Fourier Transform of Anisotropic Hardy Spaces Associated with Ball Quasi-Banach Function Spaces and Its Applications to Hardy-Littlewood Inequalities

IF 0.9 4区 数学 Q3 MATHEMATICS, APPLIED
Chao-an Li, Xian-jie Yan, Da-chun Yang
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引用次数: 0

Abstract

Let A be a general expansive matrix and X be a ball quasi-Banach function space on ℝn, whose certain power (namely its convexification) supports a Fefferman-Stein vector-valued maximal inequality and the associate space of whose other power supports the boundedness of the powered Hardy-Littlewood maximal operator. Let H AX (ℝn) be the anisotropic Hardy space associated with A and X. The authors first prove that the Fourier transform of fH AX (ℝn) coincides with a continuous function F on ℝn in the sense of tempered distributions. Moreover, the authors obtain a pointwise inequality that the function F is less than the product of the anisotropic Hardy space norm of f and a step function with respect to the transpose matrix of the expansive matrix A. Applying this, the authors further induce a higher order convergence for the function F at the origin and give a variant of the Hardy-Littlewood inequality in H AX (ℝn). All these results have a wide range of applications. Particularly, the authors apply these results, respectively, to classical (variable and mixed-norm) Lebesgue spaces, Lorentz spaces, Orlicz spaces, Orlicz-slice spaces, and local generalized Herz spaces and, even on the last four function spaces, the obtained results are completely new.

与球准巴纳赫函数空间相关的各向异性哈代空间的傅立叶变换及其在哈代-利特尔伍德不等式中的应用
设 A 是一般扩张矩阵,X 是ℝn 上的球状准巴纳赫函数空间,其某个幂(即其凸化)支持费弗曼-斯坦向量值最大不等式,其另一个幂的关联空间支持动力哈代-利特尔伍德最大算子的有界性。作者首先证明了 f∈ HAX(ℝn) 的傅里叶变换与ℝn 上的连续函数 F 重合。此外,作者还得到了一个点式不等式,即函数 F 小于 f 的各向异性哈代空间规范与关于扩张矩阵 A 的转置矩阵的阶跃函数的乘积。应用这一点,作者进一步诱导了函数 F 在原点的高阶收敛,并给出了 HAX(ℝn) 中 Hardy-Littlewood 不等式的变体。所有这些结果都有广泛的应用前景。特别是,作者将这些结果分别应用于经典(可变和混合规范)Lebesgue 空间、洛伦兹空间、Orlicz 空间、Orlicz-slice 空间和局部广义 Herz 空间,甚至在后四个函数空间上,所获得的结果也是全新的。
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来源期刊
CiteScore
1.30
自引率
0.00%
发文量
70
审稿时长
3.0 months
期刊介绍: Acta Mathematicae Applicatae Sinica (English Series) is a quarterly journal established by the Chinese Mathematical Society. The journal publishes high quality research papers from all branches of applied mathematics, and particularly welcomes those from partial differential equations, computational mathematics, applied probability, mathematical finance, statistics, dynamical systems, optimization and management science.
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