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A discrete transform and decompositions of distribution spaces. (English) Zbl 0716.46031
A representation formula of the form \(f=\sum_{Q}<f,\phi_ Q>\psi_ Q\) for a distribution f on \({\mathbb{R}}^ n\) is studied. This formula is obtained by discretizing and localizing a standard Littlewood-Paley decomposition. The map taking f to the sequence \(\{<f,\phi_ Q>\}_ Q\), with Q running over the dyadic cubes in \({\mathbb{R}}^ n\), is called the \(\phi\)-transform. The functions \(\phi_ Q\) and \(\psi_ Q\) have a particular simple form. Moreover, most of the familiar distribution spaces \((L^ p\)-spaces, \(1<p<+\infty\), \(H^ p\) spaces, \(0<p\leq 1\), Sobolev and potential spaces, BMO, Besov and Triebel-Lizorkin spaces) are characterized by the magnitude of the \(\phi\)-transform. This enables the authors 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 \(\phi_ Q's\) and \(\psi_ Q's\). Consequently, analysis on the sequence space level is often easy. With this they simplify, extend, and unify a variety of results in harmonic analysis, and 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. Certain other, more general, decomposition methods, including the “smooth atomic decomposition”, and the “generalized \(\phi\)-transform” are discussed. The smooth atomic decomposition yields a simple method for dealing with restriction and extension phenomena for hyperplanes in \({\mathbb{R}}^ n\). Pointwise multipliers are also considered. For the characteristic function of a domain boundedness results for a general class of domains which properly includes Lipschitz domains are obtained. Several interpolation methods are easily analyzed via the sequence spaces. For real interpolation, among other things, an extension to the case \(p=0\) and a new approach to the traditional atomic decomposition of Hardy spaces are given.
Reviewer: J.F.Toland

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46F12 Integral transforms in distribution spaces
43A32 Other transforms and operators of Fourier type
46F05 Topological linear spaces of test functions, distributions and ultradistributions
46M35 Abstract interpolation of topological vector spaces
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