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

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