On the boundedness of bilinear operators on products of Besov and Lebesgue spaces

We prove mapping properties of the form T : B ˙ p 1 α 1 , q 1 × L p 2 → B ˙ p 3 α 2 , q 2 and T : B ˙ p 1 α 1 , q 1 × B ˙ p 2 α 2 , q 2 → L p 3 , for certain related indices p 1 , p 2 , p 3 , q 1 , q 2 , α 1 , α 2 ∈ R , where T is a bilinear Hörmander–Mihlin multiplier or a molecular paraproduct. Applications to bilinear Littlewood–Paley theory are discussed.