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Conformally Covariant Boundary Operators and Sharp Higher Order CR Sobolev Trace Inequalities on the Siegel Domain and Complex Ball

Joshua Flynn, Guozhen Lu, Qiaohua Yang

Abstract

We first introduce an appropriate family of conformally covariant boundary operators associated to the Siegel domain ${\mathcal U}^{n+1}$ with the Heisenberg group $\mathbb{H}^{n}$ as its boundary and the complex ball $\mathbb{B}_{\mathbb{C}}^{n+1}$ with the complex sphere $\mathbb{S}^{2n+1}$ as its boundary. We provide the explicit formulas of these conformally covariant boundary operators. Second, we establish all higher order extension theorems of Caffarelli-Silvestre type for the Siegel domain and complex ball. Third, we prove all higher order CR Sobolev trace inequalities for the Siegel domain ${\mathcal U}^{n+1}$ and the complex ball $\mathbb{B}_{\mathbb{C}}^{n+1}$.In particular, we generalize the Sobolev trace inequalityfor $γ\in (0, 1)$ in the CR setting by Frank-González-Monticelli-Tan to the case for all $γ\in (0, n+1)\backslash \mathbb{N}$. The family of higher order conformally covariant boundary operators we define are naturally intrinsic to the higher order Sobolev trace inequalities on both the Siegel domain ${\mathcal U}^{n+1}$ and complex ball $\mathbb{B}_{\mathbb{C}^{n+1}}$. Finally, we give an explicit solution to the scattering problem on the complex hyperbolic ball. More precisely, we obtain an integral representation and an expansion in terms of special functions for the solution to the scattering problem.

Conformally Covariant Boundary Operators and Sharp Higher Order CR Sobolev Trace Inequalities on the Siegel Domain and Complex Ball

Abstract

We first introduce an appropriate family of conformally covariant boundary operators associated to the Siegel domain with the Heisenberg group as its boundary and the complex ball with the complex sphere as its boundary. We provide the explicit formulas of these conformally covariant boundary operators. Second, we establish all higher order extension theorems of Caffarelli-Silvestre type for the Siegel domain and complex ball. Third, we prove all higher order CR Sobolev trace inequalities for the Siegel domain and the complex ball .In particular, we generalize the Sobolev trace inequalityfor in the CR setting by Frank-González-Monticelli-Tan to the case for all . The family of higher order conformally covariant boundary operators we define are naturally intrinsic to the higher order Sobolev trace inequalities on both the Siegel domain and complex ball . Finally, we give an explicit solution to the scattering problem on the complex hyperbolic ball. More precisely, we obtain an integral representation and an expansion in terms of special functions for the solution to the scattering problem.
Paper Structure (11 sections, 24 theorems, 221 equations)

This paper contains 11 sections, 24 theorems, 221 equations.

Table of Contents

  1. Introduction and statement of results
  2. Preliminaries
  3. The Complex Ball
  4. Spherical Harmonics on $\mathbb{S}^{2n+1}$
  5. Some Special Function Theory
  6. Boundary Operators $B_{2 j}^{2\gamma}$ $(0\leq j\leq \lfloor\gamma/2\rfloor)$ and $B_{2 j+2[\gamma]}^{2\gamma}$ $(0\leq j\leq \lfloor\gamma\rfloor-\lfloor\gamma/2\rfloor-1)$
  7. Dirichlet Problem and Scattering in the CR setting
  8. Proof of Theorem \ref{['thm:boundary-to-fractional-operator']} and \ref{['thm:boundary-operator-conformal-covariance']}
  9. Proofs of Theorems \ref{['thm:dirichlet-form-symmetry']} and \ref{['th1.5']}
  10. Boundary Operators and Sobolev trace inequalities on complex hyperbolic ball: Proofs of Theorems \ref{['th1.9']}, \ref{['th1.10']} and \ref{['th1.11']}
  11. Solution to the Scattering Problem: Proof of Theorem \ref{['thm:scattering-solution']}

Key Result

Theorem 1.1

Let $\gamma\in (0,1)$. Then there exists a unique linear bounded operator $\mathcal{T}: \dot{H}^{1,\gamma}(\mathcal{U}^{n+1})\rightarrow\dot{S}^{\gamma}(\mathbb{H}^{n})$ such that $\mathcal{T}(U)=U(\cdot,0)$ for all $U\in C_{0}^{\infty}(\mathcal{U}^{n+1})$, where $\dot{H}^{1,\gamma}(\mathcal{U}^{n+1 Equality is attained if and only if $U$ is the unique solution of the extension problem (1.6) with

Theorems & Definitions (38)

  • Theorem 1.1: Frank-González-Monticelli-Tan
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 28 more