Bilateral Continuous Embeddings for Multipliers Acting in the Scale of Bessel Potential Spaces with Nonnegative Smoothness Indices

We investigate the problem of establishing bilateral embeddings of the uniformly localized Bessel potential spaces $$H^{\gamma}_{r,\>unif}(\mathbb{R}^{n})$$ into the multiplier spaces between two Bessel potential spaces with nonnegative smoothness indices. This problem is considered in the most general situation when only the natural assumptions on the indices of these Bessel potential spaces are met 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 natural character of these assumptions is demonstrated explicitly. The embedding of a uniformly localized Bessel potential space into the corresponding multiplier space is obtained via the criterion of the validity of this embedding in terms of the multiplicative functional estimate on the norms, while the functional estimate itself is derived from general multiplicative norm estimates in Lizorkin–Triebel spaces. The uniform localization principle, which holds true not only for Bessel potential spaces but also for general Lizorkin–Triebel spaces, is of the utmost importance for employing this criterion.