发表机构
School of Mathematical Sciences, Beijing Normal University（北京师范大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明粗糙极大奇异积分交换子在BMO乘子与L(logL)^2核条件下满足弱型端点估计，通过Calderón-Zygmund分解、二进线性化、核分解及解析族插值实现。
AI 中文摘要
设$d\geq 2$，$T^*_{\Omega,b}$为$\mathbb{R}^d$上由$$T^*_{\Omega,b}f(x)=\sup_{\varepsilon>0}\left|\int_{|x-y|>\varepsilon}\bigl(b(x)-b(y)\bigr)\frac{\Omega(x-y)}{|x-y|^d}f(y)\\,dy\right|$$定义的粗糙极大奇异积分交换子。假设$\Omega\in L(\log L)^2(\mathbb{S}^{d-1})$具有零均值，且$b\in\operatorname{BMO}(\mathbb{R}^d)$。我们证明，对每个$\lambda>0$，有$$\bigl|\{x\in\mathbb{R}^d:T^*_{\Omega,b}f(x)>\lambda\}\bigr|\lesssim_{\Omega,b}\int_{\mathbb{R}^d}\frac{|f(x)|}{\lambda}\log\\!\left(e+\frac{|f(x)|}{\lambda}\right)\\,dx.$$论证从$f$的Calderón-Zygmund分解和极大截断的二进线性化开始。按大小对$\Omega$进行分解，随后对空间截断进行正则化并采用微局部分解，将问题归结为具有定量衰减的估计族。所需的衰减通过插值局部$L^1$和$L^3$界，并对交换子使用解析族论证而得到。
英文摘要
Let $d\geq 2$ and $T^*_{Ω,b}$ be the commutator of the rough maximal singular integral on $\mathbb{R}^d$ defined by $$T^*_{Ω,b}f(x)=\sup_{\varepsilon>0}\left|\int_{|x-y|>\varepsilon}\bigl(b(x)-b(y)\bigr)\frac{Ω(x-y)}{|x-y|^d}f(y)\,dy\right|.$$ Assume that $Ω\in L(\log L)^2(\mathbb{S}^{d-1})$ has mean zero and that $b\in\operatorname{BMO}(\mathbb{R}^d)$. We prove that, for every $λ>0$, \[\bigl|\{x\in\mathbb{R}^d:T^*_{Ω,b}f(x)>λ\}\bigr|\lesssim_{Ω,b}\int_{\mathbb{R}^d}\frac{|f(x)|}λ\log\!\left(e+\frac{|f(x)|}λ\right)\,dx.\] The argument starts from a Calderón-Zygmund decomposition of $f$ and a dyadic linearization of the maximal truncation. A decomposition of $Ω$ by size, followed by regularization of the spatial cutoffs and a microlocal decomposition, reduces the problem to a family of estimates with quantitative decay. The required decay follows by interpolating localized $L^1$ and $L^3$ bounds and using an analytic-family argument for the commutators.
Comments38 pages
