L–Harmonicity in Taylor Polynomials

• L–harmonicity of Taylor polynomials is the property where each homogeneous Taylor expansion of an L–harmonic function remains L–harmonic, analogous to classic Euclidean results.
• The concept leverages sub-Riemannian quotient structures and hypoelliptic operators, with proofs employing function lifting to Carnot groups and homogeneous Taylor series.
• This framework has practical implications for local solution analysis, unique continuation, and constructing fundamental solutions in degenerate elliptic and CR geometries.

L–harmonicity of Taylor polynomials refers to the phenomenon wherein, for functions satisfying a homogeneous differential equation induced by a hypoelliptic operator $L$ on a sub-Riemannian manifold $M$ modeled as a left-quotient of a Carnot group, every homogeneous Taylor polynomial of such a function is itself exactly $L$–harmonic. This property is the direct analogue of the harmonic decomposition of Taylor expansions for solutions of the Laplace equation in Euclidean settings, now extended to the geometry and analysis of sub-Riemannian quotients (Ottazzi, 13 Dec 2025). The following sections provide a detailed account of the geometric context, analytic framework, formal statements, proof mechanisms, illustrative examples, and implications.

1. Sub-Riemannian Quotients and Horizontal Structures

Let $G$ denote a Carnot group of step $s$ with stratified Lie algebra

$\g = \g_1 \oplus \g_2 \oplus \cdots \oplus \g_s, \qquad [\g_j, \g_1] = \g_{j+1},$

admitting homogeneous dilations $\delta_\lambda$ via $\delta_\lambda|_{\g_j} = \lambda^j \mathrm{Id}$. Consider a homogeneous subgroup $H = \exp\h$ ($\h \subset \g$) with $\{w_1, ..., w_\ell\}$ an $\R$-basis for $\h$ and complementary basis $\{v_1, ..., v_m\}$ for a subspace isomorphic to $\g/\h$. Exponential coordinates of the second kind yield the identification:

$$(y_1, ..., y_\ell \mid x_1, ..., x_m) \mapsto \exp\left(\sum y_i w_i \right) \exp\left(\sum x_j v_j \right) \in G,$$

with projection $\Pi\colon G \to M = H \backslash G \cong \R^m$ defined by

$$\Pi\left(\exp\left(\sum y_i w_i\right)\exp\left(\sum x_j v_j\right)\right) = (x_1, ..., x_m).$$

The horizontal structure is induced by left-invariant vector fields $\tilde X_j$ spanning $\g_1$, which project to bracket-generating fields $X_j = \Pi_* (\tilde X_j)$ on $M$, characterizing the sub-Riemannian distribution. The associated sub-Riemannian distance $d_M$ is defined as the infimum over curves tangent to the span $\{X_1, ..., X_r\}$, and the projection $\Pi$ is a submetry between metric spaces $(G, d_G)$ and $(M, d_M)$ (Ottazzi, 13 Dec 2025).

2. Homogeneous Differential Operators and $L$–Harmonicity

The model second-order hypoelliptic operator is the sum-of-squares sub-Laplacian,

$L_{ss} = \sum_{j=1}^r X_j^2.$

More generally, any left-invariant homogeneous differential operator of total degree $\sigma$ on $M$ can be expressed as

$L = \sum_{\substack{I=(i_1,...,i_k)\d(I)=\sigma}} c_I X^I, \qquad X^I := X_{i_1} X_{i_2} \cdots X_{i_k},$

where $d(I) = \sum_j d_{i_j}$ corresponds to the degree structure inherited from the stratification. A smooth function $f : M \to \R$ is termed $L$–harmonic if $L f = 0$ pointwise; similarly, a polynomial $P : M \to \R$ is $L$–harmonic if $L P \equiv 0$ on $M$. These operators crucially depend on the bracket-generating horizontal fields and homogeneous group structure (Ottazzi, 13 Dec 2025).

3. Taylor Polynomials in Sub-Riemannian Geometry

On the Carnot group $G$, horizontal multi-indices $I = (i_1, ..., i_K)$ are assigned total homogeneous degree $d(I)$, enabling the definition of iterated derivatives $\tilde X^I = \tilde X_{i_1} \cdots \tilde X_{i_K}$. The McLaurin (homogeneous Taylor) polynomial of order $k$ for $F \in C^k_{\rm hor}(G)$ at the group identity $0$ is the unique degree $\le k$ polynomial $\tilde P_k(F, 0)$ satisfying

$$\tilde X^I\left(\tilde P_k(F, 0)\right)(0) = \tilde X^I F(0), \quad \forall d(I) \le k.$$

More generally, at $g_0 \in G$,

$$\tilde P_k(F, g_0)(g) = \left[\tilde P_k(F \circ L_{g_0}, 0)\right](L_{g_0}^{-1} g),$$

where $L_{g_0}$ denotes left translation by $g_0$.

