Calderon-Vaillancourt--type theorem for bilinear operators

Bi-parameter and bilinear Calderón–Vaillancourt theorem with subcritical order

“…When ( p , p ) is taken from each of the four regions, the corresponding upper bound for m is shown there. For (b), note that we have BS m , ⊆ BBS m , for m ≤ , so this follows directly from the result when σ(x, ξ, η) ∈ BS m , in [24].…”

“…We should mention in the bilinear (one-parameter) case, Bényi, Bernicot, Maldonado, Naibo and Torres [2] established the boundedness for m < m(p, q) and Miyachi and Tomita [24] proved the boundedness at the critical case when m = m(p, q).…”

“…Subsequently, the symbolic calculus for bilinear pseudo-differential operators was studied, e.g., in the works [2,[21][22][23] motivated by the bilinear Calderón-Zygmund theory developed in, e.g., [5,13,18] and references therein. In particular, critical order for boundedness of bilinear pseudo-differential operators with symbols BS m , has been considered in [2,24].…”

Boundedness and Compactness for the Commutators of Bilinear Operators on Morrey Spaces

“…Moreover, Calderón-Zygmund operators are "essentially the same" as pseudodifferential operators with symbols in the subclass BS 0 1,δ , 0 ≤ δ < 1, see [10]. On bilinear pseudodifferential operators, one can see [6,7,9,12,13,15,26,27,49].…”

$$L^p$$-Boundedness and $$L^p$$-Nuclearity of Multilinear Pseudo-differential Operators on $${\mathbb {Z}}^n$$ and the Torus $${\mathbb {T}}^n$$

“…The nuclearity of pseudo-differential operators on R n has been treated in details by Aoki [2] and Rempala [54]. Multilinear pseudo-differential operators studied by several authors including Bényi, Maldonado, Naibo, and Torres, [3,4], Michalowski, Rule and Staubach, Miyachi and Tomita [41,42,43,44] and references therein. It is worth mentioning that the multilinear analysis for multilinear multipliers of the form T a (f )(x) = R nr e i2πx·(η 1 +···+ηr) a(η) f 1 (η 1 ) · · · f r (η r )dη, x ∈ R n , (1.6) born with the multilinear results by Coifman and Meyer (see [12,13]), where it was shown that the condition |∂ α 1 η 1 ∂ α 2 η 2 · · · ∂ αr ηr a(η 1 , η 2 , · · · , η r )| ≤ C α (|η 1 | + |η 2 | + · · · + |η r |) −|α| , (1.7)…”