On spectral multipliers for Heisenberg and related groups
Let \(\mathcal L\) be a left invariant sub-Laplacian on a stratified Lie group \(G\). We denote by \(\overline{\mathcal L}= \int^\infty_0 \lambda dE(\lambda)\) the spectral resolution of its closure. If \(m\) is a bounded, Borel measurable function on \(\mathbb{R}^+\), one may define a bounded operator on \(L^2(G)\): \(m({\mathcal L})= \int^\infty_0 m(\lambda) dE(\lambda)\). Let \(\chi\in C^\infty_0(\mathbb{R}^+)\) be a fixed cut-off function with support contained in the interval \([1/2, 1]\), and define the scale-invariant localized Sobolev norm of order \(\alpha\) of \(m\) by \[ |m|_{L^2_{\alpha, \text{sloc}}}= \sup_{t> 0} |\chi m(t\cdot)|_{L^2_\alpha}. \] In this paper, the authors prove the following two theorems: Theorem A. Suppose that \(G= H_n\), where \(H_n\) is a Heisenberg group. If \(|m|_{L^2_{\alpha, \text{sloc}}}< \infty\) for some \(\alpha> (1/2)\dim H_n\), then \(m({\mathcal L})\) is bounded on \(L^p(H_n)\) for \(1< p< \infty\), and of weak-type \((1, 1)\). -- In the above theorem, the condition that \(\alpha> (1/2)\dim H_n\) is essentially sharp for the Heisenberg group. This is shown by proving that the weak-type \((1, 1)\) norm of \({\mathcal L}^{i^\alpha}\) is bounded from below by \(C|\alpha|^{(2n+ 1)/2}\) as \(\alpha\to \infty\). -- Theorem B. Suppose that \(G= G_1\times\cdots\times G_k\), where \(G_j\) is either a Heisenberg group \(H_{n_j}\) or a Euclidean group \(\mathbb{R}^{n_j}\). Suppose further that \({\mathcal L}_j\) is a sub-Laplacian on \(G_j\), \(j= 1,\dots, k\), and \({\mathcal L}= {\mathcal L}_1+\cdots+ {\mathcal L}_k\) is a sub-Laplacian on \(G\). If \(|m|_{L^2_{\alpha, \text{sloc}}}< \infty\) for some \(\alpha> (1/2)\dim G\), then \(m({\mathcal L})\) is bounded on \(L^p(G)\), \(1< p< \infty\) and of weak-type \((1,1)\).
scientific article; zbMATH DE number 811449
scientific article