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Weighted Triebel–Lizorkin and Besov spaces on the unit ball Bd in ℝd with weights wμ(x)=(1−|x|2)μ−1/2, μ⩾0, are introduced and explored. A decomposition scheme is developed in terms of almost exponentially localized polynomial elements (needlets) {φξ}, {ψξ} and it is shown that the membership of a distribution to the weighted Triebel–Lizorkin or Besov spaces can be determined by the size of the needlet coefficients {〈f, φξ〉} in appropriate sequence spaces.

Abstract. The Littlewood-Paley theory is extended to weighted spaces of distributions on [−1, 1] with Jacobi weights w(t) = (1−t) α (1+t) β . Almost exponentially localized polynomial elements (needlets) {ϕ ξ }, {ψ ξ } are constructed and, in complete analogy with the classical case on R n , it is shown that weighted Triebel-Lizorkin and Besov spaces can be characterized by the size of the needlet coefficients { f, ϕ ξ } in respective sequence spaces.

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Let Θ := {θ e I : e ∈ E, I ∈ D} be a decomposition system for L 2 (R d ) indexed over D, the set of dyadic cubes in R d , and a finite set E, and let Θ := { θ e I : e ∈ E, I ∈ D} be the corresponding dual functionals. That is, for everyWe study sufficient conditions on Θ, Θ so that they constitute a decomposition system for Triebel-Lizorkin and Besov spaces. Moreover, these conditions allow us to characterize the membership of a distribution f in these spaces by the size of the coefficients f, θ e I , e ∈ E, I ∈ D. Typical examples of such decomposition systems are various wavelet-type unconditional bases for L 2 (R d ), and more general systems such as affine frames.2000 Mathematics Subject Classification: 41A17, 41A20, 42B25, 42C15. [133] J (4.17) = e∈E J∈D I∈D s I (f ) a e (I, J)θ e J = e∈E J∈D I∈Dwhere all identities above are considered in the distributional sense. To justify the third equality, we note that the assumptions of the theorem guarantee that for every e ∈ E the matrix A T e is bounded onḟ s pq . Since ς := (s I (f )) I ∈ḟ s pq , the sequence (d e J ) := ( I∈D | a e (I, J)| |s I (f )|) J belongs inḟ s pq . Finally, for every η ∈ S k with k := max{[s − d/p], −1}, as in the proof of Lemma 4.1 we have J∈D |d e J | | θ e J , η | < ∞, which allows us to interchange the order of the summations. where for e ∈ E, a e (I, J) := Φ I , ψ e J , |I| = 1, |J| ≤ 1, φ I , ψ e J , |I| < 1, |J| ≤ 1, while a 0 (I, J) :=      Φ I , ψ 0 J , |I| = 1, |J| = 1, φ I , ψ 0 J , |I| < 1, |J| = 1, 0, |I| ≤ 1, |J| < 1.