The characterization of the Triebel-Lizorkin spaces forp=∞

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∞,qα 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.