Characterizations of pointwise multipliers of Besov spaces in endpoint cases with an application to the duality principle

Let p, q∈(0, ∞], s ∈R, and M(Bsp,q(Rn))denote the pointwise multiplier space of the Besov space Bsp,q(Rn). In this article, the authors first establish the characterizations of both (Bs1,∞(Rn))with s ∈R \{0}and M(Bs∞,1(Rn))with s ∈(−∞, 0]. Then, as an application, the authors give a corrected proof of the well-known duality principle for pointwise multiplier spaces of Besov spaces, namely the formula
M Bs p,q(Rn) = M B −s p,q (Rn) ,
where p, q∈[1, ∞], s ∈R, and 1/a +1/a=1for any a ∈[1, ∞], and, moreover, the authors also show that this duality principle is sharp in some sense. The proofs of all these results essentially depend on the duality theorem of Besov spaces themselves, some elaborate estimates of paraproducts as well as the relation between M(Bsp,q(Rn))and the auxiliary multiplier space M(
Bsp,q(Rn)), where Bsp,q(Rn)denotes the completion of the Schwartz function space S(Rn)in Bsp,q(Rn).