By Tao T., Vargas A.
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Extra resources for A bilinear approach to cone multipliers II
We now apply the conformal linear transformation τ + ξ1 τ + ξ1 τ − ξ1 j τ − ξ1 (ξ1 , ξ2 , τ ) → − 22j , 2 ξ2 , + 22j . 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 .
2 ! 2 t=1 m I⊂Em am,I,t,γt e−2πix·ξI . Lq (Q) Vol. 10, 2000 BILINEAR CONE MULTIPLIERS II. γ2 ! t=1 |am,I,t,γt |2 1/2 . 1 of [TV] since the points ξI are only within 1/N of the cone rather than being on the cone itself, but the argument extends trivially. γ2 ! t=1 t |φγI t ∗ fm (0)|2 1/2 m I⊂Em This formula applies when Q contains the origin. γ2 ! 2 t |φγI t ∗ fm (y)|2 t=1 1/2 m I⊂Em where Q is now arbitrary and y ∈ Q. γ2 ! Lq (Q) 2 t 2 |φγI t ∗ fm | t=1 1/2 . γ2 ! γ2 ! t=1 t 2 |φγI t ∗ fm | m I⊂Em 1/2 4 .
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. , to appear. [T2] T. 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.