Multipliers in spaces of Bessel potentials: The case of indices of nonnegative smoothness

Abstract

The aim of the paper is to study spaces of multipliers acting from the Bessel potential space H (ℝ n ) to the other Bessel potential space H (ℝ n ). We obtain conditions ensuring the equivalence of uniform and standard multiplier norms on the space of multipliers $$M\left[ {H_p^s({\mathbb{R}^n}) \to H_q^t({\mathbb{R}^n})} \right]fors,t \in \mathbb{R},p,q > 1.$$ In the case $$p,q > 1,p \leqslant q,s > \frac{n}{p},t \geqslant 0,s - \frac{n}{p} \geqslant t - \frac{n}{q}$$ , the space M[H (ℝ n ) → H (ℝ n ) can be described explicitly. Namely, we prove in this paper that the latter space coincides with the space H , unif (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.