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 w μ ( 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) {φ ξ }, {ψ ξ } 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 , φ ξ 〉} in appropriate sequence spaces.