Suppose I have a multilinear map $\Gamma(u,v)$ satisfying \begin{align} \big\| \Gamma(u,v)\big\|_{L^2} &\leq \big\| u\big\|_{L^2} \big\| v\big\|_{L^2} \\ \big\| \Gamma(u,v)\big\|_{L^\infty} &\leq \big\| u\big\|_{L^\infty} \big\| v\big\|_{L^\infty} \end{align} I would like to show that \begin{align} \big\| \Gamma(u,v)\big\|_{L^p} &\leq \big\| u\big\|_{L^p} \big\| v\big\|_{L^p} \end{align} for all $p\in (2,\infty)$. I realize this looks like a multilinear Riesz-Thorin interpolation type theorem, but am having trouble finding showing this/reference. Thanks.
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2$\begingroup$ The theorems in Section 4.4 of Bergh-Lofstrom math.chalmers.se/~bergh/Interpolation.pdf should probably do the trick (though one may have to unpack their notation a bit, as they work in a quite general context). Alternatively, one can simply repeat the proof of the classical Riesz-Thorin theorem. $\endgroup$– Terry TaoApr 8, 2016 at 19:17
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$\begingroup$ It seems like paragraph 10.1 on Calderon's paper matwbn.icm.edu.pl/ksiazki/sm/sm24/sm24110.pdf also contains what you are looking for. $\endgroup$– George ShakanApr 8, 2016 at 19:20
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$\begingroup$ Great! Thanks so much for the help. $\endgroup$– k3thompsApr 9, 2016 at 15:16
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