---
title: Spectral Multipliers on Lie Groups
url: https://www.emergentmind.com/topics/spectral-multipliers-on-lie-groups
type: topic
---

# Spectral Multipliers on Lie Groups

A spectral multiplier on a Lie group is an operator defined by applying a (typically complex-valued) function to a self-adjoint left-invariant (pseudo-)differential operator via functional calculus. The subject combines harmonic analysis, representation theory, and sub-Riemannian geometry to study the boundedness and regularity properties of such multipliers, relating these to the smoothness of the symbol and the group’s algebraic structure.

## 1. Foundational Concepts: Spectral Calculus and Lie Group Structures

Let \( G \) be a connected Lie group, often assumed unimodular and of polynomial or exponential volume growth. Spectral multipliers are usually considered for positive, self-adjoint, left-invariant differential operators \( \mathcal{L} \), such as sub-Laplacians, elliptic operators, or more generally, Rockland or weighted subcoercive operators. By the spectral theorem,
\[
\mathcal{L} = \int_0^\infty \lambda\, dE_\lambda,
\]
and for any bounded Borel \( m \), define the operator
\[
m(\mathcal{L}) = \int_0^\infty m(\lambda)\, dE_\lambda,
\]
which is bounded on \(L^2(G)\). The spectral multiplier problem is to find optimal smoothness, decay, or other conditions on \( m \) ensuring that \( m(\mathcal{L}) \) is bounded on \(L^p(G)\) or, more generally, from \( L^p \) to \( L^q \).

The algebraic structure of \( G \) governs much of the multiplier theory. In particular:

- **Stratified Lie groups (Carnot groups):** \( \mathfrak{g} = \bigoplus_{j=1}^s \mathfrak{g}_j \) with \( [\mathfrak{g}_1, \mathfrak{g}_j] = \mathfrak{g}_{j+1} \), \( \mathfrak{g}_s \neq 0 \), and homogeneous dimension \( Q = \sum_j j \dim \mathfrak{g}_j \).
- **Graded groups:** admit positive gradings, dilations, and homogeneous operators, notably Rockland operators (injective on nontrivial smooth rep. vectors of all irreducible unitary reps), ensuring hypoellipticity and a robust functional calculus.
- **Two-step stratified (or Métivier, Heisenberg-type):** Structural results, such as the existence of a non-degenerate commutator form, yield finer control of the spectral resolution.

## 2. Mihlin–Hörmander Type Theorems: Thresholds and Sobolev Conditions

The classical Mihlin–Hörmander theorem requires that a multiplier symbol \( m \) satisfy smoothness:
\[
\|m\|_{M^s} = \sup_{\lambda>0}\sup_{0\leq k\leq s} |\lambda^k m^{(k)}(\lambda)| < \infty,
\]
for some \( s>\frac{Q}{2} \) (graded group with homogeneous dimension \( Q \)), yielding \( m(\mathcal{L}) \) bounded on all \(L^p(G)\), \(1<p<\infty\) [1610.04701]. In more generality:
- For a system of (possibly several) weighted subcoercive, self-adjoint, commuting operators \(L_1,\dots, L_n\) on a group of polynomial growth \( Q_G \), the threshold is \( s > \frac{Q_G}{2}+\frac{n-1}{q} \) in appropriate Sobolev or Besov spaces for multi-parameter (Marcinkiewicz) theory [1010.1186, 1007.1119].
- The best possible such threshold is dictated by the group structure and is generally not improvable for fully noncommutative or higher-step groups [1212.0775, 2201.12349].

**For two-step groups:** If \( G \) is a two-step stratified group with additional structure (e.g., Heisenberg/Métivier or small center), the threshold can be improved further to \( s > d/2 \), where \( d \) is the topological dimension [1306.0387, 1212.0775]. For abelian groups, this recovers \( s > n/2 \); for general stratified groups, \( s > Q/2 \) remains.

**Sharpness:** For the Heisenberg group, \( Q=2n+2 \) (homogeneous dimension), \( d=2n+1 \) (topological). Sharp Mihlin–Hörmander theory is available with \( s > d/2 \) [1212.0775, 2011.13987].

## 3. Endpoint, p-Specific, Weighted, and Oscillatory Multiplier Theorems

The classic Mihlin-type theorems guarantee \(L^p\)-boundedness for all \( 1<p<\infty \) if \( s > d/2 \), but recent work obtains **p-specific sharp thresholds**:
\[
s > d\left|\frac{1}{p} - \frac{1}{2}\right|
\]
guarantee boundedness of \( m(\mathcal{L}) \) on \(L^p(G)\), which is generally sharp [2501.16262].

On groups with degenerate two-step structure, new approaches such as restriction estimates and spectral decompositions into central “caps” yield \(L^p\) bounds in a range \(1<p\leq p_*\), where \(p_*\) depends on group invariants and degeneracy [2501.16262].

**Weighted and Sparse Bounds:** Modern techniques exploit sparse domination for oscillatory (e.g., \( m(\lambda)=\lambda^{i\theta}e^{i\lambda^\beta} \)) and

Source: https://www.emergentmind.com/topics/spectral-multipliers-on-lie-groups