We develop a new method of proving vector-valued estimates in harmonic analysis, which we like to call "the helicoidal method". As a consequence of it, we are able to give affirmative answers to several questions that have been circulating for some time. In particular, we show that the tensor product BHT ⊗ Π between the bilinear Hilbert transform BHT and a paraproduct Π satisfies the same L p estimates as the BHT itself, solving completely a problem introduced in [MPTT04]. Then, we prove that for "locally L 2 exponents" the corresponding vector-valued − −− → BHT satisfies (again) the same L p estimates as the BHT itself. Before the present work there was not even a single example of such exponents.Finally, we prove a bi-parameter Leibniz rule in mixed norm L p spaces, answering a question of Kenig in nonlinear dispersive PDE.

We extend the helicoidal method from [BM15] to the quasi-Banach context, proving in this way multiple Banach and quasi-Banach vector-valued inequalities for paraproducts Π and for the bilinear Hilbert transform BHT . As an immediate application, we obtain mixed norm estimates for Π ⊗ Π in the whole range of Lebesgue exponents.One of the novelties in the quasi-Banach framework (that is, when 0 < r < 1), which we expect to be useful in other contexts as well, is the "linearization" of the operator k |T (f k , g k )| r 1/r , achieved by dualizing its weak-L p quasinorms through L r (see Proposition 8). Another important role is played by the sharp evaluation of the operatorial norm TI 0 (f · 1F , g · 1G) · 1 H ′ r , which is obtained by dualizing the weak-L p quasinorms through L τ , with τ ≤ r. In the Banach case, the linearization of the operator and the sharp estimates for the localized operatorial norm can be both achieved through the classical (generalized restricted type) L 1 dualization.

We use the very recent approach developed by Lacey in [23] and extended by Bernicot-Frey-Petermichl in [3], in order to control Bochner-Riesz operators by a sparse bilinear form. In this way, new quantitative weighted estimates, as well as vector-valued inequalities are deduced