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

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.