Multipliers in Bessel potential spaces. The case of different sign smooth indices

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The objective of this paper is to describe the space of multipliers acting from a Bessel potential space Hˢₚ(Rⁿ) into another space H⁻ᵗq(Rⁿ), provided that the smooth indices of these spaces have different signs, i.e. s, t 0. This space of multipliers consists of distributions u, such that for all φ∈ Hˢₚ(Rⁿ) the product φ· u is well-defined and belongs to the space H⁻ᵗq(Rⁿ). We succeed to describe this space explicitly, provided that p q and one of the following conditions s t 0, s > n/p or t s 0, t > n/q' (where 1/q +1/q' = 1), holds. In this case one has M[Hˢₚ(Rⁿ) → H⁻ᵗq(Rⁿ)] = H⁻ᵗq, unif(Rⁿ) ∩ H⁻ˢp', unif(Rⁿ), where Hγr, unif(Rⁿ), γ∈ R, r > 1 is the scale of uniformly localized Bessel potential spaces. In particular but important case s = t < n/ (p,q') we prove two-sided continuous embeddings H⁻ˢr₁, unif(Rⁿ) ⊂ M[Hˢₚ(Rⁿ) → H⁻ˢq(Rⁿ)] ⊂ H⁻ˢr₂, unif(Rⁿ), where r₂ = (p', q), r₁ =[s/n-(1/p -1/q)]⁻¹.