AI 中文总结
研究一维双线性粗糙奇异积分的端点定理，通过刻画相关角乘子结构，在临界核假设下建立两个有界性准则，包括\(L\log L\)结果及临界方向指标下的结果，且两类临界核不可比，方法涉及多种数学工具。
AI 中文摘要
我们证明了一维双线性粗糙奇异积分的端点定理。我们从相关角乘子的精确结构特征出发，对于每个均值为零的\(\Omega\in L^1(\mathbb{S}^1)\)，与\(T_\Omega\)相关的有限部分角乘子具有有界变差，当且仅当\(\Omega\)的对映偶部分属于\(H^1(\mathbb{S}^1)\)。然后在临界核假设下建立了两个有界性准则。一是若\(\Omega\in L\log L(\mathbb{S}^1)\)，则在特定条件下\(T_\Omega\)有界，且对数指数\(1\)在\(L(\log L)^A\)尺度下是最优的。二是在临界方向指标下，对于\(\Omega\in\mathcal{K}_{1/2,\beta}(\mathbb{S}^1)\)且满足一定条件时同样有界。两个临界核类不可比。\(L\log L\)结果通过将乘子简化为有界变差的有限部分角轮廓得到，方向结果由端点傅里叶衰减、乘积小波分解和插值得出。
英文摘要
We prove endpoint theorems for one-dimensional bilinear rough singular integrals. Our starting point is a sharp structural characterization of the associated angular multiplier. For every mean-zero $Ω\in L^1(\mathbb{S}^1)$, the finite-part angular multiplier associated with $T_Ω$ has bounded variation if and only if the antipodal even part of $Ω$ belongs to $H^1(\mathbb{S}^1)$. This characterization identifies the precise rotational regularity required in the one-dimensional bilinear setting. It also yields a Stieltjes decomposition compatible with uniform estimates for the bilinear Hilbert transform. We then establish two boundedness criteria under critical kernel assumptions. First, if $Ω\in L\log L(\mathbb{S}^1)$, then $T_Ω$ is bounded from $ L^{p_1}(\mathbb{R})\times L^{p_2}(\mathbb{R})\text{to} L^p(\mathbb{R})$ whenever $1<p_1,p_2,p<\infty$ and $ \frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}.$ Moreover, the logarithmic exponent $1$ is optimal within the scale $L(\log L)^A$. Second, at the critical directional index, the same boundedness holds for $Ω\in\mathcal{K}_{1/2,β}(\mathbb{S}^1)$, provided that $β>\frac{3}{2}\max\bigl\{p_1,p_1',p_2,p_2'\bigr\}-1.$The two critical kernel classes are incomparable. The $L\log L$ result is obtained by reducing the multiplier to a finite-part angular profile of bounded variation. The directional result follows from endpoint Fourier decay, product wavelet decompositions, and interpolation.
Comments40 pages
