On \(L^ p\) spectral multipliers for a solvable Lie group
type: Item
instance of: scholarly article
title: On \(L^ p\) spectral multipliers for a solvable Lie group
zbMATH Open document ID: 0878.43008
DOI: 10.1007/BF02246787
author: Michael Christ
author: Detlef Müller
published in: Geometric and Functional Analysis. GAFA
publication date: 1997-12-15T00:00:00Z
full work available at URL: https://eudml.org/doc/58249
review text: 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 \(G\).
Mathematics Subject Classification ID: 43A22
Mathematics Subject Classification ID: 43A15
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 22E27
zbMATH DE Number: 954722
zbMATH Keywords: Laplacian
zbMATH Keywords: Lie group
zbMATH Keywords: Lie algebra
zbMATH Keywords: operator
zbMATH Keywords: spectral theorem
zbMATH Keywords: Heisenberg group
zbMATH Keywords: unitary representations
Wikidata QID: Q115392664
reviewed by: Grigori L. Litvinov
MaRDI profile type: Publication
cites work: A Lower Estimate for Central Probabilities on Polycyclic Groups
cites work: Spectral Multipliers on Lie Groups of Polynomial Growth
cites work: \(\mathbb{L}_ p\) Fourier multipliers on Riemannian symmetric spaces of the noncompact type
cites work: Connected groups with polynomially induced dual.
cites work: L p -multipliers for Noncompact Symmetric Spaces
cites work: Q5848416
cites work: Spectral multipliers for a distinguished Laplacian on certain groups of exponential growth
cites work: Analysis of a Distinguished Laplacian on Solvable Lie Groups
cites work: The Subalgebra of L 1 (AN) Generated by the Laplacian
cites work: Q3897665
cites work: Subalgebra of $L_1(G)$ associated with Laplacian on a Lie group
cites work: Q5526485
cites work: Symmetry and nonsymmetry for locally compact groups
cites work: Curvatures of left invariant metrics on Lie groups
cites work: Representation of Elliptic Operators in an Enveloping Algebra
cites work: Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. (AM-63)
cites work: Q3142876
cites work: \(L^ p\)-estimates on functions of the Laplace operator
cites work: Diffusion on Lie Groups
Recommended article: Spectral multipliers on exponential growth solvable Lie groups
Recommended article: Spectral multipliers for sub-Laplacians on solvable extensions of stratified groups
Recommended article: L p Bounds for Spectral Multipliers on Nilpotent Groups
Recommended article: \(L^p\) Fourier multipliers on compact Lie groups
Recommended article: \(L^p\)-multipliers sensitive to the group structure on nilpotent Lie groups
Recommended article: Spectral Multipliers on Lie Groups of Polynomial Growth
Recommended article: Multilinear spectral multipliers on Lie groups of polynomial growth
Recommended article: An update on the \(L^p-L^q\) norms of spectral multipliers on unimodular Lie groups
Recommended article: Spectral multipliers for sub-Laplacians with drift on Lie groups
description: scientific article; zbMATH DE number 954722