Multipliers in Bessel potential spaces with smoothness indices of different sign

The subject of the present paper is the study of multipliers from the Bessel potential space H^s_p(\mathbb{R}^n) to the Bessel potential space H^{-t}_q(\mathbb{R}^n) for the case in which the smoothness indices of these spaces have different signs, i.e., s, t \geq 0. The space of such multipliers consists of distributions u such that for all \varphi \in H^s_p(\mathbb{R}^n) the product \varphi \cdot u is well defined and belongs to H^{-t}_q(\mathbb{R}^n). It turns out that these multiplier spaces can be described explicitly in the case where p \leq q and one of the following conditions is fulfilled: \displaystyle s \geq t \geq 0, \ s > n/p \ \displaystyle \text { or } \ \ t \geq s \geq 0, \ t > n/q' \quad (\text {where } \ 1/q +1/q' = 1). Namely, in this case we have \displaystyle M[H^s_p(\mathbb{R}^n) \to H^{-t}_{q}(\mathbb{R}^n)] = H^{-t}_{q, \mathrm {unif}}(\mathbb{R}^n) \cap H^{-s}_{p', \mathrm {unif}}(\mathbb{R}^n), where H^\gamma _{r, \mathrm {unif}}(\mathbb{R}^n) is the space of uniformly localized Bessel potentials. For the important case where s = t < n/\max (p,q'), we prove the two-sided embeddings \displaystyle H^{-s}_{r_1, \mathrm {unif}}(\mathbb{R}^n) \subset M[H^s_p(\mathb... ...\to H^{-s}_q(\mathbb{R}^n)] \subset H^{-s}_{r_2, \mathrm {unif}}(\mathbb{R}^n), where r_2 = \max (p', q), r_1 =[s/n-(1/p -1/q)]^{-1}.