Quasi-Banach Valued Inequalities via the Helicoidal method

Abstract

We extend the helicoidal method that we previously developed 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 $\Pi \otimes \Pi$ 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 ${\left({\sum }_k|T(f_k,g_k){|}^r\right)}^{1/r}$ by dualizing its weak-$L^p$ quasinorms through $L^r$. Another important role is played by the sharp evaluation of the operatorial norm $\Vert T_{I_0}(f\cdot 1_F,g\cdot 1_G)\cdot 1_{H^{\prime }}{\Vert }_r$, which is obtained by dualizing the weak-$L^p$ quasinorms through $L^{\tau }$, with $\tau \le r$. 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.

46 pages, optimized certain results