The weak H p spaces on homogeneous groups
The weak H p spaces on homogeneous groups
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DOI：
10.1007/bfb0087762
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发表时间：
1991
期刊：
影响因子：
--
通讯作者：
Heping Liu
中科院分区：
文献类型：
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作者：
Heping Liu
R“上的弱HI空间是由Fefferman R.和Soria F.正如他们在文章中指出的那样，对弱HI空间的兴趣来自于对调和分析中一些基本算子的尖锐性质的仔细考虑。类似的考虑将把我们引向弱惠普空间。例如，如果T是S算子，0<；{j<；1，T*(i}=0)，则T是HP上的有界算子，只要{j>；-1)(见[1])。人们很自然地会问：如果{j=-1)7我们将证明T是从HP到弱HP的有界算子(见定理1)，那么它作为T作用在HP上有什么性质。在这种情况下，T是否是弱HPI上的有界算子仍然是一个问题。然而，在更强的条件下，T确实是弱HP上的有界算子(见定理3)。基于类似于HP的原子分解的分解定理，研究弱HPI上的奇异积分的性质。我们将在齐次群G上发展弱HP理论，G是比R“更一般的设置：例如，Heisenberg群和上三角群都是齐次群。需要指出的是，即使在G=R的情况下，本文的所有结果也是新的。我们请读者参考Folland GB和Stein EM写的书[4]。在这本书中，HP空间的理论得到了完美的发展。我们勾画了一些与我们工作有关的事实。设G是n维齐次群，G的齐次维记为Q。我们假设G有一个固定的齐次范数1·1。存在一个常数“‘I1，使得
The Weak HI space on R" has been studied by Fefferman R. and Soria F.[31. As pointed in their paper, the interest on the Weak HI space comes from the careful consideration for the sharp properties of some basic operators in harmonic analysis. Similar consideration will lead us to the Weak HP spaces. For example, if T is a s: Oaderon-Zygmund operator, 0<{j< 1, T*(I}= 0, then T is a bounded operator on HP provided that {j>-1)(see [1)). It is natural to ask: What properties does it have as T acting on HP if {j=-1) 7 We will prove that T is a bounded operator from HP to Weak HP (see Theorem 1). In this case, whether T is a bounded operator on Weak HPis still a problem. However, under a bit stronger conditions on the kernel T does be a bounded operator on Weak HP (see Theorem 3). The study for the properties of singular integrals on Weak HPis based on a decomposition theorem which is similar to the atomic decomposition of HP. We will develop the theory of Weak HP on the homogeneous group G which is a more general setting than R": For example, both Heisenberg groups and upper triangular groups are homogeneous groups. It should be pointed out that all results in this paper are new even in the case of G= R", § 1 Some preliminaries For the basic facts on homogeneous groups we refer the reader to the book [4] written by Folland GB and Stein EM in which the theory of HP spaces was developed perfectly. We sketch some facts related to our work. Let G be a n-dimensional homogeneous group. The homogeneous dimension of G is denoted by Q. We assume that G is equipped with a fixed homogeneous norm 1· 1. There is a constant"'I 1 such that