In this article, we prove a generalisation of Bochner-Godement theorem. Our result deals with Olshanski spherical pairs $(G,K)$ defined as inductive limits of increasing sequences of Gelfand pairs ${(G\left(n\right),K\left(n\right))}_{n\ge 1}$. By using the integral representation theory of G. Choquet on convex cones, we establish a Bochner type representation of any element $\varphi$ of the set ${𝒫}^{♮}\left(G\right)$ of $K$-biinvariant continuous functions of positive type on $G$.

In this note, we generalize the results in our previous paper on the Casimir operator and Berezin transform, by showing the $(L^2,L^2)$-continuity of a generalized Berezin transform associated with a branching problem for a class of unitary representations defined by invariant elliptic operators; we also show, that under suitable general conditions, this generalized Berezin transform is $(L^p,L^p)$-continuous for $1\le p\le \infty .$

In [8], we studied the problem of local solvability of complex coefficient second order left-invariant differential operators on the Heisenberg group ℍₙ, whose principal parts are "positive combinations of generalized and degenerate generalized sub-Laplacians", and which are homogeneous under the Heisenberg dilations. In this note, we shall consider the same class of operators, but in the presence of left invariant lower order terms, and shall discuss local solvability for these operators in a complete...

A result by G. H. Hardy ([11]) says that if f and its Fourier transform f̂ are $O\left(\right|x{|}^m{e}^{-\alpha x²})$ and $O\left(\right|x\left|ⁿe^{-x²/\left(4\alpha \right)}\right)$ respectively for some m,n ≥ 0 and α > 0, then f and f̂ are $P\left(x\right)e^{-\alpha x²}$ and $P^{\text{'}}\left(x\right)e^{-x²/\left(4\alpha \right)}$ respectively for some polynomials P and P’. If in particular f is as above, but f̂ is $o\left(e^{-x²/\left(4\alpha \right)}\right)$, then f = 0. In this article we will prove a complete analogue of this result for connected noncompact semisimple Lie groups with finite center. Our proof can be carried over to the real reductive groups of the Harish-Chandra class.

We give new and general sufficient conditions for a Gaussian upper bound on the convolutions $K_{m+n}\ast K_{m+n-1}\ast \cdots \ast K_{m+1}$ of a suitable sequence K₁, K₂, K₃, ... of complex-valued functions on a unimodular, compactly generated locally compact group. As applications, we obtain Gaussian bounds for convolutions of suitable probability densities, and for convolutions of small perturbations of densities.

This article deals with vector valued differential forms on $C^{\infty }$-manifolds. As a generalization of the exterior product, we introduce an operator that combines $Hom({⨂}^s\left(W\right),Z)$-valued forms with $Hom({⨂}^s\left(V\right),W)$-valued forms. We discuss the main properties of this operator such as (multi)linearity, associativity and its behavior under pullbacks, push-outs, exterior differentiation of forms, etc. Finally we present applications for Lie groups and fiber bundles.

We introduce a new method for obtaining heat kernel on-diagonal lower bounds on non- compact Lie groups and on infinite discrete groups. By using this method, we are able to recover the previously known results for unimodular amenable Lie groups as well as for certain classes of discrete groups including the polycyclic groups, and to give them a geometric interpretation. We also obtain new results for some discrete groups which admit the structure of a semi-direct product or of a wreath product....

This paper is part of a general program that was originally designed to study the Heat diffusion kernel on Lie groups.

We prove an $L^p$-boundedness result for a convolution operator with rough kernel supported on a hyperplane of a group of Heisenberg type.

The aim of this paper is to demonstrate how a fairly simple nilpotent Lie algebra can be used as a tool to study differential operators on ${ℝ}^n$ with polynomial coefficients, especially when the property studied depends only on the degree of the polynomials involved and/or the number of variables.

We prove a restriction theorem for the class-1 representations of the Heisenberg motion group. This is done using an improvement of the restriction theorem for the special Hermite projection operators proved in [13]. We also prove a restriction theorem for the Heisenberg group.

We establish the spectral gap property for dense subgroups of SU$\left(d\right)$$(d\ge 2)$, generated by finitely many elements with algebraic entries; this result was announced...