Spectral multipliers on exponential growth solvable Lie groups
Let \(M\) be a measure space, \(L\) be a positive definite operator on \(L^2(M)\) and let \(F\colon[0,\infty)\to {\mathbb C}\) be a bounded Borel function. Much effort has gone into establishing when the operator \(F(L)\), defined by the spectral theorem, extends boundedly to \(L^p(M)\) for some \(p\) in \([1,\infty]\setminus\{2\}\). This paper considers the case in which \(L\) is a sublaplacian operator on a solvable Lie group, and \(p=1\). \textit{F. M. Christ} and \textit{D. Müller} [Geom. Funct. Anal. 6, 860-876 (1996; Zbl 0878.43008)] gave an example of such an operator on a particular unimodular solvable Lie group for which the \(L^p\) boundedness of \(F(L)\) (where \(p\neq 2\)) implies that \(F\) must extend holomorphically into some \(p\)-dependent domain in \({\mathbb C}\). By contrast, Hebisch shows here that, for certain other solvable Lie groups, there is an index \(k\) such that, if \(F\) is compactly supported and \(k\)-times continuously differentiable on \([0,\infty)\), then \(F(L)\) is \(L^1\) bounded. The groups considered here all nonunimodular.
scientific article; zbMATH DE number 1336833