A localization property for $B^{s}_{pq}$ and $F^{s}_{pq}$ spaces
Volume 109 / 1994
Studia Mathematica 109 (1994), 183-195
DOI: 10.4064/sm-109-2-183-195
Abstract
Let $f^{j} = ∑_{k} a_{k} f(2^{j+1}x - 2k)$, where the sum is taken over the lattice of all points k in $ℝ^n$ having integer-valued components, j∈ℕ and $a_k ∈ ℂ$. Let $A^{s}_{pq}$ be either $B^{s}_{pq}$ or $F^{s}_{pq}$ (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on $ℝ^n.$ The aim of the paper is to clarify under what conditions $∥f^{j} | A^{s}_{pq}∥$ is equivalent to $2^{j(s-n/p)} (∑_{k} |a_k|^p)^{1/p} ∥f | A^{s}_{pq}∥$.
