标题： Littlewood-Paley平方函数和多范数结构上的局部Hardy空间对于$\mathbb{R}^{d}$ 显示英文标题

标题： Littlewood-Paley square functions and the local Hardy space for Multi-Norm Structures on $\mathbb{R}^{d}$

摘要： 多范数奇异积分和傅里叶乘子在[29]中被引入，这些概念的一个应用是对适应于$n$不同缩放族的Calderón-Zygmund核的卷积算子的复合的精确描述。 该结果算子的描述是通过由矩阵$\mathbf E$指定的微分不等式，以及通过核和乘子的二进分解来给出的。 在本文中，我们通过研究频率空间上的诱导Littlewood-Paley分解和各种相关平方函数，扩展了对$\mathbb{R}^d$上多范数结构的分析。 在建立了它们的$L^1$-等价性之后，我们利用这些平方函数定义了一个局部多范数Hardy空间$\mathbf{h}^{1}_{\mathbf{E}}(\mathbb{R}^d)$。 我们给出了这个空间的几种等价描述，包括一个原子刻画。 其他作者最近有工作，但仅限于$2$-缩放情况。 本文处理的一般$n$-缩放情况要困难得多，需要新的想法和更系统的方法。 显示英文摘要

摘要： Multi-norm singular integrals and Fourier multipliers were introduced in [29], and one application of these notions was a precise description of the composition of convolution operators with Calder\'on-Zygmund kernels adapted to $n$ different families of dilations. The description of the resulting operators was given in terms of differential inequalities specified by a matrix $\mathbf E$, and in terms of dyadic decompositions of the kernels and multipliers. In this paper we extend the analysis of multi-norm structures on $\mathbb{R}^d$ by studying the induced Littlewood-Paley decomposition of the frequency space and various associated square functions. After establishing their $L^1$-equivalence, we use these square functions to define a local multi-norm Hardy space $\mathbf{h}^{1}_{\mathbf{E}}(\mathbb{R}^d)$. We give several equivalent descriptions of this space, including an atomic characterization. There has been recent work, limited to the $2$-dilation case, by other authors. The general $n$-dilation case treated here is considerably harder and requires new ideas and a more systematic approach.