A bilinear approach to cone multipliers II by Tao T., Vargas A. PDF

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.

Vega, Schr¨ odinger Maximal Function and Restriction Properties of the Fourier transform, International Math. Research Notices 16 (1996). [MoVV2] A. Moyua, A. Vargas, L. Vega, Restriction theorems and maximal operators related to oscillatory integrals in R3 , Duke Math. , to appear. [OS] D. Oberlin, H. Smith, A Bessel function multiplier, Proc. AMS, to appear. [OSS] D. Oberlin, H. Smith, C. Sogge, Averages over curves with torsion, Math Res. , to appear. [R] J. Rubio de Francia, Estimates for some square functions of LittlewoodPaley type, Pub.

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) .

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A bilinear approach to cone multipliers II by Tao T., Vargas A.


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