The Bochner-Riesz conjecture implies the restriction conjecture
Fix \(n\geq 2\). For any \(1\leq p\leq \infty\) and \(\alpha\geq 0\), one uses \(BR(p,\alpha)\) to denote that Bochner-Riesz mean operator \(S^{\delta(p)+\alpha}\) is bounded on \(L^p(\mathbb R^n)\), where \(\delta(p)=\max(n|1/p-1/2|-1/2, 0)\) and \(S^\delta\) is the Bochner-Riesz (mean) multiplier defined by \(\widehat{S^\delta f}=(1-|\xi|^2)_+^\delta \widehat f(\xi)\). The Bochner-Riesz conjecture is the statement that \(BR(p,\alpha)\) holds for all \(1\leq p\leq\infty\), and \(\alpha>0\). One uses \(R(p,\alpha)\) to denote the local restriction estimate \(\|\mathfrak R f\|_{L^p(S^{n-1})}\leq C R^\alpha \|f\|_{L^p(B(0,R))}\) for \(f\) supported in \(B(0,R)\), where \(\mathfrak R f=\widehat f|_{S^{n-1}}\) is the sphere restriction operator. The restriction conjecture is the statement that \(R(p,0)\) holds for all \(1\leq p<2n/(n+1)\). The implication Restriction \(\Rightarrow\) Bochner-Riesz is known for the parabolic analogue of the spherical problem, etc. The author shows the implication in the reverse direction. More precisely, he shows: (i) If \(1\leq p\leq 2n/(n+1)\), then \(BR(p,\alpha)\Rightarrow R(p,2\alpha)\). (ii) If \(1<p<2\) and \(0<\alpha\) is very small, then \(R(p,\alpha)\Rightarrow R(q,0)\) whenever \(1/q>1/p+C/\log 1/\alpha\). As a corollary of these, if \(1\leq p\leq 2n/(n+1)\) and \(BR(p,\alpha)\) holds for all \(\alpha>0\), then \(R(q,0)\) holds for all \(1\leq q<p\). As a further application, he shows the equivalence of the Nikodým and Kakeya maximal operator conjectures.
scientific article; zbMATH DE number 1425182
scientific article