Spectral Multipliers on Lie Groups

• Spectral multipliers on Lie groups are operators derived from applying complex functions to self-adjoint invariant differential operators via spectral calculus.
• They integrate harmonic analysis, representation theory, and sub-Riemannian geometry to connect smoothness conditions with boundedness and regularity of operators.
• Techniques such as Mihlin–Hörmander type theorems and endpoint estimates illustrate how group structure and dimension dictate optimal multiplier thresholds.

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 $\mathcal{L}$0.

The algebraic structure of $\mathcal{L}$1 governs much of the multiplier theory. In particular:

• Stratified Lie groups (Carnot groups): $\mathcal{L}$2 with $\mathcal{L}$3, $\mathcal{L}$4, and homogeneous dimension $\mathcal{L}$5.
• 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 $\mathcal{L}$6 satisfy smoothness: $\mathcal{L}$7 for some $\mathcal{L}$8 (graded group with homogeneous dimension $\mathcal{L}$9), yielding $\mathcal{L} = \int_0^\infty \lambda\, dE_\lambda,$0 bounded on all $\mathcal{L} = \int_0^\infty \lambda\, dE_\lambda,$1, $\mathcal{L} = \int_0^\infty \lambda\, dE_\lambda,$2 (Cardona et al., 2016). In more generality:

• For a system of (possibly several) weighted subcoercive, self-adjoint, commuting operators $\mathcal{L} = \int_0^\infty \lambda\, dE_\lambda,$3 on a group of polynomial growth $\mathcal{L} = \int_0^\infty \lambda\, dE_\lambda,$4, the threshold is $\mathcal{L} = \int_0^\infty \lambda\, dE_\lambda,$5 in appropriate Sobolev or Besov spaces for multi-parameter (Marcinkiewicz) theory (Martini, 2010, Martini, 2010).
• The best possible such threshold is dictated by the group structure and is generally not improvable for fully noncommutative or higher-step groups (Martini, 2012, McDonald et al., 2022).

For two-step groups: If $\mathcal{L} = \int_0^\infty \lambda\, dE_\lambda,$6 is a two-step stratified group with additional structure (e.g., Heisenberg/Métivier or small center), the threshold can be improved further to $\mathcal{L} = \int_0^\infty \lambda\, dE_\lambda,$7, where $\mathcal{L} = \int_0^\infty \lambda\, dE_\lambda,$8 is the topological dimension (Martini et al., 2013, Martini, 2012). For abelian groups, this recovers $\mathcal{L} = \int_0^\infty \lambda\, dE_\lambda,$9; for general stratified groups, $m$0 remains.

Sharpness: For the Heisenberg group, $m$1 (homogeneous dimension), $m$2 (topological). Sharp Mihlin–Hörmander theory is available with $m$3 (Martini, 2012, Bramati et al., 2020).

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

The classic Mihlin-type theorems guarantee $m$4-boundedness for all $m$5 if $m$6, but recent work obtains p-specific sharp thresholds: $m$7 guarantee boundedness of $m$8 on $m$9, which is generally sharp (Niedorf, 27 Jan 2025).

On groups with degenerate two-step structure, new approaches such as restriction estimates and spectral decompositions into central “caps” yield $$m(\mathcal{L}) = \int_0^\infty m(\lambda)\, dE_\lambda,$$0 bounds in a range $$m(\mathcal{L}) = \int_0^\infty m(\lambda)\, dE_\lambda,$$1, where $$m(\mathcal{L}) = \int_0^\infty m(\lambda)\, dE_\lambda,$$2 depends on group invariants and degeneracy (Niedorf, 27 Jan 2025).

Weighted and Sparse Bounds: Modern techniques exploit sparse domination for oscillatory (e.g., $$m(\mathcal{L}) = \int_0^\infty m(\lambda)\, dE_\lambda,$$3) and