On \(L^ p\) spectral multipliers for a solvable Lie group
Let \(L\) be a left-invariant Laplacian on a connected Lie group \(G\), i.e. \(L=- \sum V^2_j\), where the \(V_j\) span the Lie algebra of \(G\). Then \(L\) is self-adjoint and nonnegative, and therefore to any continuous, bounded function \(m: [0,\infty) \mapsto\mathbb{C}\) there is associated a bounded operator \(m(L)\) on \(L^2(G)\) via the spectral theorem. If \(m(L)\) extends from \(L^2(G) \cap L^p (G)\) to an operator bounded on \(L^p(G)\) for some \(p\neq 2\), then we write \(m\in M_p\), assuming that \(1\leq p\leq \infty\). The authors prove the following Theorem. There exist a solvable, unimodular group \(G\) and a Laplacian \(L\) such that for any \(p\neq 2\), there exists an open neighborhood \(\Omega_p\) of \(\mathbb{R}^+\) such that any \(m\in M_p\) extends holomorphically to \(\Omega_p\). This group \(G\) is, for any positive integer \(n\), the \((2n+2)\)-dimensional group \(G_n\) obtained as a semidirect product of \(\mathbb{R}\) with the \((2n+1)\)-dimensional Heisenberg group \(H_n\) via the simplest measure-preserving action by dilations of \(\mathbb{R}\) on \(H_n\). The proof rests on the structure of the set of irreducible unitary representations of \(G\).
scientific article; zbMATH DE number 954722
scientific article