Keel-Tao implicitly used two multilinear interpolation results.

### Three end point multilinear interpolation

For general Banach spaces this is given as Exercise 3.13.5b in

*Bergh, Jöran; Löfström, Jörgen*, **Interpolation spaces. An introduction**, Grundlehren der mathematischen Wissenschaften. 223. Berlin-Heidelberg-New York: Springer-Verlag. X, 207 p. with 5 figs. DM 60.00; $ 24.60 (1976). ZBL0344.46071.

In the case used for Strichartz estimates, the function spaces involved are (vector valued) $L^p$ spaces, and a complete proof is given in

*O’Neil, Richard*, **Convolution operators and L(p, q) spaces**, Duke Math. J. 30, 129-142 (1963). ZBL0178.47701.

The statement of the Theorem is:

**Thm** Let $A_0, A_1, B_0, B_1, C_0, C_1$ be Banach spaces. Let $T$ be a bilinear mapping with the following bounds:
$$ \begin{align*}
\| T(a,b) \|_{C_0} &\lesssim \|a\|_{A_0} \|b\|_{B_0} & (T:&A_0\times B_0 \to C_0) \\
\| T(a,b) \|_{C_1} &\lesssim \|a\|_{A_0} \|b\|_{B_1} & (T:&A_0\times B_1 \to C_1) \\
\| T(a,b) \|_{C_1} &\lesssim \|a\|_{A_1} \|b\|_{B_0} & (T:&A_1\times B_0 \to C_1) \end{align*} $$
then for $\theta, \theta_A, \theta_B \in (0,1)$ with $\theta = \theta_A + \theta_B$, and $p,q,r\in [1,\infty]$ with $1 \leq \frac1p + \frac1q$, we have
$$ \|T(a,b)\|_{C_{\theta,r}} \lesssim \|a\|_{A_{\theta_A,pr}} \|b\|_{B_{\theta_B,qr}} \quad (T: A_{\theta_A,pr} \times B_{\theta_B,qr} \to C_{\theta, r}) $$
here for $X \in \{A,B,C\}$, the space $X_{\theta,r}$ is the real interpolation space $(X_0,X_1)_{\theta,r}$.

### Two end point multilinear interpolation

For general Banach spaces this is proven as Theorem 4.4.1 in Bergh and Lofstrom. It states

**Thm** Let $A_0, A_1, B_0, B_1, C_0, C_1$ be Banach spaces. Let $T$ be a bilinear mapping with the following bounds:
$$ \begin{align*}
\| T(a,b) \|_{C_0} &\lesssim \|a\|_{A_0} \|b\|_{B_0} & (T:&A_0\times B_0 \to C_0) \\
\| T(a,b) \|_{C_1} &\lesssim \|a\|_{A_1} \|b\|_{B_1} & (T:&A_1\times B_1 \to C_1) \end{align*} $$
then for $\theta \in (0,1)$, we have
$$ \|T(a,b)\|_{C_{[\theta]}} \lesssim \|a\|_{A_{[\theta]}} \|b\|_{B_{[\theta]}} \quad (T: A_{[\theta]} \times B_{[\theta]} \to C_{[\theta]}) $$
here for $X \in \{A,B,C\}$, the space $X_{[\theta]}$ is the complex interpolation space $(X_0,X_1)_{[\theta]}$.

The details of how they are applied for the proof of the Strichartz estimate is too long to include in an MO answer. I wrote up a version as Theorem 4.88 in my lecture notes.