Denseness of the spaces $Φ_V$ of Lizorkin type in the mixed $L^{p̅}(ℝ^n)$-spaces
Volume 113 / 1995
Studia Mathematica 113 (1995), 199-210
DOI: 10.4064/sm-113-3-199-210
Abstract
The spaces Φ_V(ℝ^{n}) are defined to consist of Schwartz test functions φ such that the Fourier transform φ̂ and all its derivatives vanish on a given closed set V ⊂ ℝ^{n}. Under the only assumption that m(V) = 0 it is shown that Φ_V is dense in $C_0(ℝ^{n})$ and in the space $L^{p̅}(ℝ^n)$ with the mixed norm, for $1/p̅$ in a certain pyramid. The result on the denseness for arbitrary $p̅ = (p_1,..., p_n)$, 1 < p_k < ∞, k = 1,...,n,$ is proved for so-called quasibroken sets V.
