曲线上的双线性 Hilbert-Carleson 算子：纯非零曲率情形
经典分析与常微分方程
2025-07-08 v1 动力系统
数论
摘要
本文在（纯）非零曲率情形下，给出双线性 Hilbert-Carleson 沿曲线算子的最大有界性范围（接近端点）。具体而言，我们证明算子 B H C [ a ⃗ , α ⃗ ] ( f 1 , f 2 ) ( x ) : = sup λ ∈ R ∣ p . v . ∫ R f 1 ( x − a 1 t α 1 ) f 2 ( x − a 2 t α 2 ) e i λ a 3 t α 3 d t t ∣ BHC_{[\vec{a},\vec{\alpha}]}(f_1,f_2)(x) := \sup_{\lambda\in\mathbb{R}} \left|\,p.v.\, \int_{\mathbb{R}} f_1(x - a_1 t^{\alpha_1}) \,f_2(x - a_2 t^{\alpha_2}) \,e^{i\,\lambda\,a_3 \,t^{\alpha_3}} \,\frac{dt}{t}\right| B H C [ a , α ] ( f 1 , f 2 ) ( x ) := λ ∈ R sup p . v . ∫ R f 1 ( x − a 1 t α 1 ) f 2 ( x − a 2 t α 2 ) e i λ a 3 t α 3 t d t ∥ B H C [ a ⃗ , α ⃗ ] ( f 1 , f 2 ) ∥ L r ≲ a ⃗ α ⃗ , r , p 1 , p 2 ∥ f 1 ∥ L p 1 ∥ f 2 ∥ L p 2 \|BHC_{[\vec{a},\vec{\alpha}]} (f_1,f_2)\|_{L^r} \lesssim_{\vec{a} \,\vec{\alpha},r,p_1,p_2} \|f_1\|_{L^{p_1}}\,\|f_2\|_{L^{p_2}} ∥ B H C [ a , α ] ( f 1 , f 2 ) ∥ L r ≲ a α , r , p 1 , p 2 ∥ f 1 ∥ L p 1 ∥ f 2 ∥ L p 2 a ⃗ = ( a 1 , a 2 , a 3 ) , α ⃗ = ( α 1 , α 2 , α 3 ) ∈ ( R ∖ { 0 } ) 3 \vec{a}=(a_1,a_2,a_3),\,\vec{\alpha}=(\alpha_1,\alpha_2,\alpha_3)\in (\mathbb{R}\setminus\{0\})^3 a = ( a 1 , a 2 , a 3 ) , α = ( α 1 , α 2 , α 3 ) ∈ ( R ∖ { 0 } ) 3 α ⃗ \vec{\alpha} α 1 p 1 + 1 p 2 = 1 r \frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{r} p 1 1 + p 2 1 = r 1 1 < p 1 , p 2 < ∞ 1<p_1,p_2<\infty 1 < p 1 , p 2 < ∞ 1 2 < r < ∞ \frac{1}{2}<r<\infty 2 1 < r < ∞
引用
@article{arxiv.2507.04467,
title = {The Bilinear Hilbert-Carleson operator along curves. The purely non-zero curvature case},
author = {Árpád Bényi and Bingyang Hu and Victor Lie},
journal= {arXiv preprint arXiv:2507.04467},
year = {2025}
}
备注
59 pages, no figures