A linear operator bounded in all Besov but not in Triebel-Lizorkin spaces

Abstract

We construct a linear operator $T:{\mathcal{S}}^{\prime }({\mathbb{R}}^n)\to {\mathcal{S}}^{\prime }({\mathbb{R}}^n)$ such that $T:{\mathcal{B}}_{pq}^s({\mathbb{R}}^n)\to {\mathcal{B}}_{pq}^s({\mathbb{R}}^n)$ for all $0<p,q\le \infty$ and $s\in \mathbb{R}$, but $T({\mathcal{F}}_{pq}^s({\mathbb{R}}^n))\not\subset {\mathcal{F}}_{pq}^s({\mathbb{R}}^n)$ unless $p=q$. As a result Triebel-Lizorkin spaces cannot be interpolated from Besov spaces unless $p=q$. In the appendix we purpose a question for the interpolation framework via structured Banach spaces.

11 pages. Add a new appendix section and edit acknowledgement