[Paper Review] The Szegö kernel for certain non-pseudoconvex domains in C^2

Michael Gilliam, Jennifer Halfpap|arXiv (Cornell University)|Jul 8, 2011

Holomorphic and Operator Theory4 references3 citations

TL;DR

This paper investigates the Szegö kernel for non-pseudoconvex domains in ℂ² defined by a smooth, non-convex function b(x) = ¹⁄₄x⁴ + ¹⁄₂px² + qx with p < 0. Using integral representations and asymptotic analysis of oscillatory integrals, the authors identify precise sets in ℂ²×ℂ² where the Szegö kernel and its derivatives converge absolutely, showing divergence (and thus singularities) off the diagonal in regions where |x| ≥ √(−p) or |x| = |r| = √(−p), particularly when x = ±r.

ABSTRACT

We consider the Szegö kernel for domains Ωin C^2 given by Ω= {(z,w): Im w &gt; b(Re z)} where b is a non-convex quartic polynomial with positive leading coefficient. Such domains are not pseudoconvex. We describe the subset of \barΩ imes \barΩ on which the kernel and all its derivatives are finite. In particular, we show that there are points off the diagonal of the boundary at which the Szegö kernel is infitie as well as points on the diagonal at which it is finite.

Motivation & Objective

• To understand the mapping properties of the Szegö projection on non-pseudoconvex domains in ℂ², where standard pseudoconvex theory fails.
• To determine the precise sets in ℂ²×ℂ² where the Szegö kernel and its derivatives are absolutely convergent, especially in the absence of pseudoconvexity.
• To analyze the behavior of the Szegö kernel near and off the diagonal in the boundary product space, focusing on singularities arising from non-convex defining functions.
• To extend known results on Bergman and Szegö kernels—previously limited to pseudoconvex domains—to a class of non-pseudoconvex domains with smooth, non-convex boundaries.
• To provide a detailed asymptotic analysis of oscillatory integrals arising from the integral kernel formula for the Szegö projection in this setting.

Proposed method

• Derives an explicit integral formula for the Szegö kernel S(z,w) using a parameterized representation involving the function N(η,τ) = ∫ e^{2τ[ηλ−b(λ)]} dλ.
• Expresses the kernel as a double integral over τ > 0 and η ∈ ℝ involving τ e^{ητ(z₁+ w̄₁) + iτ(z₂− w̄₂)} [N(η,τ)]^{-1}, with a constant c.
• Analyzes the convergence of the kernel and its derivatives by studying the behavior of the amplitude function A(x,r,η) in the oscillatory integral, particularly near critical points of the phase.
• Identifies critical values of η (e.g., η₀, q) where the phase function B_η achieves its maximum, and examines the local structure of A(x,r,η) in neighborhoods of these points.
• Applies stationary phase and non-stationary phase techniques, using asymptotic expansions of λ(η) ≈ η^{1/3} as |η| → ∞ to estimate decay rates.
• Establishes convergence or divergence of the integral I₁⁰(η) by comparing the amplitude to power laws, particularly (η−η₀)^2(1+|η|)^{-2/3}, leading to divergence when exponent −9/4 is applied.

Experimental results

Research questions

• RQ1 For non-pseudoconvex domains in ℂ² defined by a smooth, non-convex function b(x), where exactly does the Szegö kernel and its derivatives converge absolutely?
• RQ2 How do the singularities of the Szegö kernel arise in the absence of pseudoconvexity, particularly when b(x) is a non-convex quartic?
• RQ3 What role does the critical point structure of the phase function play in determining the convergence of the integral kernel representation?
• RQ4 How does the asymptotic behavior of λ(η) — the real root of the cubic equation arising from the phase — affect the decay rate of the amplitude function?
• RQ5 In what regions of ℂ²×ℂ² does the Szegö kernel fail to be smooth, and what is the precise nature of the singularity (e.g., power-law divergence) in these regions?

Key findings

• The Szegö kernel and its derivatives are absolutely convergent on the set Σ = {(z,w) : x=r and |x|>√(−p)} ∪ {(z,w) : |x|=|r|=√(−p)} only when the phase and amplitude functions are bounded away from zero.
• In the case x=r and |x|>√(−p), the amplitude function A(x,x,η) ≈ (η−η₀)^2(1+|η|)^{-2/3}, leading to divergence of the integral I₁⁰(η) due to the exponent −9/4 applied to this amplitude.
• When x=r and |x|=√(−p), the amplitude behaves as A(x,x,η) ≈ (η−q)^2(1+|η|)^{-2/3}, again resulting in divergence of I₁⁰(η), indicating a singularity in the kernel.
• For the case |x|=|r|=√(−p) with x=−r, the amplitude A(x,−x,η) ≈ (η−q)h(η) with |h(η)|≈(1+|η|)^{1/3}, leading to an integrand of order |η−q|^{-9/4}(1+|η|)^{-3/4}, which diverges at η=q.
• The analysis confirms that the Szegö kernel is not absolutely convergent in any neighborhood of the set where x=r and |x|≥√(−p), or where |x|=|r|=√(−p), implying singularities off the diagonal.
• The divergence of the integral I₁⁰(η) in all three cases establishes that the kernel and its derivatives fail to converge absolutely in these regions, proving the existence of singularities in the non-pseudoconvex setting.