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

By Tao T., Vargas A.

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Math. 93 (1971), 107–115. D. Foschi, S. Klainerman, Homogeneous L2 bilinear estimates for wave equations, preprint. M. Keel, T. Tao, Endpoint Strichartz estimates, Amer. Math. , to appear. S. Klainerman, M. Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math. 46:9 (1993), 1221–1268. S. Klainerman, M. Machedon, Remark on Strichartz-type inequalities, With appendices by Jean Bourgain and Daniel Tataru, Internat. Math. Res. Notices 5 (1996), 201–220.

Tao, Low regularity semilinear wave equations, Comm. , to appear. [TV] T. Tao, A. Vargas, A bilinear approach to cone multipliers I. Restriction estimates, GAFA, in this issue. [TVV] T. Tao, A. Vargas, L. Vega, A bilinear approach to the restriction and Kakeya conjectures, J. Amer. Math. Soc. 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.

The latter estimate is dual to the former (with p replaced by p ), so it suffices to prove the former estimate. By interpolation it suffices to prove this estimate for p = 1, p = 2, and p = ∞. When p = 2 the estimate is immediate from Plancherel’s theorem. When p = 1 or p = ∞ the lemma follows from the triangle inequality and the estimates Pk Fk 1 Fk 1 , Pk Fk ∞ Fk ∞ , which follow from Young’s inequality and standard estimates on the kernel of Pk . Vol. 10, 2000 BILINEAR CONE MULTIPLIERS II. APPLICATIONS 255 We also recall a pointwise estimate of Rubio de Francia, which is closely related to the standard square function estimate for equally-spaced cubes in frequency space [Co3,1].

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