arXiv 1509.075482015-09-24math.CA

积空间上具有非光滑核的奇异积分的端点估计

End-point estimates for singular integrals with non-smooth kernels on product spaces

Xuan Thinh Duong, Ji Li, Lixin Yan

备注：: arXiv admin note: text overlap with arXiv:1002.0792, arXiv:0807.4348 by other authors

摘要：

本文的主要目的是建立具有非光滑核的奇异积分在积空间上的有界性。设$L_1$和$L_2$分别是$L^2(\mathbb{R}^{n_1})$和$L^2(\mathbb{R}^{n_2})$上的非负自伴算子，其热核满足高斯上界.首先，我们得到了$H^1_{L_1,L_2}(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2})$中函数的原子分解，其中与$L_1$和$L_2$相关的Hardy空间$H^1_{L_1,L_2}(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2})$由平方函数范数定义，然后证明了该空间的一个插值性质。其次，当某些奇异积分的相关核只满足比标准Calderon-Zygmund核弱得多的正则性条件时，我们建立了奇异积分算子在Hardy空间$H^1_{L_1,L_2}(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2})$上有界的充分条件.作为应用，我们得到了与Schrdingier算子相关的双Riesz变换的端点估计和积空间上非负自伴算子的Marcinkiewicz型谱乘子定理.

英文摘要：

The main aim of this article is to establish boundedness of singular integrals with non-smooth kernels on product spaces. Let $L_1$ and $L_2$ be non-negative self-adjoint operators on $L^2(\mathbb{R}^{n_1})$ and $L^2(\mathbb{R}^{n_2})$, respectively, whose heat kernels satisfy Gaussian upper bounds. First, we obtain an atomic decomposition for functions in $H^1_{L_1,L_2}(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2})$ where the Hardy space $H^1_{L_1,L_2}(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2})$ associated with $L_1$ and $L_2$ is defined by square function norms, then prove an interpolation property for this space. Next, we establish sufficient conditions for certain singular integral operators to be bounded on the Hardy space $H^1_{L_1,L_2}(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2})$ when the associated kernels of these singular integrals only satisfy regularity conditions significantly weaker than those of the standard Calderón--Zygmund kernels. As applications, we obtain endpoint estimates of the double Riesz transforms associated to Schrdingier operators and a Marcinkiewicz-type spectral multiplier theorem for non-negative self-adjoint operators on product spaces.

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摘要：

英文摘要：

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