2003 Fiscal Year Final Research Report Summary
Characterizations of real Hardy spaces and functional analysis
| Project/Area Number |
13640148
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Basic analysis
|
|---|
| Research Institution | TOHOKU University |
|---|
Principal Investigator |
KANEKO Makoto Tohoku University, Graduate School of Information Sciences, Professor, 大学院・情報科学研究科, 教授 (10007172)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
TAYA Hisao Tohoku University, Graduate School of Information sciences, Assistant, 大学院・情報科学研究科, 助手 (40257241)
HIAI Fumio Tohoku University, Graduate School of Information Sciences, Professor, 大学院・情報科学研究科, 教授 (30092571)
NAKAMURA Makoto Tohoku University, Graduate School of Information Sciences, Assistant, 大学院・情報科学研究科, 助手 (70312634)
OHNO Yoshiki Tohoku University, Graduate School of information Sciences, Associate Professor, 大学院・情報科学研究科, 助教授 (80005777)
ARISAWA Mariko Tohoku University, Graduate School of Information Sciences, Associate Professor, 大学院・情報科学研究科, 助教授 (50312632)
|
|---|
| Project Period (FY) |
2001 – 2003
|
| Keywords | Hardy space / vertical maximal function / non-tangential maximal function / grand maximal function / Orlicz-norm / Cauchy-Riemann equation / Klein-Gordon equation / Poisson integral |
|---|
| Research Abstract |
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
|
