Quasi-Banach Valued Inequalities via the Helicoidal method

TL;DR

This paper extends the helicoidal method to the quasi-Banach setting, establishing vector-valued inequalities for paraproducts and the bilinear Hilbert transform, with applications to mixed norm estimates.

Contribution

It introduces a novel linearization technique for quasi-Banach operators and provides sharp localized operator norm estimates, advancing analysis in the quasi-Banach framework.

Findings

Established vector-valued inequalities for paraproducts and BHT in quasi-Banach spaces.

Developed a new linearization method for quasi-Banach operators using dualization.

Obtained sharp bounds for localized operator norms in the quasi-Banach setting.

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 $\| T_{I_0}(f \cdot \mathbf{1}_F, g \cdot \mathbf{1}_G) \cdot \mathbf{1}_{H'}\|_r$, which is obtained by dualizing the weak-$L^p$ quasinorms through $L^\tau$, with $\tau \leq r$. 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.