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We extend the helicoidal method that we previously developed to thequasi-Banach context, proving in this way multiple Banach and quasi-Banachvector-valued inequalities for paraproducts $\Pi$ and for the bilinear Hilberttransform $BHT$. As an immediate application, we obtain mixed norm estimatesfor $\Pi \otimes \Pi$ 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 dualizingits weak-$L^p$ quasinorms through $L^r$. Another important role is played bythe 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 bydualizing the weak-$L^p$ quasinorms through $L^\tau$, with $\tau \leq r$. Inthe Banach case, the linearization of the operator and the sharp estimates forthe localized operatorial norm can be both achieved through the classical(generalized restricted type) $L^1$ dualization.

Quasi-Banach Valued Inequalities via the Helicoidal method