Some spaces Asp,q(Rn) with A = {B, F}, s ϵ R, 0 < p, q ≤ ∞, covering Besov spaces, Hölder-Zygmund spaces and Sobolev spaces, admit characterizations in terms of Haar bases. It is the main aim of this paper to extend this observation to corresponding Morreyfied spaces Lr Asp,q(Rn). As a by-product we obtain Littlewood-Paley theorems for (homogeneous and inhomogeneous) Morrey spaces Lrp(Rn), Lrp(Rn) and, in particular, L°rp(Rn).
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We give characterizations of Besov and Triebel-Lizorkin spaces $B_{pq}^{s}(Ω)$ and $F_{pq}^s(Ω)$ in smooth domains $Ω ⊂ ℝ^n$ via convolutions with compactly supported smooth kernels satisfying some moment conditions. The results for s ∈ ℝ, 0 < p,q ≤ ∞ are stated in terms of the mixed norm of a certain maximal function of a distribution. For s ∈ ℝ, 1 ≤ p ≤ ∞, 0 < q ≤ ∞ characterizations without use of maximal functions are also obtained.
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