Abstract
We establish the characterization of the weighted Triebel-Lizorkin spaces for p=∞ by means of a “generalized” Littlewood-Paley function which is based on a kernel satisfying “minimal” moment and Tauberian conditions. This characterization completes earlier work by Bui et al. The definitions of the Ḟ α∞,q spaces are extended in a natural way to Ḟ α∞,∞ and it is proven that this is the same space as Ḃ α∞,∞ , which justifies the standard convention in which the two spaces are defined to be equal. As a consequence, we obtain a new characterization of the Hölder-Zygmund space Ḃ α∞,∞ .
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Bui, HQ., Taibleson, M.H. The characterization of the Triebel-Lizorkin spaces forp=∞. The Journal of Fourier Analysis and Applications 6, 537–550 (2000). https://doi.org/10.1007/BF02511545
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DOI: https://doi.org/10.1007/BF02511545