Quasi-Banach valued inequalities via the helicoidal method

Abstract

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 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(fk,gk)|r)1/r, achieved by dualizing its weak-Lp quasinorms through Lr (see Proposition 8). Another important role is played by the sharp evaluation of the operatorial norm ‖TI0(f⋅1F,g⋅1G)⋅1H′‖r, which is obtained by dualizing the weak-Lp quasinorms through Lτ, 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) L1 dualization.