Multipliers in Bessel potential spaces. The case of different sign smooth indices

Abstract

The objective of this paper is to describe the space of multipliers acting from a Bessel potential space $H_p^s({\mathbb{R}}^n)$ into another space $H_q^{-t}({\mathbb{R}}^n)$, provided that the smooth indices of these spaces have different signs, i.e. $s,t⩾0$. This space of multipliers consists of distributions $u$, such that for all $\phi \in H_p^s({\mathbb{R}}^n)$ the product $\phi \cdot u$ is well-defined and belongs to the space $H_q^{-t}({\mathbb{R}}^n)$. We succeed to describe this space explicitly, provided that $p⩽q$ and one of the following conditions $$s⩾t⩾0,s>n/p\text{or}t⩾s⩾0,t>n/q^{\prime }(\text{where}1/q+1/q^{\prime }=1),$$ holds. In this case one has $$M[H_p^s({\mathbb{R}}^n)\to H_q^{-t}({\mathbb{R}}^n)]=H_{q,unif}^{-t}({\mathbb{R}}^n)\cap H_{p^{\prime },unif}^{-s}({\mathbb{R}}^n),$$ where $H_{r,unif}^{\gamma }({\mathbb{R}}^n),\gamma \in \mathbb{R},r>1$ is the scale of uniformly localized Bessel potential spaces. In particular but important case $s=t<n/\max (p,q^{\prime })$ we prove two-sided continuous embeddings $$H_{r_1,unif}^{-s}({\mathbb{R}}^n)\subset M[H_p^s({\mathbb{R}}^n)\to H_q^{-s}({\mathbb{R}}^n)]\subset H_{r_2,unif}^{-s}({\mathbb{R}}^n),$$ where $r_2=\max (p^{\prime },q),r_1=[s/n-(1/p-1/q){]}^{-1}$.

22 pages, in Russian