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

**A 2007 Preprint by
G. Kyriazis,
P. Petrushev, and
Y. Xu
**

- 2007:05
Weighted Triebel-Lizorkin and Besov spaces on the unit ball $B^d$ in $R^d$ with weights $w _ \mu(x)=(1-|x|^2)^{\mu-\frac{1}{2}},$ $\mu\geq0$, are introduced and explored. A decomposition scheme is developed in terms of almost exponentially localized polynomial elements (needlets) $\{\varphi _ \xi\},\{\psi _ \xi\}$ 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 $\{\langle{f},\varphi _ \xi\rangle\}$ in appropriate sequence spaces.