$T$ and $TT^*$ are equally bounded with ${\left\VertT\right\Vert}^2={\left\VertTT^*\right\Vert}$. Thus, the two estimates are equivalent $$\begin{aligned}
 \normQRP{Tf}{q}{r}{r'}    &\leq C \normA{f}{2}, \quad\qquad \forall f \in \spaceLa{2}, \label{est: T}\\
 \normQRP{TT^*F}{q}{r}{r'} &\leq C^2 \normQRP{F}{q'}{r'}{r}, \quad \forall F \in \spaceLqrpd, \label{est: TT*}\end{aligned}$$ where ${\left\VertT\right\Vert} \leq C$. Moreover, if $\trip$ and $\tript$ are two exponent triplets for which holds, then by the factorization property of $TT^*$ we obtain the consequence $$\begin{aligned}
 \label{est: TT* 2}
 \normQRP{TT^*F}{q}{r}{p} \leq C^2 \normQRP{F}{\tilde q'}{\tilde r'}{ \tilde p'}, \quad \forall F \in \spaceLqrptd.\end{aligned}$$ By duality, is equivalent to $$\begin{aligned}
    \Int \langle TT^*[F](t), G(t) \rangle dt \leq C^2 \normQRP{F}{q'}{r'}{r}\normQRP{G}{q'}{r'}{r}, \quad \forall F, \; \forall G \in \spaceLqrpd,\end{aligned}$$ which in turn simplifies to $$\begin{aligned}
    \Int\Int \langle U(s)^*F, U(t)^*G \rangle dsdt \leq C^2 \normQRP{F}{q'}{r'}{r}\normQRP{G}{q'}{r'}{r},\end{aligned}$$ Consider the bilinear form $$B(F,G) = \iint_{s<t}  \langle U(s)^*F, U(t)^*G \rangle dsdt.$$ By symmetry, the last inequality simplifies to $$\begin{aligned}
 \label{eq: B 1}
   {\left\vertB(F,G)\right\vert} \leq C^2 \normQRP{F}{q'}{r'}{r}\normQRP{G}{q'}{r'}{r}, \quad \forall F, \; \forall G \in \spaceLqrpd.\end{aligned}$$

Analogously, we have that the inhomogeneous estimate $$\begin{aligned}
    \normQRP{W(t)F}{q}{r}{p} \lesssim \normQRP{F}{\tilde q'}{\tilde r'}{\tilde p'}, \quad \forall F \in \spaceLqrptd,\end{aligned}$$ is equivalent to the estimate $$\begin{aligned}
 \label{eq: B 2}
   {\left\vertB(F,G)\right\vert} \lesssim \normQRP{F}{\tilde q'}{\tilde r'}{\tilde p'} & \normQRP{G}{q'}{r'}{r}, \\
   &\forall F \in \spaceLqrptd ,\; \forall G \in \spaceLqrpd. 