Boundedness of bi-parameter Littlewood-Paley operators on product Hardy space
Boundedness of bi-parameter Littlewood-Paley operators on product Hardy space
复制标题
产品 Hardy 空间上双参数 Littlewood-Paley 算子的有界性
DOI：
10.1007/s13163-018-0259-4
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发表时间：
2018
影响因子：
0.8
通讯作者：
Xue Qingying
中科院分区：
文献类型：
--
作者：
Li Zhengyang;Xue Qingying
设n1 = n ≥ 1，n2 = m ≥ 1，n1 = n≥ 1，n2 = m≥ 1且λ 2> 1.对任意x=（x_1，x_2）∈ R^ n * R^ m x=（x1，x2）∈ Rn × Rm，设g和g_ λ^* g_λ是由g（f）（x）&=\left（f_0 ^ ∞ f_0 ^ ∞）定义的双参数Littlewood-Paley平方函数|θ _ t_1，t_2 f（x_1，x_2）|^ 2 dt_1 t_1 dt_2 t_2\right）^ 1/2，并且\g_ λ^*（f）（x）&=\left（\iint _ R^ m+ 1 _+\iint _ R^ n+ 1 _+\iint_ i= 1^ 2\Big（t_1 t_i+| x_i-y_i|\大）^ n_i λ _i\对。\&\左。四边形 *\，|θ _ t_1，t_2 f（y_1，y_2）|^ 2 dy_1 dt_1 t_1^ n+ 1 dy_2 dt_2 t_2^ m+ 1\right）^ 1/2，g（f）（x）= 10 ∞ 10 ∞| θ t1，t2 f（x 1，x 2）|2 dt 1 t1 dt 2 t2 1/2，g λ <$（f）（x）=<$R+ m+ 1 <$R+ n+ 1 <$i= 1 2（t1 ti+|西邑|）ni λ i×| θ t1，t2 f（y1，y2）|2dy 1dt 1t1n +1dy 2dt 2t2m + 1 1/2，其中θ t_1，t_2 f（x_1，x_2）=| iint _ R^ n * R^ m s_ t_1，t_2（x_1，x_2，y_1，y_2）f（y_1，y_2）dy_1dy_2 θ t1，t2 f（x1，x2）=<$Rn × Rmst1，t2（x1，x2，y1，y2）f（y1，y2）dy1dy2.最近Martikainen和Cao，Xue分别建立了双参数g和g_ λ^* g λ_n的L^2L2有界性.本文在核st 1，t2，st 1，t2的一定结构条件下，证明了从乘积哈代空间H^1（R^ n * R^ m）H1（Rn × Rm）到L^1（R ^n * R ^m）L1（Rn × Rm），g和g_ λ^* g λ都有界.作为推论，当1< p< 2 1< p< 2时，得到g和g_ λ^* g λ n的L^ p L p有界性.
Abstract Let n_1= n ≥ 1, n_2= m ≥ 1 n 1= n≥ 1, n 2= m≥ 1 and λ _2> 1 λ 2> 1. For any x=(x_1, x_2) ∈ R^ n * R^ m x=(x 1, x 2)∈ R n× R m, let g and g_ λ^* g λ∗ be the bi-parameter Littlewood–Paley square functions defined by g (f)(x) &=\left (∫ _0^ ∞ ∫ _0^ ∞| θ _ t_1, t_2 f (x_1, x_2)|^ 2 dt_1 t_1 dt_2 t_2\right)^ 1/2, and\g_ λ^*(f)(x) &=\left (\iint _ R^ m+ 1 _+\iint _ R^ n+ 1 _+ ∏ _ i= 1^ 2\Big (t_1 t_i+| x_i-y_i|\Big)^ n_i λ _i\right.\&\left.\quad *\,| θ _ t_1, t_2 f (y_1, y_2)|^ 2 dy_1 dt_1 t_1^ n+ 1 dy_2 dt_2 t_2^ m+ 1\right)^ 1/2, g (f)(x)=∫ 0∞∫ 0∞| θ t 1, t 2 f (x 1, x 2)| 2 dt 1 t 1 dt 2 t 2 1/2, and g λ∗(f)(x)=∬ R+ m+ 1∬ R+ n+ 1∏ i= 1 2 (t 1 ti+| xi-yi|) ni λ i×| θ t 1, t 2 f (y 1, y 2)| 2 dy 1 dt 1 t 1 n+ 1 dy 2 dt 2 t 2 m+ 1 1/2, where θ _ t_1, t_2 f (x_1, x_2)=\iint _ R^ n * 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 mst 1, t 2 (x 1, x 2, y 1, y 2) f (y 1, y 2) dy 1 dy 2. It is known that the L^ 2 L 2 boundedness of bi-parameter g and g_ λ^* g λ∗ have been established recently by Martikainen, and Cao, Xue, respectively. In this paper, under certain structural conditions assumed on the kernel s_ t_1, t_2, st 1, t 2, we show that both g and g_ λ^* g λ∗ are bounded from product Hardy space H^ 1 (R^ n * R^ m) H 1 (R n× R m) to L^ 1 (R^ n * R^ m) L 1 (R n× R m). As consequences, the L^ p L p boundedness of g and g_ λ^* g λ∗ will be obtained for 1< p< 2 1< p< 2.