Multipliers in Spaces of Bessel Potentials: The Case of Indices of Nonnegative Smoothness

The aim of the paper is to study spaces of multipliers acting from the Bessel potential space $H^s_p(\mathbb{R}^n)$ to the other Bessel potential space $H^t_q(\mathbb{R}^n)$. We obtain conditions ensuring the equivalence of uniform and standard multiplier norms on the space of multipliers
$$ M[H^s_p(\mathbb{R}^n) \to H^t_q(\mathbb{R}^n)]\qquad \text{for}\quad s,t \in \mathbb{R},\quad p,q > 1. $$
In the case
$$ p,q > 1,\qquad p \le q,\qquad s > \frac np,\qquad t \ge 0,\qquad s-\frac np \ge t-\frac nq, $$
the space $M[H^s_p(\mathbb{R}^n) \to H^t_q(\mathbb{R}^n)]$ can be described explicitly. Namely, we prove in this paper that the latter space coincides with the space $H^t_{q,\mathrm{unif}}(\mathbb{R}^n)$ of uniformly localized Bessel potentials introduced by Strichartz. It is also proved that if both smoothness indices $s$ and $t$ are nonnegative, then such a description is possible only for the given values of the indices.