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By Tao T., Vargas A.

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11 (1998), 967–1000. [V] L. Vega, Schr¨odinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc. 102 (1988), 874–878. 258 [W] T. TAO AND A. VARGAS GAFA T. Wolff, Recent work connected with the Kakeya problem, Anniversary Proceedings, Princeton 1996, to appear.

We thus have smoothing for the helix for α < 1/6 + 1/714. A sharp exponent for (45) would imply an essentially sharp result for the helix. 254 T. TAO AND A. VARGAS GAFA It is extremely likely that all these results extend to multipliers which are singular on surfaces similar to the cone. Based on the arguments of this paper, one sufficient condition seems to be that portions of the surface whose normals differ by 2−j can be parabolically rescaled by an amount 1 × 2j × 22j to be of disjoint conic type.

2 2 2 2 One may verify that the norms on both sides of (49) scale the same way under this transformation. Furthermore, the sets Em , Em map to very similar sets but with the role of N replaced by 2−j N . The claim then follows from (46). Vol. 10, 2000 BILINEAR CONE MULTIPLIERS II. APPLICATIONS 247 Applying (49) to (48) and comparing this with (47), we see that we only need to verify that Fj,k 4 2 1/2 Fj,k |Fj,k |2 4 τj,k ∼τj,k 1/2 k where |ψˆm ∗ f |2 Fj,k = 2 (50) 4 1/2 . m∈τj,k By polarizing we may majorize the left-hand side of (50) by Fj,k 4 4 1/2 .

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


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