Sub-Laplacian in Carnot Groups

• Sub-Laplacian in Carnot groups is a second-order hypoelliptic operator defined via horizontal vector fields in stratified nilpotent Lie groups.
• It underpins analysis and geometry by determining sub-Riemannian metrics, harmonicity, and spectral properties through unique eigenvalue inequalities.
• Recent research extends its framework to fractional and nonlocal variants while establishing rigidity, isometry, and unique continuation principles.

The sub-Laplacian in Carnot groups is a second-order hypoelliptic operator intrinsically defined by the geometry of stratified, nilpotent Lie groups. It arises from the horizontal vector fields generating the first layer of the group’s graded Lie algebra and underpins much of the analysis, geometry, and PDE theory on these non-Euclidean structures. The operator encodes both local and global properties: its structure determines the sub-Riemannian metric, regularity of isometries, harmonicity, spectral quantities, and function space inequalities. Recent research has elucidated its deep regularity properties, generalized analytical frameworks for fractional and nonlocal variants, and established key rigidity and spectral principles analogous to classical Riemannian settings.

1. Structural Foundations: Definition and Properties

A Carnot group $G$ is a connected, simply connected nilpotent Lie group whose Lie algebra splits as a stratification

$$\mathfrak{g} = V_1 \oplus V_2 \oplus \cdots \oplus V_r,$$

where the horizontal layer $V_1$ generates the algebra via iterated commutators. The horizontal bundle $\Delta$ is the left-translation of $V_1$ through $G$. The Carnot–Carathéodory metric $d_{CC}$ is defined as the infimal length among absolutely continuous paths tangent to $\Delta$.

The canonical sub-Laplacian is constructed from a chosen orthonormal basis $X_1, \ldots, X_m$ of $V_1$,

$\Delta_H u = \sum_{i=1}^m X_i^2 u.$

This operator is hypoelliptic, left-invariant, and homogeneous of order 2 under Lie algebra dilations $\delta_\lambda$ satisfying $\delta_\lambda(X) = \lambda^j X$ for $X \in V_j$ (Donne, 2016).

Key properties include:

• Homogeneity: $$(\Delta_H u)(\delta_\lambda(p)) = \lambda^2 (\Delta_H u_\lambda)(p)$$.
• Metric invariance under left-translations and automorphic dilations.
• The fundamental solution has explicit homogeneous structure, e.g., $E(x, y) = [d(x, y)]^{2 - Q}$, with $Q$ the homogeneous dimension (Ruzhansky et al., 2015).

2. Sub-Laplacian, Harmonicity, and Regularity

The sub-Laplacian governs harmonic functions in Carnot groups, often called $Q$-harmonic when $Q$ is the homogeneous dimension. Regularity results include:

• Solutions to $\Delta_H u = 0$ are analytic under natural boundary or smoothness hypotheses.
• 1-quasiconformal mappings, i.e., those whose Pansu differential is conformal on $V_1$, preserve $Q$-harmonic functions and are necessarily smooth (Ottazzi et al., 2010).
• The infinitesimal generators of 1-quasiconformal flows (conformal vector fields) satisfy constrained differential systems reflecting the algebraic grading and geometry (Ottazzi et al., 2010).

Intrinsic regular surfaces—locally defined as zero-level sets of horizontally differentiable functions—play an analogous geometric role to $C^1$ hypersurfaces in Euclidean theory. The intrinsic gradient

$V_G f = (X_1 f, \ldots, X_{m_1} f)$

is foundational for defining the sub-Laplacian and characterizing surface regularity (Donato, 2018).

3. Spectral Theory and Eigenvalue Inequalities

Dirichlet sub-Laplacians on bounded domains $\Omega \subset G$ are self-adjoint, possess pure point spectrum, and exhibit a spectral gap: $\lambda_1 > 0$ is simple, with a strictly positive eigenfunction (Carfagnini et al., 2022). The spectral decomposition of the associated semigroup is

$$P_t^\Omega f(x) = \sum_{n=1}^\infty e^{-t \lambda_n} (f, \varphi_n) \varphi_n(x).$$

Spectral properties facilitate precise asymptotics for quantities such as exit probabilities for hypoelliptic Brownian motion, controlled by the first Dirichlet eigenvalue (Carfagnini et al., 2022).

Crucially, strict inequalities between Dirichlet and Neumann eigenvalues have been established: $$\lambda_j(-\Delta_\Omega^D) > \lambda_{j+1}(-\Delta_\Omega^N) \quad \text{for all } j \in \mathbb{N},$$ deepening the classical principle for elliptic operators and now unified for Carnot group sub-Laplacians (Frank et al., 17 Nov 2024).

Cheeger-type inequalities hold in the subelliptic context: $$\lambda_1^D(\Omega) \geq \frac{1}{4} h_D(\Omega)^2, \quad \lambda_2^N(\Omega) \geq \frac{1}{4} h_N(\Omega)^2,$$ where the Cheeger constants are defined using horizontal perimeter and volume and estimated via geometric max flow–min cut arguments (Kluitenberg, 2023).

4. Boundary Value Problems, Layer Potentials, and Hardy Inequality

A robust potential-theoretic framework has been established for the sub-Laplacian. Layer potentials are defined via the fundamental solution, with the single layer potential

$$S_j u(x) = \int_{\partial \Omega} u(y) E(y, x) (X_j, dv(y)).$$

Double and higher layer potentials admit Plemelj-type jump relations through the boundary (Ruzhansky et al., 2015). Green’s first and second formulae and nonlocal boundary conditions, reminiscent of Kac's principle of not feeling the boundary, yield unique solutions for the Newton potential and generalize to higher powers of the sub-Laplacian.

