On The Boundedness of Bi-parameter Littlewood-Paley g_λ^*-function

2015/12/02 by Mingming Cao, Qingying Xue, Cao, Mingming +1
Mathematics · #42B25 #47G10 #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Mathematical functions and polynomials #math.AP #math.CA #msc:42B25 #msc:47G10

paper · pdf · doi:10.48550/arxiv.1512.00569

21 pages

openalex publication_date 2015/12/02 · arxiv created 2015/12/04 · arxiv updated 2015/12/07 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Let m,n≥ 1 and g_λ1,λ2^* be the bi-parameter Littlewood-Paley g_λ^*-function defined by g_λ1,λ2^*(f)(x)= (\iint_\R^m+1₊ ((t₂)/(t₂ + |x₂ - y₂|))^{m λ₂} \iint_\Rⁿ⁺¹₊ ((t₁)/(t₁ + |x₁ - y₁|))^{n λ₁}|θ_t1,t2 f(y₁,y₂)|² \fracdy₁ dt₁t₁ⁿ⁺¹ \fracdy₂ dt₂t₂^m+1 )^1/2, λ₁>1, λ₂>1 where θ_t1,t2 f is a non-convolution kernel defined on ℝ^m+n. In this paper, we showed that the bi-parameter Littlewood-Paley function g_λ1,λ2^* was bounded from L²(\R^n+m) to L²(\R^n+m). This was done by means of probabilistic methods and by using a new averaging identity over good double Whitney regions.

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