AI 中文总结
本文建立非局部算子的两类逐点端点极限，将其应用于推导多个经典公式的逐点形式，研究半群生成算子及逼近过程的端点极限，还解决了乘积空间上p=1时的弱型估计问题。
AI 中文摘要
我们建立非局部算子的两类逐点端点极限：一类基于归一化积分，另一类基于加权弱型范数的分布极限。作为应用，这些极限可导出逐点的Bourgain--Brezis--Mironescu、Maz'ya--Shaposhnikova、Brezis--Seeger--Van Schaftingen--Yung及Gu--Yung公式，还有高阶与平均振荡变体。随后，我们研究半群生成的分数阶幂算子及逼近过程的端点极限，推导调和分析中算子的精确强型与弱型端点估计。最后，Domínguez和Milman在乘积空间上得到了p>1时的弱型估计，留下p=1的端点问题未解决（见[Adv. Math. 411 (2022), Paper No. 108774, p.22]），我们证明了参数变量下的精确逐点极限，并得到p=1处的双侧迭代弱型估计。
英文摘要
We establish two pointwise endpoint limits for nonlocal operators: one based on normalized integrals and a distributional limit based on weighted weak-type norms. As applications, the limits yield pointwise Bourgain--Brezis--Mironescu, Maz'ya--Shaposhnikova, Brezis--Seeger--Van Schaftingen--Yung and Gu--Yung formulas, together with higher-order and mean-oscillation variants. We then investigate endpoint limits for fractional powers generated by semigroups and for approximation processes, and derive sharp strong and weak endpoint estimates for operators in harmonic analysis. Finally, while Domínguez and Milman obtained weak-type estimates on product spaces for $p>1$ and left the endpoint $p=1$ open (see [Adv.~Math.~411 (2022), Paper~No.~108774, p.~22]), we prove a sharp pointwise limit in the parameter variable and obtain two--sided iterated weak-type estimates at $p=1$.
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