双参数Littlewood-Paley算子在乘积Hardy空间上的有界性
经典分析与常微分方程
2016-05-03 v1
摘要
令 n 1 , n 2 ≥ 1 , λ 1 > 1 n_1,n_2\ge 1, \lambda_1>1 n 1 , n 2 ≥ 1 , λ 1 > 1 λ 2 > 1 \lambda_2>1 λ 2 > 1 x = ( x 1 , x 2 ) ∈ R n × R m x=(x_1,x_2) \in \mathbb {R}^n\times\mathbb{R}^m x = ( x 1 , x 2 ) ∈ R n × R m g g g g λ ⃗ ∗ g_{\vec{\lambda}}^* g λ ∗ g λ ⃗ ∗ ( f ) ( x ) = ( ∬ R + m + 1 ∬ R + n + 1 ∏ i = 1 2 ( t 1 t i + ∣ x i − y i ∣ ) n i λ i ∣ θ t 1 , t 2 f ( y 1 , y 2 ) ∣ 2 d y 1 d t 1 t 1 n + 1 d y 2 d t 2 t 2 m + 1 ) 1 / 2 , g_{\vec{\lambda}}^*(f)(x) = \Big(\iint_{\mathbb{R}^{m+1}_{+}} \iint_{\mathbb{R}^{n+1}_{+}} \prod_{i=1}^2\Big(\frac{t_1}{t_i + |x_i - y_i|}\Big)^{n_i \lambda_i} |\theta_{t_1,t_2} f(y_1,y_2)|^2 \frac{dy_1 dt_1}{t_1^{n+1}} \frac{dy_2 dt_2}{t_2^{m+1}} \Big)^{1/2}, g λ ∗ ( f ) ( x ) = ( ∬ R + m + 1 ∬ R + n + 1 i = 1 ∏ 2 ( t i + ∣ x i − y i ∣ t 1 ) n i λ i ∣ θ t 1 , t 2 f ( y 1 , y 2 ) ∣ 2 t 1 n + 1 d y 1 d t 1 t 2 m + 1 d y 2 d t 2 ) 1/2 , θ t 1 , t 2 f ( x 1 , x 2 ) = ∬ R n × R m s t 1 , t 2 ( x 1 , x 2 , y 1 , y 2 ) f ( y 1 , y 2 ) d y 1 d y 2 \theta_{t_1,t_2} f(x_1, x_2) = \iint_{\mathbb{R}^n\times\mathbb{R}^m} s_{t_1,t_2}(x_1,x_2,y_1,y_2)f(y_1,y_2) dy_1dy_2 θ t 1 , t 2 f ( x 1 , x 2 ) = ∬ R n × R m s t 1 , t 2 ( x 1 , x 2 , y 1 , y 2 ) f ( y 1 , y 2 ) d y 1 d y 2 g g g g λ ⃗ ∗ g_{\vec{\lambda}}^* g λ ∗ L 2 L^2 L 2 s t 1 , t 2 s_{t_1,t_2} s t 1 , t 2 g g g g λ ⃗ ∗ g_{\vec{\lambda}}^* g λ ∗ H 1 ( R n × R m ) H^1(\mathbb{R}^n\times\mathbb{R}^m) H 1 ( R n × R m ) L 1 ( R n × R m ) L^1(\mathbb{R}^n\times\mathbb{R}^m) L 1 ( R n × R m ) g g g g λ ⃗ ∗ g_{\vec{\lambda}}^* g λ ∗ 1 < p < 2 1<p<2 1 < p < 2 L p L^p L p
引用
@article{arxiv.1605.00476,
title = {Boundedness of Bi-parameter Littlewood-Paley operators on product Hardy space},
author = {Zhengyang Li and Qingying Xue},
journal= {arXiv preprint arXiv:1605.00476},
year = {2016}
}
备注
26 pages