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.

Jacobi decomposition of weighted Triebel–Lizorkin and Besov spaces

“…The proof of this embedding result can be carried out as the proof of Proposition 4.11 in [9] (the argument is similar to the one in the classical case of R n , see e.g. [19, p. 129]).…”

“…The proofs of this theorem and of Proposition 8.1 can be carried out exactly as the proofs of the respective Jackson estimate and embedding result in [9,11] and will be omitted.…”

“…This is a follow-up paper of [13]; it is closely related to [11] and [9], where needlet decompositions of Triebel-Lizorkin and Besov spaces on the unit sphere and ball are developed.…”

Sub-exponentially localized kernels and frames induced by orthogonal expansions

“…The rapid decay of needlets makes them a powerful tool for decomposition of spaces of functions and distributions in various settings. The above scheme has already been utilized for construction of needlets and needlet decomposition of L p , Sobolev, and the more general TriebelLizorkin and Besov spaces in the frameworks of spherical harmonics [15,16], Jacobi polynomials [17,12], orthogonal polynomials on the ball [18,13], and Hermite and Laguerre functions [4,19,9]. …”

Decomposition of Spaces of Distributions Induced by Hermite Expansions

“…Our development here is a part of a bigger project for needlet characterization of Triebel-Lizorkin and Besov spaces on nonclassical domains such as the unit sphere [11], the interval with Jacobi weights [8,13], and the unit ball [9,14].…”