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

Bilinear Sobolev–Poincaré Inequalities and Leibniz-Type Rules

“…Second, the identity (1.5) can be exploited to produce Leibniz-type rules (1.1) involving function spaces that interact well with J m (for example, Besov and Triebel-Lizorkin spaces) provided that mapping properties for bilinear multipliers T σ are established for such spaces. Indeed, implementations of this program (see, for instance, [7,16,25]), produce Besov, Triebel-Lizorkin, and mixed Besov-Lebesgue Leibniz-type rules. A Littlewood-Paley-free path towards Leibniz-type rules was introduced in [24] in the scales of Campanato-Morrey spaces.…”

Hardy space estimates for bilinear square functions and Calderon-Zygmund operators

“…The study of multilinear Calderón-Zygmund operators was initiated by Coifman and Meyer [12,13,14]. A fruitful theory has grown around these operators, see for example [9,26,27,23,2,43,29,36,34,41]. A multilinear and multi-parameter version of the Coifman-Meyer Fourier multiplier theorem was established in [44,45] using time-frequency analysis (see also [8] using the Littlewood-Paley analysis), and a pseudo-differential analogue was carried out in [15].…”

“…A multilinear and multi-parameter version of the Coifman-Meyer Fourier multiplier theorem was established in [44,45] using time-frequency analysis (see also [8] using the Littlewood-Paley analysis), and a pseudo-differential analogue was carried out in [15]. There has also been some work done for the operators in the context of distributions spaces (Triebel-Lizorkin and Besov spaces), see for example [28,1,43,2]. For appropriate indices, some of these distribution spaces coincide with Hardy spaces, which are the focus of this work.…”

Weighted multilinear square functions bounds

“…Square functions associated to this type of operators have been studied in a number of recent works. In Maldonado [34] and Maldonado-Naibo [35], the authors introduce the operators (1.3), and making the natural extension of Semmes's point of view in [38] to prove bounds for a Besov type relative of the square function S (1.2),…”