure of the Laplace-Beltrami Operator on 2D Almost-Riemannian Manifolds and Semi-Fredholm Properties of Differential Operators on Lie Manifolds

“…Thus we only need to look at the principal order in the asymptotic expansions. In particular, ∆ − cS will be essentially self-adjoint if and only if either u ± do not satisfy (35) or, if they do, then additionally they satisfy (34).…”

“…Compared to their work, the method presented here can determine not only when ∆ is not essentially self-adjoint but further can explicitly construct all of its self-adjoint extensions. Also, operators in [34] can be studied using the α-calculus we develop in this paper.…”