复核的相消与尖锐临界线及Forelli--Rudin算子的端点理论 I：纯超奇异情形
Cancellation of complex kernels and sharp critical lines and endpoint theory for the Forelli--Rudin operators I: the purely hypersingular case
- Auburn University（奥本大学）
- Chongqing University（重庆大学）
- Fudan University（复旦大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文对超奇异区域中Forelli--Rudin算子及其正版本进行弱型和限制弱型映射性质的完整尖锐分类，发现复核的相消现象导致临界端点处两算子有界性分离，并给出精确的端点估计。
AI中文摘要：
对于$a,b,c\in\mathbb R$，我们考虑Forelli--Rudin算子$$ T_{a,b,c}f(z):=(1-|z|^2)^a\int_{\mathbb D}\frac{(1-|w|^2)^b}{(1-z\overline w)^c}f(w)\\,dA(w) $$及其正版本$$ S_{a,b,c}f(z):=(1-|z|^2)^a\int_{\mathbb D}\frac{(1-|w|^2)^b}{|1-z\overline w|^c}f(w)\\,dA(w). $$我们在超奇异区域$$ \Omega_{\mathcal H}:=\{(p,q):1\leq p,q\leq\infty,\\ p>q\} $$中获得了它们的弱型和限制弱型映射性质的完整且尖锐的分类，从而实质性地扩展了第一和第四作者关于超奇异Bergman投影的近期工作。本工作的主要发现之一是与复Forelli--Rudin核相关的内在相消现象：在某些临界端点，相消导致两个算子之间出现尖锐分离，$T_{a,b,c}$保持有界而其正版本$S_{a,b,c}$无界。也许令人惊讶的是，这种相消在Zhao和Zhou于2022年建立的强$L^p$--$L^q$理论中不可见，在那里两个算子具有相同的有界性范围，并且仅在弱型和限制弱型层面显现。允许参数$a,b,c$变化，我们证明$\Omega_{\mathcal{H}}$中所有弱型Forelli--Rudin对的集合恰好是$$ \mathcal{FR}_w=\{(p,q)\in\Omega_{\mathcal H}:1<p\leq2\}, $$而所有限制弱型Forelli--Rudin对的集合是$$ \mathcal{FR}_{rw}=\{(p,q)\in\Omega_{\mathcal H}:p\neq\infty\}. $$这些范围连同所有相应的端点估计和失败都是尖锐的。我们的方法结合了二进分解、概率构造和弱型Hardy估计。
英文摘要：
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.
