Multipliers for a distinguished Laplacean on solvable extensions of \(H\)-type groups
Let \(N\) be a connected and simply connected Lie group whose Lie algebra is a so-called Heisenberg type algebra. Let \(\mathbb{R}^+\) act on \(N\) by homogeneous dilations and form the semidirect product \(S=\mathbb{R}^+ \ltimes N\). The author studies the functional calculus for a particular right-invariant Laplacian \(\Delta\) on \(S\) which is closely related to the Laplace-Beltrami operator \({\mathcal L}\) as follows. If \(\delta\) denotes the modular function of \(S\) and \(Q\) the homogeneous dimension of \(N\), then \(\delta^{1/2} \Delta\delta^{-1/2} f=(-{\mathcal L} +{Q^2\over 4})f\) for all radial functions \(f\) in \(C^\infty_c (S)\). Fix a function \(\psi\) in \(C_c^\infty (\mathbb{R}^+)\), supported on \(({1\over 2}, 2)\), such that \(\sum^\infty_{j= -\infty} \psi(2^{-j}\xi)=1\) for every \(\xi\in\mathbb{R}^+\). Let \(H^s\) denote \(L^2\)-Sobolev space of order \(s\) on \(\mathbb{R}\), and for a function \(m\) on \(\mathbb{R}^+\) which is locally in \(H^s\) on \((1,\infty)\), \(\text{set} |m|_{(s)}= \sup_{t\geq 1}|\psi (\cdot) m(t \cdot) |_{H^s}\). Then the main result of the paper reads as follows. Fix \(r\in (2,\infty)\) and \(s\in \mathbb{R}\) such that \(s>1+ {1\over 2} \dim N\), and suppose that \(m\) is a function on \([0, \infty)\) such that: (i) on \([0,2]\) it coincides with a function in \(H^r\); (ii) \(m\) is locally in \(H^s\) on \((1,\infty)\) and \(|m|_{(s)} <\infty\). Then the operator \(m(\Delta)\), defined by the spectral functional calculus, is bounded on \(L^p(S)\) for \(1<p< \infty\) and weak type \((1,1)\). This result and its proof is analogous to one due to \textit{M. Cowling}, \textit{S. Giulini}, \textit{A. Hulanicki} and \textit{G. Mauceri} [Stud. Math. 111, 103-121 (1994; Zbl 0820.43001)] in the case of noncompact symmetric spaces. An important step is to establish, for \(0<t \leq 1\), pointwise exponential estimates for the gradient of the heat kernel \(q_t\) associated to the operator \(e^{-t {\mathcal L}}\).
scientific article; zbMATH DE number 838718
scientific article