Collectanea Mathematica ( IF 0.7 ) Pub Date : 2021-01-01 , DOI: 10.1007/s13348-020-00307-0 Zhidan Wang , Qingying Xue , Xinchen Duan
Let \(x=(x_1,x_2)\) with \(x_1,x_2 \in \mathbb {R}^n\) and let \(K(x)={\Omega \big ({x}/{|x|}\big )}{\big |x\big |^{-2n}}\), where \(\Omega \in L^{\infty }(\mathbb {S}^{2n-1})\) and satisfies \(\int _{\mathbb {S}^{2n-1}}\Omega =0\). We show that the maximal truncated bilinear singular integrals with rough kernel \(K(x_1,x_2)\) satisfy a sparse bound by (p, p, p)-averages for all \(p>1\). As consequences, we obtain some quantitative weighted estimates for these rough singular integrals. A pointwise sparse domination for commutators of bilinear rough singular integrals were also established, which can be used to establish some weighted inqualities.
中文翻译:
稀疏支配的最大双线性粗糙奇异积分的加权估计
令\(x =(x_1,x_2)\)与\(x_1,x_2 \ in \ mathbb {R} ^ n \)并令\(K(x)= {\ Omega \ big({x} / {| x |} \ big}} {\ big | x \ big | ^ {-2n}} \),其中\(\ Omega \ in L ^ {\ infty}(\ mathbb {S} ^ {2n-1}) \)并满足\(\ int _ {\ mathbb {S} ^ {2n-1}} \ Omega = 0 \)。我们表明,对于所有\(p> 1 \),具有粗糙核\(K(x_1,x_2)\)的最大截断双线性奇异积分满足由(p, p, p)-均值约束的稀疏边界。结果,我们获得了这些粗糙奇异积分的一些定量加权估计。还建立了双线性粗糙奇异积分的交换子的逐点稀疏控制,可用于建立一些加权不等式。