By Tao T., Vargas A.
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Sogge, Local smoothing estimates related to the circular maximal theorem, Math. Res. Lett. 4 (1997) 1–15. ¨ lin, Regularity of solutions to Schr¨ [Sj] P. Sjo odinger equations, Duke Math. J. 55 (1987), 699–715. M. Stein, Harmonic Analysis, Princeton University Press, 1993. S. Strichartz, Restriction of Fourier transform to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), 705– 774. [T1] T. Tao, The Bochner-Riesz conjecture implies the Restriction conjecture, Duke Math.
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We consider the contribution of a single cube Q in (51). To begin with, let us assume that Q contains the origin. We have e−2πix·ξ φI (ξ)fˇm (ξ)dξ φˆI ∗ fm (x) = I⊂Em I⊂Em = e−2πix·ξI e−2πix·(ξ−ξI ) φ(ξ − ξI )fˇm (ξ) dξ . As in [TV], we perform a Taylor expansion of the phase e−2πix·(ξ−ξI ) = 1 γ! −2πix γ N N (ξ − ξI ) γ γ where γ is a multi-index. The term γ = 0 should be viewed as the main term. We thus have φˆI ∗ fm (x) m I⊂Em = γ 1 γ! −2πix N γ e−2πix·ξI φγ (ξ − ξI )fˇm (ξ)dξ m I⊂Em where φγ (ξ) = (N ξ)γ φ(ξ) .
A bilinear approach to cone multipliers II by Tao T., Vargas A.