Wavelet transforms and operators in various function spaces
We define a class of wavelet transforms as a continuous and micro-local version of the Littlewood-Paley decompositions. Hörmander’s wave front sets (see ) as well as the Besov and Triebel-Lizorkin spaces (see  and ) may be characterized in terms of our wavelet transforms. By using the results obtained above (see ), we characterize the wave front sets in the sense of the Besov-Triebel-Lizorkin regularity in terms of our wavelet transforms. Finally, Päivärinta’s results on the continuity of pseudodifferential operators in the Besov-Triebel-Lizorkin spaces (see ) 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.
KeywordsFunction Space Besov Space Pseudodifferential Operator Inverse Fourier Transform Inversion Formula
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