Sharp embeddings of uniformly localized Bessel potential spaces into multiplier spaces

TL;DR

This paper establishes sharp embeddings of uniformly localized Bessel potential spaces into multiplier spaces, demonstrating the precise conditions under which these embeddings hold and their limits of sharpness.

Contribution

It proves the exact conditions for embeddings of uniformly localized Bessel potential spaces into multiplier spaces, including the sharpness of the left embedding.

Findings

The embeddings $H^{-r}_{p_1, unif} o M[s, -t] o H^{-r}_{2, unif}$ are valid under specified conditions.

The left embedding is sharp and does not hold if the lower index $p_1$ is decreased slightly.

The results clarify the precise relationship between localized Bessel potential spaces and multiplier spaces.

Abstract

For $p > 1, \gamma \in \mathbb{R}$, denote by $H^{\gamma}_p(\mathbb{R}^n)$ the Bessel potential space, by $H^{\gamma}_{p, unif}(\mathbb{R}^n)$ the corresponding uniformly localized Bessel potential space and by $M[s, -t]$ the space of multipliers from $H^s_2(\mathbb{R}^n)$ into $H^{-t}_2(\mathbb{R}^n)$. Assume that $s, t \geqslant 0, n/2 > \max(s, t) > 0, r: = \min(s, t), p_1: = n/max(s, t)$. Then the following embeddings hold $$ H^{-r}_{p_1, unif}(\mathbb{R}^n) \subset M[s, -t] \subset H^{-r}_{2, unif}(\mathbb{R}^n). $$ The main result of the paper claims the sharpness of the left embedding in the following sense: it does not hold if the lower index $p_1$ is replaced by $p_1 -\varepsilon$ with any sufficiently small $\varepsilon > 0$.