Decomposition of weighted Triebel-Lizorkin and Besov spaces on the ball

Weighted Triebel–Lizorkin and Besov spaces on the unit ball B d in R d 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. Localized bases and frames allow to decompose functions and distributions in terms of building blocks of simple nature and have numerous advantages over other means of representation. In particular, they enable one to encode smoothness and other norms in terms of the coefficients of the decompositions. Meyer’s wavelets [10] and the ϕ-transform of Frazier and Jawerth [5–7] provide such building blocks for decomposition of Triebel–Lizorkin and Besov spaces in the classical case on R d . The aim of this article is to develop similar tools for decomposition of weighted Triebel– Lizorkin and Besov spaces on the unit ball B d in R d ( d> 1) with weights wμ(x ): = (1 −| x| 2 ) μ−1/2 ,μ 0, where |x| is the Euclidean norm of x ∈ B d . These include Lp(B d ,w μ), the Hardy spaces Hp(B d ,w μ), and weighted Sobolev spaces. For our purpose, we develop localized frames which can be viewed as an analog of the ϕ-transform of Frazier and Jawerth on B d .