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

@misc{hu2026cancellation,
  title = {{Cancellation of complex kernels and sharp critical lines and endpoint theory for the Forelli--Rudin operators I: the purely hypersingular case}},
  author = {Bingyang Hu and Zipeng Wang and Kenan Zhang and Xiaojing Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14552},
  eprint = {2609.14552},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14552},
}

TY  - UNPB
TI  - Cancellation of complex kernels and sharp critical lines and endpoint theory for the Forelli--Rudin operators I: the purely hypersingular case
AU  - Bingyang Hu
AU  - Zipeng Wang
AU  - Kenan Zhang
AU  - Xiaojing Zhou
PY  - 2026
DA  - 2026/09/13
PB  - arXiv
DO  - 10.48550/arxiv.2609.14552
UR  - https://arxiv.org/abs/2609.14552
AB  - For a,b,c∈ℝ, we consider the Forelli--Rudin operators T_a,b,cf(z):=(1−|z|^2)^a∫_𝔻((1−|w|^2)^b)/((1−zw̅)^c)f(w) dA(w) and their positive counterparts S_a,b,cf(z):=(1−|z|^2)^a∫_𝔻((1−|w|^2)^b)/(|1−zw̅|^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_ℋ:={(p,q):1≤ p,q≤∞, 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_ℋ is precisely ℱℛ_w={(p,q)∈Ω_ℋ:1<p≤2}, whereas the collection of all restricted weak-type Forelli--Rudin pairs is ℱℛ_rw={(p,q)∈Ω_ℋ:p≠∞}. These ranges, together with all corresponding endpoint estimates and failures, are sharp. Our approach combines dyadic decompositions, probabilistic constructions, and weak-type Hardy estimates.
ER  -