Sharp embeddings of uniformly localized Bessel potential spaces into multiplier spaces

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 $s, t \geqslant 0, n/2 > \max(s, t) > 0, r: = \min(s, t), p_1: = n/max(s, t)$\mathrm{.}Thenthefollowingembeddingshold$$ H^{-r}{p_1, unif}($\mathbb{R}$^n) \subset M[s, -t] \subset H^{-r}{2, unif}($\mathbb{R}$^n). $$ 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 $p_1 -\varepsilon$withanysufficientlysmall$\varepsilon > 0$.