Characterizations of real Hardy spaces and functional analysis

真实 Hardy 空间的表征和泛函分析

• 批准号: 13640148
• 负责人: KANEKO Makoto
• 金额: $ 2.18万
• 依托单位: TOHOKU University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640148/
• 关键词: Hardy space; vertical maximal function; non-tangential maximal function; grand maximal function; Orlicz-norm; Cauchy-Riemann equation; Klein-Gordon equation; Poisson integral; ハーディー空関; オルリッツ空間; コーシー・リーマンの方程式; クレイン・ゴードン方程式; 渦度方程式; 直行最大関数; 振動レ

We have been interesting in the methods to judge whether a given tempered distribution f on n-dimensional Euclidean space is in a Hardy space or not. One of the methods is to investigate the integrability of one of the maximal functions made from f. There are many kinds of maximal functions which might be the tools for the purpose. Among them, we have picked up the following maximal functions. We take a test function on the same space where f is given and consider the dilations of it with dilation rates t. Then we have a family of the functions which are the convolutions of f and the dilations of the test function with the dilation rates t. Then the components of the family have parameters t. The vertical maximal function of f with respect to the given test function is defined by the supremum of the family taken over all t > 0. The non-tangential maximal function is such a function whose value at a point x is the supremum of the values of the convolution over such t and y that the dist … More ance from x to y is in less than t. The modified maximal function is the improvement of the non-tangential maximal function to reflect the behavior of the convolution in the long distant area from x. The above maximal functions are determined by the given test function. Now we consider the all test functions satisfying certain conditions. For each test function, we can get a corresponding vertical maximal function. Among them we take the largest one and call it the grand maximal function of f. We have also treated the vertical maximal function and the non-tangential maximal function made from the Poisson integral of f.In our research, we have investigated the integral estimates of such functions that are obtained by putting a function of lower p type over the above maximal functions. These integrals contain the Orlicz-norms and the p-th integral means.We have proved the equivalence between the finiteness of these integrals and given some improvements of proofs appearing in the papers treating the related topics. Less

We have been interesting in the methods to judge whether a given tempered distribution f on n-dimensional Euclidean space is in a Hardy space or not. One of the methods is to investigate the integrability of one of the maximal functions made from f. There are many kinds of maximal functions which might be the tools for the purpose. Among them, we have picked up the following maximal functions. We take a test function on the same space where f is given and consider the dilations of it with dilation rates t. Then we have a family of the functions which are the convolutions of f and the dilations of the test function with the dilation rates t. Then the components of the family have parameters t. The vertical maximal function of f with respect to the given test function is defined by the supremum of the family taken over all t > 0. The non-tangential maximal function is such a function whose value at a point x is the supremum of the values of the convolution over such t and y that the dist … More ance from x to y is in less than t. The modified maximal function is the improvement of the non-tangential maximal function to reflect the behavior of the convolution in the long distant area from x. The above maximal functions are determined by the given test function. Now we consider the all test functions satisfying certain conditions. For each test function, we can get a corresponding vertical maximal function. Among them we take the largest one and call it the grand maximal function of f. We have also treated the vertical maximal function and the non-tangential maximal function made from the Poisson integral of f.In our research, we have investigated the integral estimates of such functions that are obtained by putting a function of lower p type over the above maximal functions. These integrals contain the Orlicz-norms and the p-th integral means.We have proved the equivalence between the finiteness of these integrals and given some improvements of proofs appearing in the papers treating the related topics. Less

项目成果

田谷久雄, 福田隆: "岩澤不変量の計算"日本応用数理学会論文誌. 12 4. 293-306 (2002)

Hisao Taya，Takashi Fukuda：“岩泽不变量的计算”日本应用数学学会杂志 12 4. 293-306 (2002)。

F.Hiai: "Concavity of certain matrix trace functions"Taiwan J. Math.. 5・3. 535-554 (2001)

F.Hiai：“某些矩阵迹函数的凹性”台湾 J. Math.. 5・3（2001）

F.Hiai, M.Miguo, D.Petg: "Free relative entropy for measures and a corresponding perturbation theory"J.Math.Soc.Japan. 54・3. 679-718 (2002)

F.Hiai、M.Miguo、D.Petg：“测度的自由相对熵和相应的微扰理论”J.Math.Soc.Japan 54・3（2002）。

M.Nakamura, T.Ogawa: "Small solutions to nonlinear wave equations in the sobolev spaces"Houston J. Math.. 27・3. 613-632 (2001)

M.Nakamura、T.Okawa：“索博列夫空间中非线性波动方程的小解”Houston J. Math.. 27・3（2001）。

田谷 久雄, 福田 隆: "岩澤不変量の計算"日本応用数理学会論文誌. 12・4. 293-306 (2002)

Hisao Taya，Takashi Fukuda：“岩泽不变量的计算”日本应用数学学会杂志12・4（2002）。