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_p^s(ℝⁿ) to the other Bessel potential space H_q^t(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_p^s(ℝⁿ) → H_q^t(ℝ_n) can be described explicitly. Namely, we prove in this paper that the latter space coincides with the space H_{q, unif}^t(ℝⁿ) 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.