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Asymptotic behaviour of Dirichlet eigenvalues for homogeneous Hörmander operators and algebraic geometry approach

Hua Chen, Hong-Ge Chen, Jin-Ning Li

Abstract

We study the Dirichlet eigenvalue problem of homogeneous Hörmander operators $\triangle_{X}=\sum_{j=1}^{m}X_{j}^{2}$ on a bounded open domain containing the origin, where $X_1, X_2, \ldots, X_m$ are linearly independent smooth vector fields in $\mathbb{R}^n$ satisfying Hörmander's condition and a suitable homogeneity property with respect to a family of non-isotropic dilations. Suppose that $Ω$ is an open bounded domain in $\mathbb{R}^n$ containing the origin. We use the Dirichlet form to study heat semigroups and subelliptic heat kernels. Then, by utilizing subelliptic heat kernel estimates, the resolution of singularities in algebraic geometry, and employing some refined analysis involving convex geometry, we establish the explicit asymptotic behavior $λ_k \approx k^{\frac{2}{Q_0}}(\ln k)^{-\frac{2d_0}{Q_0}}$ as $k \to +\infty$, where $λ_k$ denotes the $k$-th Dirichlet eigenvalue of $\triangle_{X}$ on $Ω$, $Q_0$ is a positive rational number, and $d_0$ is a non-negative integer. Furthermore, we provide optimal bounds of index $Q_0$, which depend on the homogeneous dimension associated with the vector fields $X_1, X_2, \ldots, X_m$.

Asymptotic behaviour of Dirichlet eigenvalues for homogeneous Hörmander operators and algebraic geometry approach

Abstract

We study the Dirichlet eigenvalue problem of homogeneous Hörmander operators on a bounded open domain containing the origin, where are linearly independent smooth vector fields in satisfying Hörmander's condition and a suitable homogeneity property with respect to a family of non-isotropic dilations. Suppose that is an open bounded domain in containing the origin. We use the Dirichlet form to study heat semigroups and subelliptic heat kernels. Then, by utilizing subelliptic heat kernel estimates, the resolution of singularities in algebraic geometry, and employing some refined analysis involving convex geometry, we establish the explicit asymptotic behavior as , where denotes the -th Dirichlet eigenvalue of on , is a positive rational number, and is a non-negative integer. Furthermore, we provide optimal bounds of index , which depend on the homogeneous dimension associated with the vector fields .
Paper Structure (21 sections, 49 theorems, 301 equations)

This paper contains 21 sections, 49 theorems, 301 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. $\delta_{t}$-homogeneous functions and vector fields
  4. Properties of Nagel-Stein-Wainger polynomials
  5. Subunit metric and volume estimates of subunit balls
  6. Classification of homogeneous Hörmander vector fields
  7. Weighted Sobolev spaces of Hörmander vector fields
  8. Subelliptic estimates and Poincaré inequality
  9. Estimates of the subelliptic heat kernels
  10. Abstract theory of Dirichlet forms, heat semigroups and heat kernels
  11. Subelliptic global heat kernel
  12. Subelliptic Dirichlet heat kernel
  13. Comparison of heat kernels
  14. Asymptotic behaviour of integral of $\Lambda(x,r)^{-1}$
  15. Simplification procedures for the integral $J_{\Omega}(r)$
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $X=(X_{1},X_{2},\ldots,X_{m})$ be the homogeneous Hörmander vector fields defined on $\mathbb{R}^n$. Suppose that $\Omega$ is a bounded open domain in $\mathbb{R}^n$ containing the origin. Then the Dirichlet heat kernel $h_D(x,y,t)$ of $\triangle_{X}$ on $\Omega$ satisfies where $|B_{d_{X}}(x,r)|$ denotes the $n$-dimensional Lebesgue measure of subunit ball $B_{d_{X}}(x,r)$.

Theorems & Definitions (115)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 2.1: $\delta_{t}$-homogeneous function
  • Proposition 2.1: Smooth $\delta_{t}$-homogeneous functions
  • proof
  • Definition 2.2
  • ...and 105 more