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Square function inequality for a class of Fourier integral operators satisfying cinematic curvature conditions

  • Chuanwei Gao , Changxing Miao EMAIL logo and Jianwei-Urbain Yang
Published/Copyright: July 16, 2020
Forum Mathematicum
From the journal Forum Mathematicum Volume 32 Issue 6

Abstract

In this paper, we establish an improved variable coefficient version of the square function inequality, by which the local smoothing estimate Lαp→Lp for the Fourier integral operators satisfying cinematic curvature condition is further improved. In particular, we establish almost sharp results for 2<p⩽3 and push forward the estimate for the critical point p=4. As a consequence, the local smoothing estimate for the wave equation on the manifold is refined. We generalize the results in [S. Lee and A. Vargas, On the cone multiplier in ℝ3, J. Funct. Anal. 263 2012, 4, 925–940; J. Lee, A trilinear approach to square function and local smoothing estimates for the wave operator, preprint 2018, https://arxiv.org/abs/1607.08426v5] to its variable coefficient counterpart. The main ingredients in the argument includes multilinear oscillatory integral estimate [J. Bennett, A. Carbery and T. Tao, On the multilinear restriction and Kakeya conjectures, Acta Math. 196 2006, 2, 261–302] and decoupling inequality [D. Beltran, J. Hickman and C. D. Sogge, Variable coefficient Wolff-type inequalities and sharp local smoothing estimates for wave equations on manifolds, Anal. PDE 13 2020, 2, 403–433].

Keywords: Fourier integral operator; cinematic curvature condition; local smoothing; decoupling inequality
MSC 2010: 35S30; 35L05

Communicated by Christopher D. Sogge

Funding source: China Postdoctoral Science Foundation

Award Identifier / Grant number: 8206300279

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11831004

Award Identifier / Grant number: 11901032

Funding statement: C. Gao was supported by Chinese Postdoc Foundation grant 8206300279. C. Miao was supported by NSFC Grant 11831004. J. Yang was supported by NSFC grant 11901032, and Beijing Institute of Technology Research Fund Program for Young Scholars.

A Appendix

In this section, we will prove Lemma 5.4.

Proof of Lemma 5.4.

Due to the fast decay of the weight wR away from |z|⩾R12+ε4, it suffices to consider

(∑ν∥𝟏{|x|⩽R12+ε4}⁢(Ez¯ν⁢f)∥L2⁢(wBR)12)12.

Freeze time t0, and note that

(A.1)Ez¯ν⁢f⁢(x,t0)=∫ei⁢〈x,η〉⁢χz¯,ν⁢(η)⁢(Ez¯⁢f)∧⁢(η,t0)⁢d⁢η,

where

Ez¯⁢f⁢(x,t0)=∫ℝ2ei⁢(〈x,η〉+t0⁢hz¯⁢(η))⁢a2,z¯⁢(η)⁢f⁢(η)⁢d⁢η,χz¯,ν⁢(η)=χν⁢(Ψλ⁢(z¯,η)).

We further decompose

(A.2)Ez¯⁢f⁢(⋅,t0)=𝟏{|x|⩽R12+ε2}⁢(⋅)⁢Ez¯⁢f⁢(⋅,t0)+𝟏{|x|>R12+ε2}⁢(⋅)⁢Ez¯⁢f⁢(⋅,t0).

It remains to estimate

∫ei⁢〈x,η〉⁢χz¯,ν⁢(η)⁢(𝟏{|x|⩽R12+ε2}⁢(⋅)⁢Ezk⁢f⁢(⋅,t0))∧⁢(η)⁢d⁢η.

In fact, for |x~|⩽R12+ε4, the contribution of the second term in (A.2) to (A.1) equals

∫χ^z¯,ν⁢(x~-y)⁢𝟏{|x|>R12+ε2}⁢(y)⁢Ez¯⁢f⁢(y,t0)⁢d⁢y⩽R-ε⁢N⁢∥f∥L2.

Now, unfreezing t0, by Plancherel’s theorem, we have

(∑ν∥∫ei⁢〈x,η〉⁢χz¯,ν⁢(η)⁢(𝟏{|x|⩽R12+ε2}⁢(⋅)⁢Ez¯⁢f⁢(⋅,t))∧⁢(η)⁢d⁢η∥L2⁢(wBR)2)12≲∥Ez¯⁢f∥L2⁢(wBR).

This completes the proof of Lemma 5.4. ∎

Acknowledgements

The authors would like to thank David Beltran and Christopher Sogge for their helpful discussion and suggestions. The authors are also deeply grateful to the anonymous referees for their invaluable comments which helped improve the paper greatly.

References

[1] D. Beltran, J. Hickman and C. D. Sogge, Sharp local smoothing estimates for Fourier integral operators, preprint (2018), https://arxiv.org/abs/1812.11616v1. 10.1007/978-3-030-72058-2_2Search in Google Scholar

[2] D. Beltran, J. Hickman and C. D. Sogge, Variable coefficient Wolff-type inequalities and sharp local smoothing estimates for wave equations on manifolds, Anal. PDE 13 (2020), no. 2, 403–433. 10.2140/apde.2020.13.403Search in Google Scholar

[3] J. Bennett, A. Carbery and T. Tao, On the multilinear restriction and Kakeya conjectures, Acta Math. 196 (2006), no. 2, 261–302. 10.1007/s11511-006-0006-4Search in Google Scholar

[4] J. Bourgain, Lp-estimates for oscillatory integrals in several variables, Geom. Funct. Anal. 1 (1991), no. 4, 321–374. 10.1007/BF01895639Search in Google Scholar

