Quasi-Banach valued inequalities via the helicoidal method

We extend the helicoidal method from [1] to the quasi-Banach context, proving in this way multiple Banach and quasi-Banach vector-valued inequalities for paraproducts Pi and for the bilinear Hilbert transform BHT. As an immediate application, we obtain mixed norm estimates for Pi circle times Pi 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 (Sigma(k) vertical bar T(f(k),g(k))vertical bar(r))(1/r), achieved by dualizing its weak-L-P quasi norms through L-T (see Proposition 8). Another important role is played by the sharp evaluation of the operatorial norm parallel to T-I0 (f . 1(F), (g) . 1(G)) . 1(H') parallel to(T), which is obtained by dualizing the weak-L-P quasinorms through L-T, with T <= 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. (C) 2017 Elsevier Inc. All rights reserved.