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 p s (ℝ n ) to the Bessel potential space H q -t (ℝ n ) for the case in which the smoothness indices of these spaces have different signs, i.e., s, t ≥ 0. The space of such multipliers consists of distributions u such that for all Φ ∈ H p s (ℝ n ) the product Φ · u is well defined and belongs to H q -t (ℝ n ). It turns out that these multiplier spaces can be described explicitly in the case where p ≤ q and one of the following conditions is fulfilled: s ≥ t ≥ 0, s > n/p or t ≥ s ≥ 0, t > n/q' (where 1/q + 1/q'= 1). Namely, in this case we have M[H p s (ℝ n ) → H q -t (ℝ n )] = H q,unif -t (ℝ n ) ∩ H p',unif -s (ℝ n ), where H r,unif γ (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 H r1,unif -s (ℝ n ) ⊂ M[H p s (ℝ n ) → H q -s (ℝ n )] ⊂ H r2,unif -s (ℝ n ), where r 2 = max(p', q), r 1 = [s/n - (1/p - 1/q)] -1 . © 2019 American Mathematical Society.