  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.
