A note on harmonic polynomials on Heisenberg and Carnot groups

Abstract

In this paper, we consider homogeneous $\Delta_H$-harmonic polynomials on the first Heisenberg group $\mathbb H$ and their traces on the unit sphere $S_\rho$ associated with the Kor\'anyi--Folland homogeneous norm $\rho$. We prove that $L^2(S_\rho,\sigma)$ decomposes as the orthogonal Hilbert direct sum of finite-dimensional spaces $H_m(S_\rho)$ of spherical harmonics of degree $m$, in direct analogy with the classical Euclidean spherical harmonic decomposition. We also show that, for the polynomial gauge $\eta_+^2(z,t)=|z|^2+4t$, every homogeneous polynomial on $\mathbb H$ admits a unique decomposition $$ P_m(\mathbb H) = H_m(\mathbb H)\oplus \eta_+^2 P_{m-2}(\mathbb H). $$ 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 $\Delta_G$. The traces on $S_N$ of homogeneous $\Delta_G$-harmonic polynomials of degree $m$ form pairwise orthogonal eigenspaces of the spherical operator on $S_N$, and their span is dense in $L^2(S_N,\sigma_N)$.