A new proof of the bilinear T(1) Theorem

Abstract

A new simple proof of the bilinear T(1) Theorem in the spirit of the proof of Coifman-Meyer of the celebrated result of David and Journé in the linear case is presented. This new proof is obtained independently of the linear T(1) Theorem by combining recent bilinear square function bounds and a paraproduct construction.

Linear and bilinear 𝑇(𝑏) theorems à la Stein

“…We note that it suffices to verify (2.4) for x 1 = x 2 ; see [11]. The statement of the T (1) theorem of David and Journé [4] is the following.…”

A bilinear T(b) theorem for singular integral operators

“…Following the same idea, David-Jounrné-Semmes [13] proved the Tb theorem by constructing a para-accretive version of the Bony paraproduct. In [24], we constructed a bilinear Bony-type paraproduct, which allowed us to transition from a reduce bilinear T1 theorem to a full T1 theorem. Here we construct a bilinear paraproduct in a para-accretive function setting.…”

“…The need of Theorem 1.2 to prove Theorem 1.1 comes about in a paraproduct construction used to decompose our bilinear singular integral operator T (as in Theorem 1.1). To prove Theorem 1.1, we follow the ideas in [12,13,24] to write T = S + L 0 + L 1 + L 2 , where M b 0 S(b 1 , b 2 ) = M b 1 S * 1 (b 0 , b 2 ) = M b 2 S * 2 (b 1 , b 0 ) = 0 and L 0 , L 1 , L 2 are bilinear paraproducts. We construct these paraproducts in Section 6 so that they satisfy…”

Multilinear local Tb theorem for square functions

“…This type of decomposition was originally done by Coifman-Meyer in [5], and then in the bilinear setting in [16]. Since 1 < p 1 , .…”