[Paper Review] Local regularity of the Bergman projection on a class of pseudoconvex domains of finite type

Tran Vu Khanh, Andrew Raich|arXiv (Cornell University)|Jun 25, 2014

Holomorphic and Operator Theory21 references4 citations

TL;DR

This paper establishes local $L^p$-Sobolev and Hölder regularity estimates for the Bergman projection on a class of pseudoconvex domains of finite type in $\mathbb{C}^n$ that satisfy Bell-Ligocka's Condition R and admit a good anisotropic dilation structure. The key result is that such domains—specifically $h$-extendible domains—exhibit uniform $L^2$ pseudolocal estimates under anisotropic scalings, which implies local regularity in $L^p_s$ and $\Lambda_s$ spaces.

ABSTRACT

The purpose of this paper is to prove that if a pseudoconvex domains $Ω\subset\mathbb{C}^n$ satisfies Bell-Ligocka's Condition R and admits a ``good" dilation, then the Bergman projection has local $L^p$-Sobolev and Hölder estimates. The good dilation structure is phrased in terms of uniform $L^2$ pseudolocal estimates for the Bergman projection on a family of anisotropic scalings. We conclude the paper by showing that $h$-extendible domains satisfy our hypotheses.

Motivation & Objective

• To establish local $L^p$-Sobolev and Hölder regularity estimates for the Bergman projection on pseudoconvex domains of finite type.
• To identify a class of domains satisfying Bell-Ligocka's Condition R and admitting a 'good' anisotropic dilation structure.
• To show that $h$-extendible domains satisfy the required geometric and analytic conditions for local regularity.
• To prove uniform $L^2$ pseudolocal estimates for the Bergman projection under anisotropic scalings, linking them to Catlin's multitype.
• To extend known regularity results beyond strictly pseudoconvex and convex domains to a broader class including $h$-extendible domains.

Proposed method

• Introduce a local version of Bell-Ligocka's Condition R via $L^2$ pseudolocal estimates for the Bergman projection in a neighborhood $U$.
• Define 'good anisotropic dilation' at boundary points using uniform scaling behavior of the defining function and multitype invariants.
• Use the scaling structure to derive uniform $L^2$ pseudolocal estimates for $B_{\Omega_{p,\delta}}$ across dilated domains $\Omega_{p,\delta}$, independent of $\delta$.
• Apply subelliptic estimates from Catlin's theory to the scaled domains, leveraging lower bounds on the Bergman metric.
• Establish uniform control on the Bergman kernel's growth via the Jacobian of the scaling maps $\Phi_{q,\eta}$ and the multitype $m_{q,j}$.
• Connect the scaling behavior to the local regularity of $B$ by showing that uniform pseudolocal estimates imply $L^p_s$ and $\Lambda_s$ bounds.

Experimental results

Research questions

• RQ1 Under what geometric and analytic conditions does the Bergman projection on a finite type pseudoconvex domain admit local $L^p$-Sobolev and Hölder regularity?
• RQ2 How does the existence of a 'good' anisotropic dilation structure relate to the uniformity of $L^2$ pseudolocal estimates for the Bergman projection?
• RQ3 Do $h$-extendible domains satisfy the necessary conditions for local regularity of the Bergman projection?
• RQ4 Can uniform pseudolocal estimates under anisotropic scalings be used to derive global regularity properties in $L^p_s$ and $\Lambda_s$ spaces?
• RQ5 What is the role of Catlin's multitype in characterizing the scaling behavior that ensures uniform regularity estimates?

Key findings

• The Bergman projection on domains satisfying Bell-Ligocka's Condition R and admitting a good anisotropic dilation structure has local $L^p$-Sobolev and Hölder regularity estimates.
• Uniform $L^2$ pseudolocal estimates for $B_{\Omega_{p,\delta}}$ hold across all dilated domains $\Omega_{p,\delta}$, with constants independent of $\delta$.
• $h$-extendible domains satisfy the hypotheses of the main theorem, including the existence of a good anisotropic dilation and uniform pseudolocal estimates.
• The lower bound on the Bergman metric in scaled domains implies subelliptic estimates uniformly in $\delta$, which in turn yield the desired pseudolocal estimates.
• The scaling maps $\Phi_{q,\eta}$ and the multitype $m_{q,j}$ control the growth of the Bergman kernel, leading to the estimate $B_{\Omega_{p,\delta}}(z,X) \gtrsim d^{-\kappa}_{\Omega_{p,\delta}}(z)|X|$.
• The main result extends known regularity results to a broader class of finite type domains, including $h$-extendible domains, beyond strictly pseudoconvex, convex, or decoupled domains.