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Characterizations of real Hardy spaces and functional analysis

真实 Hardy 空间的表征和泛函分析

基本信息

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

项目摘要

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

关于判定n维欧氏空间上给定的调和分布f是否在Hardy空间中的方法，我们一直很感兴趣。其中一种方法是研究由f构成的极大值函数的可积性。有很多种极大值函数可以作为工具。其中，我们选取了以下极大函数。我们在给定f的同一空间上取一个测试函数，并考虑它的伸缩率为t.然后我们有一族函数族，它们是f的卷积和测试函数随伸缩率t的伸缩性.然后这个族的分量有参数t.f关于给定的测试函数的垂直极大函数由包含所有t&gt；0的族的上确界来定义.非切向极大函数是这样的函数，其在点x的值是t和y上的卷积值的上确界，使得Dist…修正的极大值函数是对非切向极大值函数的改进，以反映从x到y的长距离区域的卷积行为。上述极大值函数由给定的检验函数确定。现在我们考虑所有的测试函数都满足一定的条件。对于每个检验函数，我们可以得到相应的垂直极大值函数。我们还讨论了由f的Poisson积分构成的垂直极大函数和非切向极大函数。在我们的研究中，我们研究了通过在上述极大函数上加一个下p型函数而得到的这类函数的积分估计。这些积分包含Orlicz范数和p次积分平均，我们证明了这些积分的有限性之间的等价性，并对相关文献中的证明进行了改进。较少