Wavelet transforms and operators in various function spaces

Abstract

We define a class of wavelet transforms as a continuous and micro-local version of the Littlewood-Paley decompositions. Hormander’s wave front sets (see [3]) as well as the Besov and Triebel-Lizorkin spaces (see [6] and [7]) may be characterized in terms of our wavelet transforms. By using the results obtained above (see [4]), we characterize the wave front sets in the sense of the Besov-Triebel-Lizorkin regularity in terms of our wavelet transforms. Finally, Paivarinta’s results on the continuity of pseudodifferential operators in the Besov-Triebel-Lizorkin spaces (see [9]) may be microlocalized. In other words, we show the pseudo-microlocal properties in the sense of the Besov-Triebel-Lizorkin regularity. We remark that the components of our decompositions are not linearly independent but can be treated as if they were.