A discrete transform and decompositions of distribution spaces
A discrete transform and decompositions of distribution spaces
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DOI：
10.1016/0022-1236(90)90137-a
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发表时间：
1990-10
影响因子：
1.7
通讯作者：
Michael Frazier;B. Jawerth
中科院分区：
文献类型：
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作者：
Michael Frazier;B. Jawerth
我们研究了R n上一个分布f的形式为φ =∑Q< φ, φ Q> ψ Q的表示公式。这个公式是通过对标准Littlewood-Paley分解的离散化和局部化得到的。取f到序列{< f, φ Q>} Q的映射，其中Q遍历R n中的二进立方体，称为ϑ-transform。函数φ Q和φ Q有一个特别简单的形式。此外，大多数常见的分布空间（L - p空间，1< p<+∞，H - p空间，0< p≤1，Sobolev和势空间，BMO， Besov和triiebel - lizorkin空间）都以ϑ-transform的大小为特征。这使我们能够对与这些分布空间相对应的序列空间进行离散Littlewood-Paley理论。序列空间范数只与量有关；消去是在∑Q和∑Q中得到的。因此，在序列空间级别上进行分析通常很容易。由此可以简化、推广和统一谐波分析中的各种结果。通过考虑相关序列空间上矩阵的有界条件，得到了这些分布空间上线性算子有界的条件。应用程序包括Hörmander（傅里叶）乘数定理的一般版本和Calderón-Zygmund类型的核算子的结果。我们还讨论了其他一些更一般的分解方法，包括“平滑原子分解”和“广义ϑ-transform”。光滑原子分解为处理rn中超平面的限制和扩展现象提供了一种简单的方法。我们还考虑了点乘子。对于定义域的特征函数，我们得到了一类适当包含Lipschitz定义域的定义域的有界性结果。通过序列空间分析了几种插值方法。对于实插值，我们得到了p= 0的一个扩展。这反过来又为Hardy空间的传统原子分解提供了一种新的方法。
We study a representation formula of the form ƒ=∑ Q< ƒ, ϑ Q> ψ Q for a distribution ƒ on R n. This formula is obtained by discretizing and localizing a standard Littlewood-Paley decomposition. The map taking ƒ to the sequence {< ƒ, ϑ Q>} Q, with Q running over the dyadic cubes in R n, is called the ϑ-transform. The functions ϑ Q and ψ Q have a particularly simple form. Moreover, most of the familiar distribution spaces (L p-spaces, 1< p<+∞, H p spaces, 0< p⩽ 1, Sobolev and potential spaces, BMO, Besov and Triebel-Lizorkin spaces) are characterized by the magnitude of the ϑ-transform. This enables us to carry out a discrete Littlewood-Paley theory on the sequence spaces corresponding to these distribution spaces. The sequence space norms depend only on magnitudes; cancellation is accounted for in the ϑ Q's and ψ Q's. Consequently, analysis on the sequence space level is often easy. With this we can simplify, extend, and unify a variety of results in harmonic analysis. We obtain conditions for the boundedness of linear operators on these distribution spaces by considering corresponding conditions for matrices on the associated sequence spaces. Applications include a general version of the Hörmander (Fourier) multiplier theorem and results for kernel operators of Calderón-Zygmund type. We discuss certain other, more general, decomposition methods, including the “smooth atomic decomposition,” and the “generalized ϑ-transform.” The smooth atomic decomposition yields a simple method for dealing with restriction and extension phenomena for hyperplanes in R n. We also consider pointwise multipliers. For the characteristic function of a domain, we obtain boundedness results for a general class of domains which properly includes Lipschitz domains. Several interpolation methods are easily analyzed via the sequence spaces. For real interpolation, we obtain, among other things, an extension to the case p= 0. This in turn gives a new approach to the traditional atomic decomposition of Hardy spaces.