Local regularity of the Bergman projection on a class of pseudoconvex domains of finite type

TL;DR

The paper addresses local regularity of the Bergman projection $B_\,Omega$ on bounded smooth pseudoconvex domains of finite type by combining Bell–Ligocka Condition $R$ with a good anisotropic dilation tied to Catlin multitype. It develops uniform, scale-invariant pointwise and derivative estimates for the Bergman kernel under anisotropic dilations (Theorem main1) and then derives local $L^p_s$ and Hölder regularity for the Bergman projection (Theorem main2), using a combination of kernel bounds, transformation laws, and pseudolocal estimates. It then specializes to domains satisfying these hypotheses, notably $h$-extendible domains, showing global Condition $R$ and that the boundary is a set of good anisotropic dilation points, which yields local regularity in $L^p_s$ and $\\Lambda_s$, plus a lower semicontinuity property for the multitype sum $T(p)$. The contributions extend known finite-type regularity results to a broad class of domains by linking geometric dilation structures to operator regularity, providing tools for boundary analysis via Catlin multitype. Overall, the work clarifies how nonisotropic scaling governed by the boundary geometry governs local Bergman-projection regularity and broadens the class of domains where sharp local estimates are available.

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.

Theorems & Definitions (46)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 1.7
• Theorem 2.1
• proof
• Theorem 2.2