A note on harmonic polynomials on Heisenberg and Carnot groups

In this paper, we consider homogeneous $Δ_H$-harmonic polynomials on the first Heisenberg group $\mathbb H$ and their traces on the unit sphere $S_ρ$ associated with the Korányi–Folland homogeneous norm $ρ$. We prove that $L^2(S_ρ,σ)$ decomposes as the orthogonal Hilbert direct sum of finite-dimensional spaces $H_m(S_ρ)$ of spherical harmonics of degree $m$, in direct analogy with the classical Euclidean spherical harmonic decomposition. We also show that, for the polynomial gauge $η_+^2(z,t)=|z|^2+4t$, every homogeneous polynomial on $\mathbb H$ admits a unique decomposition $$ P_m(\mathbb H) = H_m(\mathbb H)\oplus η_+^2 P_{m-2}(\mathbb H). $$ Finally, we extend the spherical $L^2$-decomposition to general Carnot groups $G$ equipped with a canonical homogeneous norm $N$ associated with a fundamental solution of a fixed sub-Laplacian $Δ_G$. The traces on $S_N$ of homogeneous $Δ_G$-harmonic polynomials of degree $m$ form pairwise orthogonal eigenspaces of the spherical operator on $S_N$, and their span is dense in $L^2(S_N,σ_N)$.

💡 Research Summary

This paper develops a concrete, polynomial‑based framework for spherical harmonic analysis associated with the sub‑Laplacian on the first Heisenberg group ℍ₁ and, more generally, on arbitrary Carnot groups. The authors begin by recalling the group law of ℍ₁, the left‑invariant vector fields, and the sub‑Laplacian Δ_H expressed both in complex and real coordinates. They introduce the Korányi–Folland homogeneous norm ρ(z,t) = (|z|⁴ + t²)^{1/4}, which is homogeneous of degree one with respect to the Heisenberg dilations δ_r(z,t) = (rz, r²t) and smooth away from the origin. Using the Folland–Stein polar integration formula, they decompose the ambient space into radial and angular parts, thereby defining a spherical operator L_{S_ρ} on the unit Korányi sphere S_ρ = {ρ = 1}.

The first main result (Theorem 1.2) shows that the restriction to S_ρ of any homogeneous Δ_H‑harmonic polynomial of degree m is an eigenfunction of L_{S_ρ} with eigenvalue –m(m+2). The spaces H_m(S_ρ) of such traces are finite‑dimensional, mutually orthogonal in L²(S_ρ,σ), and their direct sum exhausts L²(S_ρ,σ). The proof avoids representation theory; instead it relies on a density argument based on the Stone–Weierstrass theorem and the fact that homogeneous polynomials are dense in the space of continuous functions on the sphere.

The second major contribution (Theorem 1.3) is an analogue of the Euclidean decomposition P_m(ℝⁿ) = H_m(ℝⁿ) ⊕ |x|² P_{m‑2}(ℝⁿ). In the Heisenberg setting the authors introduce the polynomial gauge η₊(z,t) = (|z|² + 4t)^{1/2} (so η₊² = |z|² + 4t) and prove the unique decomposition
 P_m(ℍ) = H_m(ℍ) ⊕ η₊² P_{m‑2}(ℍ).
The key technical step is solving the equation Δ_H