双线性粗糙交换子的稀疏界与联合振荡
Sparse bounds and joint oscillation for bilinear rough commutators
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中文总结 AI 辅助
本文研究双线性粗糙奇异积分交换子的联合振荡条件，证明二阶混合与重复位置交换子的下界，并建立稀疏控制，得到Banach与拟Banach范围内的强、弱界。
中文摘要 AI 辅助
我们研究具有有界零均值角核的双线性粗糙奇异积分交换子的联合振荡条件。对于具有两个可能不同的复值符号的二阶交换子，我们在几何非退化假设下证明了其下界。这些下界区分了交换位置：混合交换子控制各二次振荡的乘积，而部分转置的重复位置交换子额外控制中心符号乘积的二次平均值。我们还建立了迭代交换子的稀疏控制，将符号振荡保留在局部平均值内。对于每个固定的交换子阶数，当平均指数s递减至1时，稀疏常数为O(s/(s-1))。由此得到的联合振荡条件在Banach范围内产生强界，在拟Banach范围内产生弱界。在自然指数三元组处，必要条件和充分条件在振荡指数上相差任意小的增量。
英文摘要
We study joint oscillation conditions for commutators of bilinear rough singular integrals with bounded mean-zero angular kernels. For second-order commutators with two possibly distinct complex-valued symbols, we prove lower bounds under a geometric non-degeneracy assumption. These bounds distinguish the commutation positions: mixed commutators control the product of the separate quadratic oscillations, while repeated-position commutators of the partial transposes additionally control the quadratic average of the product of the centered symbols. We also establish sparse domination for iterated commutators, retaining the symbol oscillations inside the local averages. For each fixed commutator order, the sparse constant is $O(s/(s-1))$ as the averaging exponent $s$ decreases to $1$. The resulting joint oscillation conditions yield strong bounds in the Banach range and weak bounds in the quasi-Banach range. At the natural exponent triples, the necessary and sufficient conditions differ by an arbitrarily small increase in the oscillation exponents.
发表机构
- Center for Applied Mathematics, Tianjin University（天津大学应用数学中心）
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