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A note on harmonic polynomials on Heisenberg and Carnot groups

Francesco Paolo Maiale

Abstract

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 $L^2(S_N,σ_N)$.

A note on harmonic polynomials on Heisenberg and Carnot groups

Abstract

In this paper, we consider homogeneous -harmonic polynomials on the first Heisenberg group and their traces on the unit sphere associated with the Korányi--Folland homogeneous norm . We prove that decomposes as the orthogonal Hilbert direct sum of finite-dimensional spaces of spherical harmonics of degree , in direct analogy with the classical Euclidean spherical harmonic decomposition. We also show that, for the polynomial gauge , every homogeneous polynomial on admits a unique decomposition Finally, we extend the spherical -decomposition to general Carnot groups equipped with a canonical homogeneous norm associated with a fundamental solution of a fixed sub-Laplacian . The traces on of homogeneous -harmonic polynomials of degree form pairwise orthogonal eigenspaces of the spherical operator on , and their span is dense in .
Paper Structure (22 sections, 16 theorems, 160 equations)

This paper contains 22 sections, 16 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Heisenberg groups and the sub-Laplacian operator
  3. Homogeneity and statements of the main results
  4. A Euclidean-style decomposition on H
  5. Contributions and organization of the paper
  6. Homogeneous norms and metric structure on Hk
  7. Polar coordinates and surface measure
  8. Surface measure and Gauss--Green
  9. Radial derivative and radial sub-Laplacians
  10. Spherical decomposition on the Heisenberg group
  11. Homogeneous harmonic polynomials, traces, and orthogonality
  12. Density and $L^2$-decomposition on $S_\rho$
  13. Higher-dimensional Heisenberg groups Hk
  14. An eta-based decomposition of polynomials on H
  15. A Dirichlet-type algebraic lemma
  16. ...and 7 more sections

Key Result

Theorem 1.2

Let $\mathbb H=\mathbb H_1$, let $\rho$ be the Korányi--Folland homogeneous norm and let $\sigma$ be the associated surface measure on $S_\rho=\{\rho=1\}$. For $m\ge0$ set Then each $H_m(S_\rho)$ is finite-dimensional, the spaces $H_m(S_\rho)$ are pairwise orthogonal in $L^2(S_\rho,\sigma)$, and

Theorems & Definitions (39)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1: Cygan
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4: Geometry of the Korányi sphere
  • Theorem 3.1: Folland--Stein polar coordinates
  • Remark 3.2
  • ...and 29 more