The characterization of the Triebel-Lizorkin spaces forp=∞
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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 ∞,q α 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 .
Math Subject ClassificationsPrimary 42B25 secondary 46E35
Keywords and PhrasesLittlewood-Paley functions BMO, A∞ weights Besov-Lipschitz spaces Triebel-Lizorkin spaces
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