Abstract. We summarize our recent work on linear and bilinear restriction conjectures in three dimensions.

THREE-DIMENSIONAL RESTRICTION THEOREMS

TERENCE TAO, ANA VARGAS, AND LUIS VEGA

1. Elliptic surfaces

Let S be the unit sphere or compact subset of the paraboloid in R^{3}, and let R^{∗}(p → q) denote the adjoint restriction estimate

∥f̂dσ∥_{q} ≲ ∥f∥_{p}

where dσ is surface measure on S. We also let R^{∗}(p × p → q/2) denote the bilinear version

∥f̂dσĝdσ∥_{q/2} ≲ ∥f∥_{p}∥g∥_{p}

where f and g have unit-seperated supports.

Conjecture 1.1. (Restriction conjecture) R^{∗}(p → q) holds whenever q ≥ 2p′, q > 3.

Conjecture 1.2. (Bilinear restriction conjecture) R^{∗}(p×p → q/2) holds whenever q ≥ \frac{5}{3}p′, q ≥ 3, \frac{5}{2q} + \frac{1}{p} ≤ 2. In other words, we conjecture R^{∗}(3 × 3 → 3/2) and R^{∗}(2 × 2 → 5/3).

These exponents are best possible.

Existing progress on these conjectures:

• (Tomas-Stein) The restriction conjecture is true whenever q ≥ 4.

• (Bourgain, Wolff, MVV) The restriction conjecture is true whenever p > 7/3 and q > 4 − 2/11.

• (Bourgain, MVV) We have R^{∗}(p × p → 2) if and only if p ≥ 12/7. In particular, the bilinear restriction conjecture is true whenever q ≥ 4.

R^{∗}(p → q) trivially implies R^{∗}(p × p → q/2). Conversely, if R^{∗}(p × p → q/2 − ε) holds and q ≥ 2p′, then R^{∗}(p → q) holds. In particular, the bilinear restriction conjecture implies the restriction conjecture. The proof involves a Whitney decomposition of f̂dσf̂dσ, parabolic scaling and the inequality

∥ ∑_{k} f_{k}∥_{p} ≲ (∑_{k} ∥f_{k}∥_{p}^{p^{∗}})^{1/p^{∗}}

1991 Mathematics Subject Classification. 42B10, 42B25.

which holds whenever f_{k} have frequency supports in disjoint rectangles and 1 ≤ p ≤ ∞; here p^{∗} = min(p, p′).

In [12] we used bilinear versions of arguments in [2, 6, 14] to prove the following:

• The restriction conjecture is true whenever q > 4 − 5/27.

• The restriction conjecture is true whenever p > 170/77 and q > 4 − 2/9.

• We have R^{∗}(2 × 2 → q) for all q > 2 − 5/69.

In [13] we combined the above techniques with the ideas from [5] and the x-ray estimate in [15] to improve these results to

• The restriction conjecture is true whenever q > 4 − 8/31.

• The restriction conjecture is true whenever p > 26/11 and q > 4 − 2/7.

• We have R^{∗}(2 × 2 → q) for all q > 2 − 2/17.

To the best of our knowledge, these results (together with what can be obtained by interpolation) are currently the best known restriction theorems.

As an application, we show that the solution to the free Schrödinger equation converges a.e. to its initial data whenever the data is in H^{s} for s > 15/32. This improves upon the earlier result of [8], which showed it for s > (164 + \sqrt{2})/339; for comparison, the classical results were for s > 1/2.

2. Conic surfaces

We now repeat the preceding discussion, but with S replaced by a small subset of the light cone in R^{3}.

The linear estimates on the cone are completely known:

Theorem 2.1. [1] R^{∗}(p → q) holds if and only if q ≥ 3p′, q > 4.

However, the exact range of bilinear estimates are still open:

Conjecture 2.2. R^{∗}(p × p → q/2) holds whenever q ≥ \frac{5}{3}p′, q ≥ 10/3. In other words, we conjecture R^{∗}(2 × 2 → 5/3).

Thee exponents are sharp.

It is easy to show (by a variant of an argument of Sjölin) that R^{∗}(2×2 → 2). By Radon transform L^{p} mapping properties one can improve this to R^{∗}(12/7×12/7 → 2); this is the same argument which gave the corresponding estimate for elliptic surfaces.

In [5] the estimate R^{∗}(p × p → 2) was proven for all p > 2 − τ for some τ > 0 (we conservatively estimate τ = 13/2408). In [13] we prove this estimate for all p > 2 − 1/20.

In [5] Bourgain observed that this estimate could be used to improve Mockenhaupt’s 1/8 cone multiplier result on L^{4}. Indeed we can lower the order of the cone multiplier from 1/8 + ε to 1/8 − 1/238 + ε.

Similar considerations apply to Sogge’s local smoothing conjecture, the 1/8 + ε loss in derivatives for the wave equation from L_{x}^{4} to L_{x,t}^{4} can be improved to 1/8 − 1/238 + ε. As in [5], the idea is to first improve the square function estimate in [7].

One can also improve the sharp (L^{p}, L^{q}) local smoothing estimates for the wave equation of Schlag and Sogge, with q = 3p′ from q ≥ 5 to q ≥ 5 − 1/20.

References

1. B. Barcelo, On the restriction of the Fourier transform to a conical surface, Trans. Amer. Math. Soc. 292 (1985), 321–333.

2. J. Bourgain, Besicovitch-type maximal operators and applications to Fourier analysis, Geom. and Funct. Anal. 22 (1991), 147–187.

3. J. Bourgain, On the restriction and multiplier problem in R^{3}, Lecture notes in Mathematics, no. 1469. Springer Verlag, 1991.

4. J. Bourgain, A remark on Schrodinger operators, Israel J. Math. 77 (1992), 1–16.

5. J. Bourgain, Estimates for cone multipliers, Operator Theory: Advances and Applications, 77 (1995), 41–60.

6. J. Bourgain, Some new estimates on oscillatory integrals, Essays in Fourier Analysis in honor of E. M. Stein, Princeton University Press (1995), 83–112.

7. G. Mockenhaupt, A note on the cone multiplier, Proc. AMS 117 (1993), 145–152.

8. A. Moyua, A. Vargas, L. Vega, Schrödinger Maximal Function and Restriction Properties of the Fourier transform, International Math. Research Notices 16 (1996).

9. A. Moyua, A. Vargas, L. Vega, Restriction theorems and Maximal operators related to oscillatory integrals in R^{3}, to appear, Duke Math. J.

10. W. Schlag, C. D. Sogge, Local smoothing estimates related to the circular maximal theorem, Math. Res. Lett. 4 (1997): 1–15.

11. C. D. Sogge, Propogation of singularities and maximal functions in the plane, Invent. Math. 104 (1991), 349–376.

12. T. Tao, A. Vargas, L. Vega, A bilinear approach to the restriction and Kakeya conjectures, to appear, JAMS.

13. T. Tao, A. Vargas, A bilinear approach to cone multipliers and applications, in preparation.

14. T. H. Wolff, An improved bound for Kakeya type maximal functions, Revista Mat. Iberoamericana. 11 (1995). 651–674.

15. T. H. Wolff, A mixed norm estimate for the x-ray transform, to appear in Revista Mat. Iberoamericana.

Department of Mathematics, UCLA, Los Angeles, CA 90024 E-mail address: tao@math.ucla.edu

Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid (Spain).

E-mail address: ana.vargas@uam.es

Departamento de Matemáticas, Universidad del País Vasco, Apartado 644, 48080, Bilbao (Spain).

E-mail address: mtpvegol@lg.ehu.es