From Complex-Analytic Models to Dyadic Methods: A Real-Variable Approach to Hypersingular Operators

Motivated by the work of Cheng-Fang-Wang-Yu on the hypersingular Bergman projection, we develop a real-variable framework for hypersingular operators in regimes where strong-type bounds fail on the critical line. Our main new ingredient is the Forelli-Rudin method: a dyadic mechanism, inspired by complex-analytic Forelli-Rudin type arguments, that yields sharp critical-line and endpoint estimates. On the unit disc, for $1<t<3/2$, we give a complete $(p,q)$-mapping characterization for the dyadic hypersingular maximal operator $\mathcal M_t^{\mathcal D}$, including sharp bounds on the critical line $1/q-1/p=2t-2$ and a weighted endpoint criterion in the radial setting. We also prove a novel two-weight estimate for $\mathcal M_t^{\mathcal D}$ in the range $p>q$, valid for all $t>0$. For the hypersingular Bergman projection $$K_{2t}f(z)=\int_{\mathbb D}\frac{f(w)}{(1-z\overline w)^{2t}}\,dA(w),$$ we establish sharp critical-line bounds, with emphasis on the endpoint weak-type estimate at $(p,q)=\bigl(\tfrac{1}{3-2t},1\bigr)$. In particular, this result resolves an open question on the critical-line behavior of the Bergman projection in the hypersingular regime. Finally, we introduce a class of hypersingular cousins of sparse operators in $\mathbb R^n$ associated with graded sparse families, quantified by the sparseness $\eta$ and a new structural parameter (the degree) $K_{\mathcal S}$. We characterize the corresponding sharp strong- and weak-type regimes in terms of $(n,t,\eta,K_{\mathcal S})$. This real-variable perspective addresses an inquiry of Cheng-Fang-Wang-Yu on developing effective real-analytic tools in the hypersingular regime for both $\mathcal M_t^{\mathcal D}$ and $K_{2t}$, and it also provides a new route to critical-line analysis for Forelli-Rudin type and related hypersingular operators in both real and complex settings.