Multipliers in the Bessel potential spaces with positive smoothness indices: bilateral continuous embeddings and their exact character

TL;DR

This paper studies the conditions under which certain Bessel potential spaces with positive smoothness can be continuously embedded into multiplier spaces, providing optimal index characterizations in a complex functional analysis setting.

Contribution

It offers new insights into bilateral embeddings of localized Bessel potential spaces into multiplier spaces, including optimal index characterizations and limitations of existing description theorems.

Findings

Established bilateral continuous embeddings under specific conditions.

Identified the optimal indices for these embeddings.

Highlighted limitations in existing description theorems.

Abstract

We investigate the problem of establishing bilateral continuous embeddings of the uniformly localized Bessel potential spaces $H^{\gamma}_{r, \: unif}(\mathbb{R}^n)$ into the multiplier spaces between Bessel potential spaces with positive smoothness indices. This problem is considered in the model situation when the natural norms of both of these Bessel potential spaces are generated by some inner product yet the description theorems for the corresponding multiplier space in terms of the spaces $H^{\gamma}_{r, \: unif}(\mathbb{R}^n)$ can not be established. The optimal character of the indices figuring in these embeddings is also examined.