1. Fatou theorem and its converse for positive eigenfunctions of the Laplace-Beltrami operator on Harmonic NA groups(arXiv)
Abstract : We prove a Fatou-type theorem and its converse for certain positive eigenfunctions of the Laplace-Beltrami operator L on a Harmonic NA group. We show that a positive eigenfunction u of L with eigenvalue β2−ρ2, β∈(0,∞), has an admissible limit in the sense of Korányi, precisely at those boundary points where the strong derivative of the boundary measure of u exists. Moreover, the admissible limit and the strong derivative are the same. This extends a result of Ramey and Ullrich regarding nontangential convergence of positive harmonic functions on the Euclidean upper half space.
2. Approximation of the spectral fractional powers of the Laplace-Beltrami Operator(arXiv)
Abstract : We consider numerical approximations of spectral fractional Laplace-Beltrami problems on closed surfaces. The proposed numerical algorithms rely on their Balakrishnan integral representation and consist of a sinc quadrature coupled with standard finite element methods for parametric surfaces. Possibly up to a log term, optimal rates of convergence are observed and derived analytically when the discrepancies between the exact solution and its numerical approximations are measured in L2 and H1. The performances of the algorithms are illustrated in different settings including the approximation of Gaussian fields on surfaces.
3. Product formulas and convolutions for two-dimensional Laplace-Beltrami operators: beyond the trivial case(arXiv)
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 L=∂2x+12x∂x+1x∂2θ on R+×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.