New Characterizations of Besov-Triebel-Lizorkin-Hausdorff Spaces Including Coorbits and Wavelets

摘要

In this paper, the authors establish new characterizations of the recently introduced Besov-type spaces B ~(S,τ) _(p,q)(? ~n) and Triebel-Lizorkin-type spaces F ~(S,τ) _(p,q)(? ~n) with p∈(0,~∞], s∈?, τ∈[0,~∞), and q∈(0,~∞], as well as their preduals, the Besov-Hausdorff spaces BH ~(S,τ) _(p,q)(? ~n) and Triebel-Lizorkin-Hausdorff spaces FH ~(S,τ) _(p,q)(? ~n), in terms of the local means, the Peetre maximal function of local means, and the tent space (the Lusin area function) in both discrete and continuous types. As applications, the authors then obtain interpretations as coorbits in the sense of Rauhut (Stud. Math. 180:237-253, 2007) and discretizations via biorthogonal wavelet bases for the full range of parameters of these function spaces. Even for some special cases of this setting such as F ~S _(∞,q)(? ~n) for s∈?, q∈(0,~∞] (including BMO(? ~n) when s=0 and q=2), the Q space Q _α(? ~n), the Hardy-Hausdorff space HH _(-α)(? ~n) for α∈(0,min{n/2,1}), the Morrey space M ~u _p(? ~n) for 1 < p ≤ u < ~∞, and the Triebel-Lizorkin-Morrey space E ~S _(upq)(? ~n) for 0 < p ≤ u < ~∞, s ∈ ? and q∈(0,~∞], some of these results are new.