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

We investigate the problem of establishing bilateral continuous embeddings of the uniformly localized Bessel potential spaces $H_{r,unif}^{\gamma }({\mathbb{R}}^n)$ 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 $H_{r,unif}^{\gamma }({\mathbb{R}}^n)$ can not be established. The optimal character of the indices figuring in these embeddings is also examined.