On the quotient $M$, functions $f \in C^k_{\rm hor}(M)$ are lifted to $F = f \circ \Pi \in C^k_{\rm hor}(G)$, allowing the Taylor machinery of the group to be ported to $M$ through projection and appropriate coordinate mapping:

$$P_k(f, p)(x) = \tilde P_k\big(f \circ \Pi \circ L_{\Phi^{-1}(p)}, 0\big)\big(L_{\Phi^{-1}(p)}^{-1} \Phi^{-1}(x)\big),$$

with $P_k(f, p)$ the Taylor polynomial of $f$ at $p$ of homogeneous degree at most $k$ (Ottazzi, 13 Dec 2025).

4. The $L$–Harmonicity Theorem: Statement and Proof Constructs

Theorem (L–harmonicity of Taylor polynomials):

Let $M = H \backslash G$ be a sub-Riemannian quotient as above, and $L$ a homogeneous differential operator of degree $\sigma$, constructed from the horizontal frame $\{X_j\}$. For $f \in C^\infty(M)$ such that $L f = 0$ on $M$, the Taylor polynomial $P_n(f,p)$ at every base point $p \in M$ and every order $n \ge 1$ satisfies:

$L(P_n(f,p)) \equiv 0 \quad \text{on } M.$

Specifically, for the canonical sub-Laplacian $L = \sum_{j=1}^r X_j^2$, every Taylor polynomial of an $L$–harmonic function is an $L$–harmonic polynomial solution.

Proof strategy:

The argument proceeds by lifting the function $f$ to $F = f \circ \Pi$ on $G$, where $F$ is smooth and satisfies $\tilde L F = 0$ for the lifted operator $\tilde L = \sum c_I \tilde X^I$. By Bonfiglioli's theorem on Carnot groups, the homogeneous Taylor polynomial $\tilde P_n(F, g_0)$ at any $g_0 \in G$ is $\tilde L$–harmonic. The projection of $\tilde P_n(F, g_0)$ under $\Pi$ yields $P_n(f, p)$ for $p = \Pi(g_0)$, and since $\Pi_*$ and $\Pi^*$ intertwine the operators, $L$–harmonicity transfers directly. No delicate remainder estimates are required; the submetry property and group Taylor theory suffice (Ottazzi, 13 Dec 2025).

5. Explicit Example: The Grushin Plane

Consider the Grushin plane $M = \R^2$ with vector fields

$$X = \partial_{x_1}, \qquad Y = x_1^\ell \partial_{x_2},$$

and sub-Laplacian $L = X^2 + Y^2$. For an $L$–harmonic $f$, the homogeneous Taylor polynomial of degree $2$ at the origin has the form

$$P_2(f, 0)(x_1, x_2) = f(0) + Xf(0) x_1 + \tfrac12 X^2 f(0) x_1^2,$$

with direct computation verifying

$(X^2 + Y^2) P_2(f, 0) \equiv 0.$

This confirms the exact $L$–harmonicity property for Taylor expansions in this degenerate sub-Riemannian setting (Ottazzi, 13 Dec 2025).

6. Analytic and Geometric Implications

The $L$–harmonic decomposition of Taylor polynomials generalizes Euclidean potential theory, where each homogeneous piece of a harmonic function's Taylor expansion is itself a harmonic polynomial. In the sub-Riemannian scenario, this provides a decomposition into $L$–harmonic polynomials for solutions of hypoelliptic equations on these quotients.

Applications include:

• Power-series expansion and local analysis of solutions;
• Unique continuation properties for $L$–harmonic functions;
• Construction of fundamental solutions and Poisson kernels tailored to sub-Riemannian geometries.

Further research directions encompass the convergence of Taylor series (real-analyticity criteria), Gevrey or ultradifferentiable regularity, and boundary phenomena in CR geometry and degenerate elliptic PDEs. Ottazzi's results extend classical theory to non-Euclidean settings, enabling granular harmonic analysis in sub-Riemannian contexts (Ottazzi, 13 Dec 2025).