Almost orthogonality and a class of bounded bilinear pseudodifferential operators

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MODULATION SPACES AND A CLASS OF BOUNDED MULTILINEAR PSEUDODIFFERENTIAL OPERATORS (2005, background): “It was shown in [ 2 ] that condition (4.1) does not necessarily yield an operator Ts that is bounded from L2 ◊ L2 into L1, or more generally from Lp ◊ Lq into Lr for 1 p + 1 q = 1 r . Here”
Rough Bilinear Singular Integrals (2015, background): “We prove the assertion by showing that the multiplierσ j = K̂ j associated withTj satisfies the conditions of the Coifman-Meyer multiplier theorem [6], which was extended to the casep 1 by Kenig and Stein [18] and by Grafakos and Torres [16].”
Weighted Norm Inequalities for Paraproducts and Bilinear Pseudodifferential Operators with Mild Regularity (2009, unknown): “Without that factor the result is not true, since there exist x-independent symbols with bounded derivatives that do not produce ΨDOs mapping any L × L into L for 1/r = 1/p + 1/q, 1 ≤ p, q, r < ∞, see Proposition 1 in [5].”