A discrete transform and decompositions of distribution spaces

Abstract

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.