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）我们获得了多线性算子的H^p估计，这些算子是单数积分的点式产物的有限总和。所考虑的奇异积分类别包括Calderon-Zygmund奇异积分和任意顺序的分数集成。我们还获得了其他耐铁型空间中那些多线性操作员的估计值。2）使用尖锐的最大函数和洛伦兹空间，我们定义了一个功能空间，该功能空间相对于函数的点乘积形成代数。可以将此功能空间视为具有临界索引的Sobolev空间的变体。3）使用纯粹的真实变量方法，我们表明，可以定义加权耐药空间，以在r^n的任意开放子集上进行加倍措施。这一结果扩展了Stromberg和Torchinsky的结果，这些结果在Springer讲座注释＃1381，1989.4）证明了两个结果，这些结果巩固了A.Seeger广义的Triebel-Lizorkin Spaces中的功能扩展。5）我们获得了Hardy Clasical Throrem on the Fourier Transform的概括。

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: "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: "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: "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。

Kazuaki Kobayashi: "On links from the “splittness" view point" Science Reports of Tokyo Woman's Christian University. 49-3. 1513-1522 (1998)

Kazuaki Kobayashi：“从“分裂”角度来看链接”，东京女子基督教大学科学报告 49-3 1513-1522（1998）。