Sub-Laplacians of holomorphic \(L^p\)-type on rank one AN-groups and related solvable groups
Let \(G=\exp{\mathfrak g}\) be an exponential solvable Lie group endowed with a left-invariant Haar measure and \(L=\int^\infty_0 \lambda dE_\lambda\) the spectral resolution of a sub-Laplacian \(L\) on \(G\). We call a bounded function \(m\) on \([0,\infty[\) an \(L^p\)-multiplier for \(L\) \((1\leq p<\infty)\) if \(m(L)=\int^\infty_0 m(\lambda)d E_\lambda\) extends from \(L^p(G)\cap L^2(G)\) to a bounded operator on \(L^p(G)\). We say that \(L\) is of holomorphic \(L^p\)-type if there exists some non-isolated point \(\lambda_0\in \text{Spec}_{L^2 (G)}L= [0, \infty[\) and an open complex neighborhood \(\Omega\) of \(\lambda_0\) such that every continuous \(L^p\)-multiplier \(m(\lambda)\) vanishing at infinity extends holomorphically to \(\Omega\). The authors prove that, if the Boidol condition holds for some \(l\in{\mathfrak g}^*\) and \({\mathfrak g}={\mathfrak g}(l)+ [{\mathfrak g},{\mathfrak g}]\), \({\mathfrak g}(l)\) is the stabilizer of \(l\), then \(L\) is of holomorphic \(L^1\)-type; and moreover, if the assumption on \(l\) holds on a whole neighborhood of \(l\), then \(G\) is unimodular and \(L\) is of holomorphic \(L^p\)-type for \(1<p< \infty\), \(p\neq 2\). Let \({\mathfrak n}\) be a nilpotent ideal of \({\mathfrak g}\) containing \([{\mathfrak g}, {\mathfrak g}]\) and \({\mathfrak a}\) a linear subspace of \({\mathfrak g}\) such that \({\mathfrak g}={\mathfrak a}\oplus {\mathfrak n}\). For \(l=(\tau, l_0)\in {\mathfrak g}^* ={\mathfrak a}^* \oplus{\mathfrak n}^*\), let \({\mathfrak p}(l)\) denote a Vergne-polarization for \(l\) and put \(P(l)=\exp {\mathfrak p}(l)\). We denote by \(\pi_l=\text{ind}^G_{P (l)} \chi_l\), where \(\chi_l(p) =e^{il(\log p)}\) \((p\in P(l))\), the induced representation of \(G\) realized on \(L^2(\mathbb{R}^\nu)\). Then, by applying continuous perturbation theory, the Boidol condition yields that an eigenvalue \(\lambda (l)\) and an eigenfunction \(\varphi_l\) in \(L^2(\mathbb{R}^\nu)\) to \(\pi_l(L)\) depend continuously on \(l\) in a neighborhood \({\mathcal F}\times {\mathcal V}\) of a given point \(l^0=(\tau_0, l_0)\), where \({\mathcal F}\) is a complex neighborhood of \(\tau_0\) in \({\mathfrak a}^*_\mathbb{C}\) and \({\mathcal V}\) a neighborhood of \(l_0\); more precisely, \(\lambda: {\mathcal F}\times {\mathcal V}\to \mathbb{C}\) and \(\varphi: {\mathcal F} \times {\mathcal V}\to {\mathcal S}(\mathbb{R}^\nu)\) are holomorphic with respect to \(\tau\) and they satisfy \(\pi_l(L) \varphi_l= \lambda(l) \varphi_l\) for all \(l\in{\mathcal F} \times{\mathcal V}\). We choose \(\psi\) in \({\mathcal S}(\mathbb{R}^\nu)\) such that \(\langle \psi,\varphi_l \rangle\neq 0\) for every \(l\in{\mathcal F}\times {\mathcal V}\). Then, if we put \(\Phi_f(l)= \langle\pi_l (f)\psi, \varphi_l\rangle\), we see that \(\Phi_{m (L)F}(l)= m(\lambda(l)) \Phi_F(l)\) for real \(l\) and \(F\in C_0^\infty (G)\) such that \(\Phi_F (l^0)\neq 0\). In order to obtain the desired result, applying the Plancherel theorem, the authors show that the function \(\Phi_f\) is well defined on \({\mathcal F}\times {\mathcal V}\) and the \(L^{p'}\)-norm of \(\Phi_f\), \(p'\) is the conjugate exponent of \(p\), is dominated by the \(L^p\)-norm of \(f\). The principal strategy of the proof is the same as in [\textit{M. Christ} and \textit{D. Müller}, Geom. Funct. Anal. 6, 860-876 (1996; Zbl 0878.43008)], however, the proof in this paper is more general and based on a variety of facts from the representation theory of exponential Lie groups.
scientific article; zbMATH DE number 1419626
scientific article