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

Abstract

Weighted Triebel-Lizorkin and Besov spaces on the unit ball $B^d$ in $\Rd$ with weights $\W(x)= (1-|x|^2)^{\mu-1/2}$, $\mu \ge 0$, are introduced and explored. A decomposition scheme is developed in terms of almost exponentially localized polynomial elements (needlets) $\{\phi_\xi\}$, $\{\psi_\xi\}$ 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 $\{\ip{f,\phi_\xi}\}$ in appropriate sequence spaces.