Refined Hardy inequalities are available: $$\int_\Omega d^\alpha |V u|^2\, dv \geq \frac{(Q+\alpha-2)^2}{4} \int_\Omega d^{\alpha-2} |u|^2\, dv + \text{(boundary term)}.$$ Such inequalities support uncertainty principles and sharpen classical estimates relevant for regularity and analysis (Ruzhansky et al., 2015).

5. Fractional and Nonlocal Sub-Laplacians

The fractional sub-Laplacian, $L^{\gamma/2}$, is defined spectrally or via singular kernels and exhibits nonlocality: $$L^{\alpha/2} u(x) = PV \int_G [u(y) - u(x)] \widetilde{R}_{-\alpha}(y^{-1} x) dy.$$ Abstract extension techniques, generalizing Caffarelli–Silvestre, lift $u$ to a degenerate elliptic PDE in $G \times \mathbb{R}$ using the heat kernel. The explicit Poisson kernel has the form

$$P_G(x, y) = C_a y^{1-a} \int_0^\infty t^{(a-3)/2} e^{-y^2/(4t)} h(t, x) dt.$$

Invariant Harnack inequalities extend classical regularity results to the fractional setting (Ferrari et al., 2012).

Commutator estimates for fractional powers result in fractional Leibniz rules; e.g., the 3-commutator

$$H_\alpha(u, v) = (-\Delta_b)^{\alpha/2}(uv) - u(-\Delta_b)^{\alpha/2}v - v(-\Delta_b)^{\alpha/2}u$$

admits sharp pointwise and mixed-norm bounds, with scaling dictated by the homogeneous dimension (Maalaoui, 2017).

For the 1-Laplacian, both local (least gradient) and nonlocal variational problems converge under uniform total variation bounds, capturing the passage of discrete difference quotients to horizontal derivatives via group dilations (Górny, 2020).

6. Rigidity, Isometry, and Classification: Sub-Laplacian Determines Geometry

The sub-Laplacian encodes the sub-Riemannian structure; if two Carnot groups have equivalent sub-Laplacians, their sub-Riemannian metrics are isometric. Any smooth map intertwining sub-Laplacians must be a sub-Riemannian conformal submersion: a contact map whose horizontal differential is homothetic (Kijowski et al., 31 Dec 2024). This rigidity parallels Helgason and Fuglede’s theorems for Laplacian-commuting maps in the Riemannian context.

Maps commuting with sub-Laplacians are characterized as conformal submersions—locally, their differentials scale inner products on horizontal spaces by a function $\lambda^2$, with explicit intertwining relations for the operators. For Carnot groups of equal dimension, such maps must be isometries up to translation and dilation, firmly linking the analytic operator to the geometric structure (Kijowski et al., 31 Dec 2024).

7. Homogeneous Norms, Polar Coordinates, and Special Carnot Structures

Homogeneous quasi-norms, e.g., those derived from fundamental solutions, play a dual analytic and geometric role. In polarizable Carnot groups (e.g., Heisenberg groups), powers of the homogeneous norm solve the $p$-sub-Laplacian for all $1<p\le\infty$: $$u_p(g) = |g|_u^{\frac{p-Q}{p-1}},\quad \Delta_p u_p = 0,\quad \Delta_\infty |g|_u = 0,$$ and admit canonical horizontal polar coordinate systems with

$$\int_G f(g)\, d\mu(g) = \int_S \int_0^\infty f(\gamma_\xi(s)) s^{Q-1} ds\, d\sigma(\xi),$$

replacing standard dilation polar coordinates by a foliation of $G$ by horizontal curves (Tyson, 2022, Tyson, 2022).

In H-type (Heisenberg-type) Carnot groups, analytic characterizations connect the sub-Laplacian, the homogeneous quasi-norm, and horizontal gradients, quantifying deviation from the Heisenberg case via explicit formulae (Tyson, 2022).

Carleman estimates for the sub-Laplacian with singular weights yield sharp vanishing order results for stationary Schrödinger equations, relying on discrepancy conditions between horizontal and radial derivatives, thereby extending unique continuation principles to the subelliptic setting (Arya et al., 2022).

References to Key Research

• Liouville-type rigidity via Tanaka prolongation: (Ottazzi et al., 2010)
• Harnack and fractional extensions: (Ferrari et al., 2012, Maalaoui, 2017, Corni et al., 2023)
• Reverse Poincaré, isoperimetry, Riesz transforms: (Baudoin et al., 2015)
• Layer potentials, Hardy inequalities, boundary value problems: (Ruzhansky et al., 2015, Baldi et al., 2016)
• Intrinsic differentiability and geometric regularity: (Donato, 2018)
• Local/nonlocal 1-Laplacian and variational methods: (Górny, 2020)
• Spectral theory and eigenvalue gaps: (Carfagnini et al., 2022, Frank et al., 17 Nov 2024)
• Cheeger inequalities, min–cut/max–flow viewpoints: (Kluitenberg, 2023)
• Rigidity and equivalence of sub-Laplacians: (Kijowski et al., 31 Dec 2024)
• Polarizability, horizontal polar coordinates, H-type Carnot groups: (Tyson, 2022, Tyson, 2022)
• Carleman estimates and unique continuation: (Arya et al., 2022)

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The sub-Laplacian of Carnot groups thus forms the foundation for analysis, geometry, and regularity theory in sub-Riemannian spaces, possessing a rich interplay between its algebraic, metric, and analytic features. Its intrinsic structure determines not only function-theoretic and spectral properties but also the underlying geometry, classification, and rigidity of Carnot and related Lie groups.