Let ${{F}_{2n,2}}$ be the free nilpotent Lie group of step two on $2n$ generators, and let $\mathbf{P}$ denote the affine automorphism group of ${{F}_{2n,2}}$. In this article the theory of continuous wavelet transform on ${{F}_{2n,2}}$ associated with $\mathbf{P}$ is developed, and then a type of radial wavelet is constructed. Secondly, the Radon transform on ${{F}_{2n,2}}$ is studied, and two equivalent characterizations of the range for Radon transform are given. Several kinds of inversion Radon transform formulae are established. One is obtained from the Euclidean Fourier transform; the others are from the group Fourier transform. By using wavelet transforms we deduce an inversion formula of the Radon transform, which does not require the smoothness of functions if the wavelet satisfies the differentiability property. In particular, if $n\,=\,1$, ${{F}_{2,2}}$ is the 3-dimensional Heisenberg group ${{H}^{1}}$, the inversion formula of the Radon transform is valid, which is associated with the sub-Laplacian on ${{F}_{2,2}}$. This result cannot be extended to the case $n\,\ge \,2$.

This paper deals with the Schrödinger equation $i{\partial }_{s} u(\mathbf{z} , t; s)- \mathcal{L} u(\mathbf{z} , t; s)= 0, $ where $ \mathcal{L} $ is the sub-Laplacian on the Heisenberg group. Assume that the initial data $f$ satisfies $\vert f(\mathbf{z} , t)\vert \lesssim {q}_{\alpha } (\mathbf{z} , t), $ where ${q}_{s} $ is the heat kernel associated to $ \mathcal{L} . $ If in addition $\vert u(\mathbf{z} , t; {s}_{0} )\vert \lesssim {q}_{\beta } (\mathbf{z} , t), $ for some ${s}_{0} \in \mathbb{R} \setminus \{ 0\} , $ then we prove that $u(\mathbf{z} , t; s)= 0$ for all $s\in \mathbb{R} $ whenever $\alpha \beta \lt { s}_{0}^{2} . $ This result holds true in the more general context of $H$-type groups. We also prove an analogous result for the Grushin operator on ${ \mathbb{R} }^{n+ 1} . $

We construct the heat kernels of the sub-Laplacian and the Laplacian on theh-Heisenberg group and compute the limits as h → 0 ofthe heat kernels.

Let $L$ denote the sub-Laplacian on the Heisenberg group $\mathbb{H}_n$ and $T_r^\lambda := (1 - r L)^\lambda_+$ the corresponding Bochner-Riesz operator. Let $Q$ denote the homogeneous dimension and $D$ the Euclidean dimension of $\mathbb{H}_n$. We prove convergence a.e. of the Bochner-Riesz means $T_r^\lambda f$ as $r \rightarrow 0$ for $\lambda > 0$ and for all $f \in L^p(\mathbb{H}_n)$, provided that \frac{Q-1}{Q} \Big(\frac{1}{2} - \frac{\lambda}{D-1} \Big) < 1/p \le 1/2. Our proof is based on explicit formulas for the operators $\partial_{\omega^a}$ with $a \in \mathbb{C}$, defined on the dual of $\mathbb{H}_n$ by $\partial_{\omega^a} \widehat{f} := \widehat{\omega^a f}$, which may be of independent interest. Here $\omega$ is given by $\omega(z,u) := |z|^2 - 4iu$ for all $(z,u) \in \mathbb{H}_n$.

2000 Mathematical Subject Classification: 22E30, 43A80.