How is interpolation used in the proof of Lemma 4.1 in Tao's article Endpoint Strichartz Estimates?

Question

In the proof of Lemma 4.1, pp. 962–963 in "Endpoint Strichartz Estimates" by Tao and Keel (1997) (see MR1646048 or Zbl 0922.35028), the authors first proved the statements hold for some boundary and vertex values of $(a,b)$, and then it was claimed that the statements of the Lemma hold true for points in some neighborhood of $(r,r)$ since $(r,r)$ is in the convex hull determined by those boundary and vertex values of $(a,b)$. This is due to interpolation for bilinear forms, according to the expression in the article.

Question. I wonder what is interpolation for bilinear forms and how it is used here. Is there some books including this? I merely heard of interpolation for operators between function spaces, and interpolation for functionals of single variable as well. But never saw interpolation for bilinear forms before...

Answer 1

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 $(X_0,X_1)_{[\theta]}$.

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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.