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

TL;DR

This paper characterizes the space of multipliers between Bessel potential spaces with different sign indices, providing explicit descriptions under certain conditions and establishing embeddings for the case when indices are equal.

Contribution

It explicitly describes the multiplier space between Bessel potential spaces with different sign indices, extending understanding of their structure and embeddings.

Findings

Multiplier space equals intersection of uniformly localized Bessel potential spaces.

Established two-sided embeddings for the case s = t < n/max(p,q').

Provided explicit conditions for the description of the multiplier space.

Abstract

The objective of this paper is to describe the space of multipliers acting from a Bessel potential space $H^s_p(\mathbb R^n)$ into another space $H^{-t}_q(\mathbb R^n)$, provided that the smooth indices of these spaces have different signs, i.e. $s, t \geqslant 0$. This space of multipliers consists of distributions $u$, such that for all $\varphi \in H^s_p(\mathbb R^n)$ the product $\varphi \cdot u$ is well-defined and belongs to the space $H^{-t}_q(\mathbb R^n)$. We succeed to describe this space explicitly, provided that $p \leqslant q$ and one of the following conditions $$ s \geqslant t \geqslant 0, \ s > n/p \ \ \, \text{or} \ \ \, t \geqslant s \geqslant 0, \ t > n/q' \quad (\: \text{where} \; 1/q +1/q' = 1), $$ holds. In this case one has $$ M[H^s_p(\mathbb{R}^n) \to H^{-t}_{q}(\mathbb{R}^n)] = H^{-t}_{q, \: unif}(\mathbb{R}^n) \cap H^{-s}_{p', \: unif}(\mathbb{R}^n), $$ where $H^\gamma_{r, \: unif}(\mathbb{R}^n), \: \gamma \in \mathbb{R}, \: r > 1$ is the scale of uniformly localized Bessel potential spaces. In particular but important case $s = t < n/\max (p,q')$ we prove two-sided continuous embeddings $$ H^{-s}_{r_1, \: unif}(\mathbb{R}^n) \subset M[H^s_p(\mathbb{R}^n) \to H^{-s}_q(\mathbb{R}^n)] \subset H^{-s}_{r_2, \: unif}(\mathbb{R}^n), $$ where $r_2 = \max (p', q), \ r_1 =[s/n-(1/p -1/q)]^{-1}$.