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In some old lecture notes on the Restriction and Kakeya conjectures (Notes 5, specifically), Terence Tao gives a proof of the restriction conjecture (for the sphere) in two dimensions via a bilinear argument. Specifically, he proves the weak-type estimate $$\|\widehat{\chi_{\Omega}d\sigma}\|_{L^{q}(\mathbb{R}^{2})}\lesssim |\Omega|^{1/p},$$ where $\Omega\subset S^{1}$ and $q>4$, $q=3p'$.

Briefly, the idea is to square the above estimate, and instead consider estimates for the bilinear quantities $\|\widehat{fd\sigma}\widehat{gd\sigma}\|_{L^{\infty}(\mathbb{R}^{2})}$ and $\|\widehat{fd\sigma}\widehat{gd\sigma}\|_{L^{2}(\mathbb{R}^{2})}$, where $f,g$ are functions on $S^{1}$. The first is trivial. The second is proven by first showing that if $f,g$ are supported on distinct $\theta$-arcs with separation $\sim\theta$, then $$\|\widehat{fd\sigma}\widehat{gd\sigma}\|_{L^{2}(\mathbb{R}^{2})}\lesssim\theta^{-1/2}\|f\|_{L^{2}(S^{1})}\|g\|_{L^{2}(S^{1})}$$ To exploit this estimate, he then introduces a Whitney decomposition of $S^{1}$ into dyadic arcs, defining distinct arcs $I\simeq J$ if $I$ and $J$ belong to the same generation $A_{n}$, are not adjacent, but their parents are. From the observation $$\widehat{\chi_{\Omega}d\sigma}\widehat{\chi_{\Omega}d\sigma}=\sum_{n>1}\sum_{I,J\in A_{n}: I\sim J}\widehat{\chi_{\Omega}d\sigma_{I}}\widehat{\chi_{\Omega}d\sigma_{J}}$$ one obtains via almost orthogonality that $$\|\sum_{I,J\in A_{n} : I\sim J}\widehat{\chi_{\Omega}d\sigma_{I}}\widehat{\chi_{\Omega}d\sigma_{J}}\|_{L^{2}(\mathbb{R}^{2})}\lesssim\left(\sum_{I,J\in A_{n} : I\sim J}\|\widehat{\chi_{\Omega}d\sigma}\widehat{\chi_{\Omega}d\sigma_{J}}\|_{L^{2}(\mathbb{R}^{2})}^{2}\right)^{1/2}\lesssim 2^{n/2}\left(\sum_{I,J\in A_{n} : I\sim J}|\Omega\cap I||\Omega\cap J|\right)^{1/2}$$ One concludes by Holder's inequality and summing over $n$.

At the end of this proof, Tao briefly mentions that one can prove the Bochner-Riesz conjecture in two dimensions by a similar argument. Can anyone provide me with a reference for where this is sketched out?

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Lee's 2006 Duke Math. J. paper "Improved bounds for Bochner-Riesz and maximal Bochner-Riesz operators" deals with the bilinear approach to Bochner-Riesz multipliers, using the Whitney decomposition argument described in Tao's notes. It is mainly concerned with the higher dimensional case, but it is easy to work through things in the plane. The main ingredient is a bilinear multiplier estimate, which can be proved directly from the bilinear restriction estimate via a Fubini-type argument.

It should be mentioned that the bilinear approach is implicit in many of the early works in the subject, such as the foundational papers of Carleson-Sjolin and Hormander.

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  • $\begingroup$ Welcome to mathoverflow. Your answer looks nice. If you could add a link to Lee's paper (journal or arXiv version), it would be even better. $\endgroup$ Mar 1, 2017 at 20:53

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