Cancellation of complex kernels and sharp critical lines and endpoint theory for the Forelli--Rudin operators I: the purely hypersingular case

Abstract

For $a,b,c\in\mathbb R$, we consider the Forelli--Rudin operators $$ T_{a,b,c}f(z):=(1-|z|^2)^a\int_{\mathbb D}\frac{(1-|w|^2)^b}{(1-z\overline w)^c}f(w)\,dA(w) $$ and their positive counterparts $$ S_{a,b,c}f(z):=(1-|z|^2)^a\int_{\mathbb D}\frac{(1-|w|^2)^b}{|1-z\overline w|^c}f(w)\,dA(w). $$ We obtain a complete and sharp classification of their weak- and restricted weak-type mapping properties in the hypersingular regime $$ Ω_{\mathcal H}:=\{(p,q):1\leq p,q\leq\infty,\ p>q\}, $$ thereby substantially extending the recent work of the first and fourth authors on hypersingular Bergman projections. One of the main discoveries of this work is an intrinsic cancellation phenomenon associated with the complex Forelli--Rudin kernel: at certain critical endpoints, cancellation creates a sharp separation between the two operators, with $T_{a,b,c}$ remaining bounded while its positive counterpart $S_{a,b,c}$ fails to be bounded. Perhaps surprisingly, this cancellation is invisible in the strong $L^p$--$L^q$ theory established by Zhao and Zhou in 2022, where the two operators have the same boundedness range, and emerges only at the weak- and restricted weak-type levels. Allowing the parameters $a,b,c$ to vary, we show that the collection of all weak-type Forelli--Rudin pairs in $Ω_{\mathcal{H}}$ is precisely $$ \mathcal{FR}_w=\{(p,q)\inΩ_{\mathcal H}:1<p\leq2\}, $$ whereas the collection of all restricted weak-type Forelli--Rudin pairs is $$ \mathcal{FR}_{rw}=\{(p,q)\inΩ_{\mathcal H}:p\neq\infty\}. $$ These ranges, together with all corresponding endpoint estimates and failures, are sharp. Our approach combines dyadic decompositions, probabilistic constructions, and weak-type Hardy estimates.