[5] J. Bourgain and C. Demeter, The proof of the l2 decoupling conjecture, Ann. of Math. (2) 182 (2015), no. 1, 351–389. 10.4007/annals.2015.182.1.9Search in Google Scholar

[6] J. Bourgain, C. Demeter and L. Guth, Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three, Ann. of Math. (2) 184 (2016), no. 2, 633–682. 10.4007/annals.2016.184.2.7Search in Google Scholar

[7] J. Bourgain and L. Guth, Bounds on oscillatory integral operators based on multilinear estimates, Geom. Funct. Anal. 21 (2011), no. 6, 1239–1295. 10.1007/s00039-011-0140-9Search in Google Scholar

[8] C. Gao, C. Miao and J.-U. Yang, Improved variable coefficient square functions and local smoothing of Fourier integral operators, preprint (2019), https://arxiv.org/abs/1901.01487. Search in Google Scholar

[9] L. Guth, J. Hickman and M. Iliopoulou, Sharp estimates for oscillatory integral operators via polynomial partitioning, Acta Math. 223 (2019), no. 2, 251–376. 10.4310/ACTA.2019.v223.n2.a2Search in Google Scholar

[10] L. Guth, H. Wang and R. Zhang, A sharp square function estimate for the cone in ℝ3, preprint (2019), https://arxiv.org/abs/1909.10693. 10.4007/annals.2020.192.2.6Search in Google Scholar

[11] L. Hörmander, Oscillatory integrals and multipliers on F⁢Lp, Ark. Mat. 11 (1973), 1–11. 10.1007/BF02388505Search in Google Scholar

[12] J. Lee, A trilinear approach to square function and local smoothing estimates for the wave operator, preprint (2018), https://arxiv.org/abs/1607.08426v5. 10.1512/iumj.2020.69.8057Search in Google Scholar

[13] S. Lee, Linear and bilinear estimates for oscillatory integral operators related to restriction to hypersurfaces, J. Funct. Anal. 241 (2006), no. 1, 56–98. 10.1016/j.jfa.2006.05.011Search in Google Scholar

[14] S. Lee and A. Vargas, On the cone multiplier in ℝ3, J. Funct. Anal. 263 (2012), no. 4, 925–940. 10.1016/j.jfa.2012.05.010Search in Google Scholar

[15] G. Mockenhaupt, A. Seeger and C. D. Sogge, Local smoothing of Fourier integral operators and Carleson–Sjölin estimates, J. Amer. Math. Soc. 6 (1993), no. 1, 65–130. 10.2307/2152795Search in Google Scholar

[16] W. Schlag and C. D. Sogge, Local smoothing estimates related to the circular maximal theorem, Math. Res. Lett. 4 (1997), no. 1, 1–15. 10.4310/MRL.1997.v4.n1.a1Search in Google Scholar

[17] C. D. Sogge, Propagation of singularities and maximal functions in the plane, Invent. Math. 104 (1991), no. 2, 349–376. 10.1007/BF01245080Search in Google Scholar

Received: 2020-03-06
Revised: 2020-05-31
Published Online: 2020-07-16
Published in Print: 2020-11-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Square function inequality for a class of Fourier integral operators satisfying cinematic curvature conditions
  3. Homological epimorphisms, homotopy epimorphisms and acyclic maps
  4. Summation formulae involving Stirling and Lah numbers
  5. Two-weighted inequalities for maximal operators related to Schrödinger differential operator
  6. 𝐿𝑝-estimates for rough bi-parameter Fourier integral operators
  7. Variation and oscillation inequalities for commutators in two-weight setting
  8. Convexity of sets in metric Abelian groups
  9. Operations that preserve integrability, and truncated Riesz spaces
  10. Very degenerate elliptic equations under almost critical Sobolev regularity
  11. Higher integrability near the initial boundary for nonhomogeneous parabolic systems of 𝑝-Laplacian type
  12. When the image of a derivation on a uniformly complete 𝑓-algebra is contained in the radical
  13. Dyadic bilinear estimates and applications to the well-posedness for the 2D Zakharov–Kuznetsov equation in the endpoint space 𝐻−1/4
  14. Indefinite Einstein metrics on nice Lie groups
  15. Automorphic Schwarzian equations
  16. Smoothing theorems for Radon transforms over hypersurfaces and related operators
  17. Metrical universality for groups
https://doi.org/10.1515/forum-2020-0061
Keywords for this article
Fourier integral operator; cinematic curvature condition; local smoothing; decoupling inequality

Articles in the same Issue

  1. Frontmatter
  2. Square function inequality for a class of Fourier integral operators satisfying cinematic curvature conditions
  3. Homological epimorphisms, homotopy epimorphisms and acyclic maps
  4. Summation formulae involving Stirling and Lah numbers
  5. Two-weighted inequalities for maximal operators related to Schrödinger differential operator
  6. 𝐿𝑝-estimates for rough bi-parameter Fourier integral operators
  7. Variation and oscillation inequalities for commutators in two-weight setting
  8. Convexity of sets in metric Abelian groups
  9. Operations that preserve integrability, and truncated Riesz spaces
  10. Very degenerate elliptic equations under almost critical Sobolev regularity
  11. Higher integrability near the initial boundary for nonhomogeneous parabolic systems of 𝑝-Laplacian type
  12. When the image of a derivation on a uniformly complete 𝑓-algebra is contained in the radical
  13. Dyadic bilinear estimates and applications to the well-posedness for the 2D Zakharov–Kuznetsov equation in the endpoint space 𝐻−1/4
  14. Indefinite Einstein metrics on nice Lie groups
  15. Automorphic Schwarzian equations
  16. Smoothing theorems for Radon transforms over hypersurfaces and related operators
  17. Metrical universality for groups
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