Taylor polynomials on left-quotients of Carnot groups

Technical Summary: Taylor Polynomials on Left-Quotients of Carnot Groups

Problem Statement
The paper addresses the challenge of establishing classical Taylor polynomial theorems for sub-Riemannian manifolds that arise as submetric images (quotients) of Carnot groups. While Rothschild and Stein [8] established a foundational framework for hypoelliptic operators by lifting vector fields to a manifold with a free nilpotent Lie algebra structure, their approach generally requires lifting to a high-dimensional space. In explicit examples, such as non-equiregular sub-Riemannian manifolds (where the rank of the horizontal distribution or the growth vector varies), it is desirable to minimize the dimension of the ambient space. The specific problem tackled here is to generalize and clarify methods (referenced from [4]) to define Taylor approximations on manifolds obtained by projecting a stratified nilpotent Lie group (a Carnot group) onto a quotient space. These quotient spaces may exhibit degeneracy, such as non-constant rank or varying growth vectors, which complicates the direct application of standard sub-Riemannian analysis.

Methodology
The core methodology relies on the concept of submetries. The author considers a Carnot group $G$ and a closed subgroup $H$ such that the dilation group preserves $H$ ($\delta_\lambda(H) = H$). The quotient space $M = H \backslash G$ (right cosets) is equipped with a distance $d_M$ defined by the infimum of distances in $G$. The canonical projection $\pi: G \to M$ is a submetry, meaning it maps metric balls in $G$ onto metric balls in $M$ of the same radius.

The analysis proceeds through the following steps:

1. Lifting and Projection: Functions and vector fields on the quotient manifold $M$ are related to their counterparts on the Carnot group $G$ via the projection $\pi$ and a specific right inverse inclusion $\iota$. A function $f$ on $M$ is lifted to a left $H$-invariant function $\tilde{f} = f \circ \Pi$ on $G$.
2. Transfer of Definitions: The definition of Taylor polynomials on $M$ is constructed by transferring the well-established definitions from Carnot groups (specifically from [6] and [2]). A polynomial $P$ on $M$ is defined by matching the derivatives of a function $f$ with respect to the projected horizontal vector fields $X_j = \Pi_*(\tilde{X}_j)$.
3. Equivalence of Suprema: A crucial technical tool is Lemma 4.2, which establishes that the supremum of a left $H$-invariant function over a ball in $G$ is equal to the supremum of the corresponding function on $M$ over the projected ball. This allows estimates derived on the group $G$ to be directly transferred to the quotient $M$.
4. Invariance Arguments: The proofs utilize the left $H$-invariance of the lifted functions and the commutation properties of right-invariant vector fields to ensure that the Taylor polynomials defined on the group are well-defined on the quotient and independent of the choice of representative in the coset.

Key Contributions and Results
The paper provides a rigorous framework for Taylor approximations on a class of sub-Riemannian manifolds that are not necessarily equiregular. The main results include:

• Taylor Theorems on Quotients: The author proves a Taylor theorem (Theorem 4.4) and a Lagrange mean value theorem (Theorem 4.3) for functions $f \in C^k_{hor}(M)$. These theorems provide remainder estimates of the form $|f(p) - P_k(f, q)(p)| \leq C d_M(p, q)^k \eta_M$, where the error term depends on the modulus of continuity of the $k$-th order horizontal derivatives.
• Well-Posedness: Proposition 4.1 demonstrates that the Taylor polynomial on the quotient $M$ is the restriction of the Taylor polynomial of the lifted function on $G$. It proves that this definition is independent of the choice of the representative point in the coset.
• Analyticity Condition: Theorem 5.1 establishes a sufficient condition for a smooth function on $M$ to be real analytic. If the horizontal derivatives satisfy a specific growth bound involving factorials ($K^{d(I)} k!$), the function is real analytic. This result is derived by lifting the condition to $G$ and applying known results from [2].
• L-Harmonicity: Theorem 5.2 proves that if a function $f$ is annihilated by a differential operator $L$ (specifically one formed by horizontal vector fields), then its Taylor polynomials are also annihilated by $L$. This extends the property of $L$-harmonicity of Taylor polynomials from stratified groups to their quotients.

Significance and Claims
The paper claims to generalize and clarify the method used in previous works (specifically [4]) to obtain Taylor approximation theorems for sub-Riemannian manifolds that may lack constant rank or uniform growth vectors. By utilizing the submetry property of the projection from a Carnot group, the author successfully transfers analytic properties (such as Taylor expansions, analyticity criteria, and harmonicity of polynomials) from the structured setting of stratified Lie groups to more general, potentially degenerate, quotient manifolds.

The significance lies in providing a unified approach that avoids the need to lift vector fields to a high-dimensional "free" nilpotent algebra (as in the Rothschild-Stein framework) when a specific, lower-dimensional quotient structure is available. The results confirm that the sub-Riemannian geometry of these quotients retains enough structure to support classical analysis tools like Taylor expansions, provided the underlying projection is a submetry. The paper explicitly notes that these results are applied to examples such as the Grushin plane and specific CR structures, demonstrating the utility of the theory in non-equiregular settings.