From Complex-Analytic Models to Sparse Domination: A Dyadic Approach of Hypersingular Operators via Bourgain's Interpolation Method

TL;DR

This work develops a real-variable, dyadic framework for hypersingular operators on the unit disk in the regime $1<t<3/2$, where strong-type bounds on the critical line fail. By combining a hypersingular sparse domination principle with Bourgain's interpolation, the authors obtain sharp critical-line and endpoint estimates for the hypersingular Bergman projection $K_{2t}$ and establish a dyadic model of hypersingular operators via graded sparse families. They prove a full $L^p$ mapping theory for the dyadic hypersingular maximal operator and derive endpoint bounds, including $K_{2t}: L^{\frac{1}{3-2t},1}(\mathbb{D}) \to L^{1,\infty}(\mathbb{D})$, through a synthesis of sparse domination and interpolation, with the graded-sparse operator framework providing a geometry-driven $(p,q)$-range dependent on $n,t,\eta,K_{\mathcal{S}}$. The results bridge real-variable harmonic analysis with Forelli–Rudin type hypersingular operators, offering new tools for weighted estimates and potential extensions to higher dimensions and related operators.

Abstract

Motivated by the work of Cheng--Fang--Wang--Yu on the hypersingular Bergman projection, we develop a real-variable and dyadic framework for hypersingular operators in regimes where strong-type estimates fail at the critical line. The main new input is a hypersingular sparse domination principle combined with Bourgain's interpolation method, which provides a flexible mechanism to establish critical-line (and endpoint) estimates. In the unit disc setting with $1<t<3/2$, we obtain a full characterization of the $(p,q)$ mapping theory for the dyadic hypersingular maximal operator $\mathcal M_t^{\mathcal D}$, in particular including estimates on the critical line $1/q-1/p=2t-2$ and a weighted endpoint criterion in the radial setting. We also prove endpoint estimates for the hypersingular Bergman projection \[ K_{2t}f(z)=\int_{\mathbb D}\frac{f(w)}{(1-z\overline w)^{2t}}\,dA(w), \] including a restricted weak-type bound at $(p,q)=\bigl(\tfrac{1}{3-2t},1\bigr)$. Finally, we introduce a class of hypersingular cousin of sparse operators in $\mathbb R^n$ associated with \emph{graded} sparse families, quantified by the sparseness $η$ and a new structural parameter (the \emph{degree}) $K_{\mathcal S}$, and we characterize the corresponding strong/weak/restricted weak-type regimes in terms of $(n,t,η,K_{\mathcal S})$. Our real-variable perspective addresses to an inquiry raised by Cheng--Fang--Wang--Yu on developing effective real-analytic tools in the hypersingular regime for $K_{2t}$, and it also provides a new route toward the critical-line analysis of Forelli--Rudin type operators and related hypersingular operators in both real and complex settings.

Figures (3)

• Figure 1: Boundedness of ${\mathcal{M}}_t^{{\mathcal{D}}}$ for $1<t<3/2$: the red line and the shaded region indicate strong $(p,q)$ bounds, while the blue line indicates weak $(p,q)$ bounds.
• Figure 2: Boundedness of $K_{2t}$ for $1<t<3/2$: the red line and the shaded region indicate strong $(p,q)$ bounds, the blue line indicates weak $(p,q)$ bounds, and the green dot indicates restricted $(p,q)$ bounds.
• Figure 3: Boundedness of $\mathbb A_{{\mathcal{S}}}^t$ for $1<t<1-\frac{\log_2(1-\eta)}{nK_{{\mathcal{S}}}}$: the red line and the shaded region indicate strong $(p,q)$ bounds, the blue line indicates weak $(p,q)$ bounds, and the green dot indicates restricted $(p,q)$ bounds.

Theorems & Definitions (38)

• Remark 1.1
• Remark 1.2
• Definition 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Remark 1.7
• Example 1.8
• Remark 1.9
• proof