Sharp embeddings of uniformly localized Bessel potential spaces into multiplier spaces

Abstract

For $p>1,\gamma \in \mathbb{R}$, denote by $H_p^{\gamma }({\mathbb{R}}^n)$ the Bessel potential space, by $H_{p,unif}^{\gamma }({\mathbb{R}}^n)$ the corresponding uniformly localized Bessel potential space and by $M[s,-t]$ the space of multipliers from $H_2^s({\mathbb{R}}^n)$ into $H_2^{-t}({\mathbb{R}}^n)$. Assume that $s,t⩾0,n/2>\max (s,t)>0,r:=\min (s,t),p_1:=n/max(s,t)$. Then the following embeddings hold $$H_{p_1,unif}^{-r}({\mathbb{R}}^n)\subset M[s,-t]\subset H_{2,unif}^{-r}({\mathbb{R}}^n).$$ The main result of the paper claims the sharpness of the left embedding in the following sense: it does not hold if the lower index $p_1$ is replaced by $p_1-\epsilon$ with any sufficiently small $\epsilon >0$.

21 pages