Uniform bounds for the bilinear Hilbert transforms, I

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On a Conjecture of E. M. Stein on the Hilbert Transform on Vector Fields (2007, background): “The reader can find the details on how to construct such a central grid structure in [11].”
Weighted multilinear Hardy operators and commutators (2014, background): “For a more complete account on multilinear operators, we refer to [6], [11], [17] and the references therein.”
Rough Bilinear Singular Integrals (2015, background): “The boundedness of Hθ1,θ2 was proved by Lacey and Thiele [19], [20] while the more relevant, for this problem, uniform in θ1,θ2 boundedness of Hθ1,θ2 was addressed by Thiele [28], Grafakos and Li [15], and Li [21 ].”
Lp theory for outer measures and two themes of Lennart Carleson united (2013, background): “1 with a constant independent of β (see [9]), but only at the expense of considerable additional work.”
Extrapolation for multilinear Muckenhoupt classes and applications (2018, unknown): “Lacey and Thiele [27, 28] (see also [22]) showed that BH maps L1 × L2 → L for 1 < p1, p2 ≤ ∞ and 1 p = 1 p1 + 1 p2 < 3 2 .”
Uniform Bounds for the Bilinear Hilbert Transforms, II (2006, methodology): “In the sequel we will adopt the terminology and notation introduced in [8].”