### A calculus of symbols and convolution semigroups on the Heisenberg group

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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 $O\left(\right|x{|}^{m}{e}^{-\alpha x\xb2})$ and $O\left(\right|x\left|\u207f{e}^{-x\xb2/\left(4\alpha \right)}\right)$ respectively for some m,n ≥ 0 and α > 0, then f and f̂ are $P\left(x\right){e}^{-\alpha x\xb2}$ and ${P}^{\text{'}}\left(x\right){e}^{-x\xb2/\left(4\alpha \right)}$ respectively for some polynomials P and P’. If in particular f is as above, but f̂ is $o\left({e}^{-x\xb2/\left(4\alpha \right)}\right)$, 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.

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

Let $N$ be an $H$-type group and $S\simeq N\times {\mathbb{R}}^{+}$ be its harmonic extension. We study a left invariant Hardy–Littlewood maximal operator ${M}_{\rho}^{\mathcal{R}}$ on $S$, obtained by taking maximal averages with respect to the right Haar measure over left-translates of a family $\mathcal{R}$ of neighbourhoods of the identity. We prove that the maximal operator ${M}_{\rho}^{\mathcal{R}}$ is of weak type $(1,1)$.

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 ${\mathbb{R}}^{n}$ 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 that every homogeneous Carnot group can be lifted to a free homogeneous Carnot group. Though following the ideas of Rothschild and Stein, we give simple and self-contained arguments, providing a constructive proof, as shown in the examples.

It is an interesting open problem to establish Paley-Wiener theorems for general nilpotent Lie groups. The aim of this paper is to prove one such theorem for step two nilpotent Lie groups which is analogous to the Paley-Wiener theorem for the Heisenberg group proved in [4].

Let N be an H-type group and consider its one-dimensional solvable extension NA, equipped with a suitable left-invariant Riemannian metric. We prove a Paley-Wiener theorem for nonradial functions on NA supported in a set whose boundary is a horocycle of the form Na, a ∈ A.

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.