Approximate identities from Laguerre functions and singular integrals on the Heisenberg group
In this paper the authors show that a suitably scaled Laguerre function acts as an approximate identity for multiplicative convolution with test functions on the half line: Theorem A: For each \(\alpha > -1\), \[ \lim_{n\rightarrow \infty}\{ \frac{(-1)^n}{2} l_{n}^{(\alpha)}(x)-\delta(x-4n)\}=0 \] in the sense of distributions where \(l_{n}^{(\alpha)}\) is the Laguerre function. As an application the authors derive a precise connection between the Mikhlin-type expansion of a singular integral operator on the Heisenberg group \(H_{n}={\mathbb R}^{2n} \times {\mathbb R}=\{ (x, t) \}\) and its natural restriction to \(H_{n}\) modulo the center: Theorem B: Suppose that \(F \in C^{\infty}(H_{n})\) is homogeneous of degree \(2n-2\) with respect to the Heisenberg dilation \[ F(\lambda x, \lambda^{2}t) = \lambda^{-2n-2}F(x,t), \quad \lambda > 0, \quad (x, t)\neq (0, 0) \] and has mean value zero. Let \[ s_{j}=\frac{\xi_{j}^{2}+\xi_{n+j}^{2}}{2}, \;\;\theta_{j}=\tan^{-1} \Biggl(\frac{\xi_{n+j}}{\xi_{j}} \Biggr). \] When each \(s_{j}\) is positive, the coefficients \(\widehat{a}_{p}(s)\) of the expansion \[ \widehat{F}(\xi, 0)=\frac{1}{(2\pi)^{n}}\sum_{p \in {\mathbb Z}^{n}}\widehat{a}_{p}(s) e^{ip\cdot \theta} \] of the Fourier transform \(\widehat{F}(\xi, \tau)\) restricted to \(\tau = 0\) can be recovered from the coefficients \(a_{jk}^{+}\) of the Laguerre expansion as limits \[ \widehat{a}_{p}(s)=\lim_{k \rightarrow \infty}\frac{(-1)^{ k }}{2^{n}} a_{pk}^{+}, \quad k=k_{1}+ \cdots +k_{n}, \] where the limit is taken over any sequence of multi-indices \(k\) such that \[ \lim_{ k \rightarrow \infty}\frac{k_{j}}{k}=s_{j}, \quad j=1, \ldots , n. \]
scientific article; zbMATH DE number 1997289
scientific article