On the Relationship and the Norm Discretization for Lizorkin-Triebel Spaces with Generalized Smoothness

Abstract

Homogeneous and non-homogeneous Lizorkin-Triebel spaces with generalized smoothness $\dot{F}}_{pq}^{\lambda(.)}(\mathbb{R}^{n}) $ and $ F_{pq}^{\lambda(.)}(\mathbb{R}^{n}) $ have been considered. In particular, under some assumptions of the function $ \lambda(t):\mathbb{R}^{+} \to \mathbb{R}^{+},\lambda(1)=1 $ determining the generalized smoothness properties, the discretization procedure is realized and the relationship is established between these spaces on $\mathbb{R}^{n}$ and their discrete analogues.