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

Abstract

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.