A restriction theorem for oscillatory integral operator with certain polynomial phase

Abstract

We consider the oscillatory integral operator {{\displaystyle T}}_{\alpha ,m}f(x)={\int }_{{ℝ}^n}{\mathrm{e}}^{\mathrm{i}({{\displaystyle x}}_{\mathrm{1}}^{{\alpha }_1}{{\displaystyle y}}_{\mathrm{1}}^m+\cdots +{{\displaystyle x}}_n^{{\alpha }_n}{{\displaystyle y}}_n^m)}f(y)\mathrm{d}y, where the function f is a Schwartz function. In this paper, the restriction theorem on S^{n-\mathrm{1}} for this operator is obtained. Moreover, we obtain a necessary condition which ensures validity of the restriction theorem.

Key words： Restriction theorem; oscillatory integral operator; L² boundedness; optimal estimate; necessary condition

References

1 BennettJ, CarberyA, TaoT. On the multilinear restriction and Kakeya conjectures.Acta Math, 2006, 196(2): 261–302

2 BourgainJ. Harmonic analysis and combinatorics: how much may they contribute to each other.In: Arnold V, Atiyah M, Lax P, Mazur B, eds. Mathematics: Frontiers and Perspectives. Providence: Amer Math Soc, 2000, 13–32

3 BourgainJ, GuthL. Bounds on oscillatory integral operators based on multilinear estimates.Geom Funct Anal, 2011, 21(6): 1239–1295

4 GuthL. A restriction estimate using polynomial partitioning.J Amer Math Soc, 2016, 29(2): 371–413

5 ŁabaI. From harmonic analysis to arithmetic combinatorics.Bull Amer Math Soc (NS), 2008, 45(1): 77–115

6 PhongD H, SteinE M. The Newton polyhedron and oscillatory integral operators.Acta Math, 1997, 179(1): 105–152

7 ShiZ S H, YanD Y. Sharp L^p-boundedness of oscillatory integral operators with polynomial phases.arXiv: 1602.06123

8 SteinE M, MurphyT S. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals.Princeton: Princeton Univ Press, 1993

9 TaoT. The Bochner-Riesz conjecture implies the restriction conjecture.Duke Math J, 1999, 96(2): 363–376

10 TaoT. A sharp bilinear restriction estimate for paraboloids.Geom Funct Anal, 2003, 13(6): 1359–1384

11 TaoT. Some recent progress on the restriction conjecture.In: Brandolini L, Colzani L, Iosevich A, Travaglini G, eds. Fourier analysis and convexity. Applied and Numerical Harmonic Analysis. Boston: Birkhäuser, 2004, 217–243

12 TaoT, VargasA, VegaL. A bilinear approach to the restriction and Kakeya conjectures.J Amer Math Soc, 1998, 11(4): 967–1000

13 TomasP A. A restriction theorem for the Fourier transform.Bull Amer Math Soc, 1975, 81(2): 477–478

14 WolffT. Recent work connected with the Kakeya problem.In: Prospects in Mathematics (Princeton, 1996). Providence: Amer Math Soc, 1999, 129–162

15 WolffT. A sharp bilinear cone restriction estimate.Ann of Math (2), 2001, 153(3): 661–698

16 YangC W. Sharp L^p estimates for some oscillatory integral operators in ℝ1.Illinois J Math, 2004, 48(4): 1093–1103