Boundedness of Bi-parameter Littlewood-Paley operators on product Hardy space

2016/05/02 by Zhengyang Li, Qingying Xue, Li, Zhengyang +1
Mathematics · #42B25 #47G10 #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.1605.00476

openalex publication_date 2016/05/02 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/01

Abstract

Let n₁,n₂≥ 1, λ₁>1 and λ₂>1. For any x=(x₁,x₂) ∈ \mathbb Rⁿ×ℝ^m, let g and g_λ^* be the bi-parameter Littlewood-Paley square functions defined by g(f)(x)= (∫₀^∞∫₀^∞|θ_t1,t2 f(x₁,x₂)|² (dt₁)/(t₁) (dt₂)/(t₂) )^1/2, \hboxand g_λ^*(f)(x) = (\iint_ℝ^m+1₊ \iint_ℝⁿ⁺¹₊ ∏_i=1²((t₁)/(t_i + |x_i - y_i|))^{n_i λ_i} |θ_t1,t2 f(y₁,y₂)|² \fracdy₁ dt₁t₁ⁿ⁺¹ \fracdy₂ dt₂t₂^m+1 )^1/2, \noindent where θ_t1,t2 f(x₁, x₂) = \iint_{ℝⁿ×ℝ^m} s_t1,t2(x₁,x₂,y₁,y₂)f(y₁,y₂) dy₁dy₂. It is known that the L² boundedness of bi-parameter g and g_λ^* have been established recently by Martikainen, and Cao, Xue, respectively. In this paper, under certain structure conditions assumed on the kernel s_t1,t2, we show that both g and g_λ^* are bounded from product Hardy space H¹(ℝⁿ×ℝ^m) to L¹(ℝⁿ×ℝ^m). As consequences, the L^p boundedness of g and g_λ^* will be obtained for 1

Citations

• A T(1) theorem on product spaces

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