Research on singular integral operators and function spaces

奇异积分算子与函数空间的研究

• 批准号: 09640224
• 负责人: MIYACHI Akihiko
• 金额: $ 1.92万
• 依托单位: Tokyo Woman's Christian University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640224/
• 关键词: Hardy space; Singular integral; Multilinear operator; Weighted Hardy space; Fractional integration; Sharp maximal function; Triebel-Lizorkin space; Hardy's theorem; ハ-ディ空間; 特異積分作用素; ソボレフ空間; フーリエ変換; ハ-ディの定理

1) We obtained H^p estimate for the multilinear operators which are finite sums of pointwise products of singular integrals. The class of singular integrals considered include the Calderon-Zygmund singular integrals and fractional integrations of arbitrary orders. We also obtained the estimates of those multilinear operators in other Hardy type spaces.2) Using a sharp maximal function and the Lorentz spaces, we defined a function space which forms an algebra with respect to pointwise product of functions. This function space can be considered as a variant of the Sobolev space with the critical index.3) Using purely real variable method, we showed that one can define the weighted Hardy spaces with respect to doubling measures on arbitrary open subset of R^n. This result extends those of Stromberg and Torchinsky which were given in the Springer Lecture Note #1381, 1989.4) We proved two results conserning the extension of functions in A.Seeger's generalized Triebel-Lizorkin spaces.5) We obtained a generalization of Hardy's clasical theorem on the Fourier transform.

1)本文给出了奇异积分点态积的有限和的多线性算子的H^p估计。所考虑的奇异积分类包括Calderon-Zygmund奇异积分和任意阶的分数次积分。2）利用一个尖锐的极大函数和Lorentz空间，定义了一个函数空间，该函数空间关于函数的点态积构成一个代数. 3）利用纯真实的变量方法，证明了在R^n的任意开子集上可以定义关于加倍测度的加权哈代空间. 4）证明了A.Seeger广义Triebel-Lizorkin空间中函数扩张的两个结果。5）推广了哈代关于Fourier变换的经典定理。

项目成果

Akihiko Miyachi: "Atomic decomposition for Sobolev spaces and for the ^α_p spaces on general domains" Tsukuba Journal of Mathematics. Vol.21,No.1. 59-96 (1997)

Akihiko Miyachi：“Sobolev 空间和一般域上的 ^α_p 空间的原子分解”《筑波数学杂志》第 21 卷，第 59-96 期（1997 年）。

Akihiko Miyachi: "Atomic decomposition for Sobolev spaces and for the C^<alpha>_ spaces on general domains" Tsukuba J.Math.Vol.21. 59-96 (1997)

Akihiko Miyachi：“Sobolev 空间和一般域上的 C^<alpha>_ 空间的原子分解”Tsukuba J.Math.Vol.21。

Akihiko Miyachi: "A generalization of a theorem og Hardy" 調和解析セミナー(研究集会報告集). 44-51 (1997)

Akihiko Miyachi：“Hardy 定理的推广”调和分析研讨会（研究会议报告集）44-51（1997）。

Akihiko Miyachi: "A generalization of a theorem of Hardy" Proceedings of the Harmonic Analysis Seminar. 13. 44-51 (1997)

Akihiko Miyachi：“Hardy 定理的推广”调和分析研讨会论文集。

Akihiko Miyachi: "On the estimate for product of functions (in Japanese)" RIMS Kokyuroku, No.1059, Research Institute for Mathematical Sciences, Kyoto University. 40-50 (1998)

Akihiko Miyachi：“关于函数乘积的估计（日语）”RIMS Kokyuroku，No.1059，京都大学数学科学研究所。