The characterization of the Triebel-Lizorkin spaces for p = ∞

We establish the characterization of the weighted Triebel-Lizorkin spaces for p = infinity 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 (F) over dot (alpha)(infinity,q) spaces are extended in a natural way to (F) over dot (alpha)(infinity,infinity) and it is proven that this is the same space as (B) over dot (alpha)(infinity,infinity) 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 Holder-Zygmund space Lambda(alpha) = (B) over dot (alpha)(infinity,infinity).