Restricted type estimates on the Bergman projection of some singular domains

TL;DR

This work studies the Bergman projection on monomial polyhedra, a class of domains generalizing the Hartogs triangle, and reveals a restricted range phenomenon for $L^p$-boundedness. The authors develop weighted restricted type estimates for the positive Bergman operator $P^+_{\mathcal{U}_B}$, with weights involving $(-\log w)$ and an invariant $m$, and define the left endpoint $p_*$ from the domain data; they show $P^+_{\mathcal{U}_B}$ is of restricted type $(p_*,p_*)$ from $(\mathcal{U}_B,(-\log w)^{(m-1)(p_*-1)})$ to $(\mathcal{U}_B,w^{p_*-2})$, yielding endpoint results by duality. When $m=1$, these give unweighted restricted type at $p_*$ and, via interpolation, strong-type results for $p_*>p>p_*$ up to the right endpoint $q_*$; for $m>1$ there are monomial polyhedra where $P_{\mathcal{U}_B}$ fails to be of restricted weak type $(p_*,p_*)$, though a weighted restricted type result remains available and, in particular, strong-type bounds hold for $p\in(p_*,q_*)$ via interpolation. The approach combines duality, interpolation, and pullbacks along ramified monomial coverings to reduce to polydisc estimates, and it yields a concrete counterexample at $p_*=\tfrac{4}{3}$ in dimension three, contradicting a broader weak-type claim in the literature. Overall, the paper extends CJM’s endpoint analysis to weighted restricted-type frameworks, clarifies endpoint irregularities for certain monomial polyhedra, and suggests further study of endpoint phenomena on quotient domains and in low dimensions. The results have implications for $\overline{\partial}$-Neumann problems and boundary regularity on non-smooth domains.

Abstract

We obtain (weighted) restricted type estimates for the Bergman projection operator on monomial polyhedra, a class of domains generalizing the Hartogs triangle. From these estimates, we recapture $L^p$ boundedness results of the Bergman projection on these domains. On some monomial polyhedra, we also discover that the Bergman projection could fail to be of weak type $(q_*,q_*)$ where $q_*$ is the right endpoint of the interval of $L^p$-regularity of the domain.

Theorems & Definitions (38)

• Theorem 1.1
• Proposition 1.2
• Corollary 1.3
• Theorem 1.4
• Proposition 1.5
• Corollary 1.6
• Proposition 2.1
• proof
• proof : Proof of Proposition \ref{['prop-sawyer']}
• proof : Proof of Corollary \ref{['cor-m=1']}