Abstract
This paper is devoted to \(L^2\) estimates for trilinear oscillatory integrals of convolution type on \({\mathbb {R}}^2\). The phases in the oscillatory factors include smooth functions and polynomials. We shall establish sharp \(L^2\) decay estimates of trilinear oscillatory integrals with smooth phases, and then give \(L^2\) uniform estimates for these integrals with polynomial phases.
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References
Carbery, A., Christ, M., Wright, J.: Multidimensional van der Corput and sublevel estimates. J. Am. Math. Soc. 12, 981–1015 (1999)
Carbery, A., Wright, J.: What is van der Corput’s lemma in higher dimensions? Publ. Mat. (2002). https://doi.org/10.5565/PUBLMAT_ESCO02_01
Christ, M.: Hilbert transforms along curves: I. Nilpotent groups. Ann. Math. 122, 575–596 (1985)
Christ, M.: Bounds for multilinear sublevel sets via Szemeredi’s theorem (2011). arXiv:1107.2350
Christ, M.: Multilinear oscillatory integrals via reduction of dimension (2011). arXiv:1107.2352
Christ, M., Li, X., Tao, T., Thiele, C.: On multilinear oscillatory integrals, nonsingular and singular. Duke Math. J. 130, 321–351 (2005)
Christ, M., Oliveira e Silva, D.: On trilinear oscillatory integrals. Rev. Mat. Iberoam. 30, 667–684 (2014)
Greenblatt, M.: A direct resolution of singularities for functions of two variables with applications to analysis. J. Anal. Math. 92, 233–257 (2004)
Greenblatt, M.: Sharp \(L^2\) estimates for one-dimensional oscillatory integral operators with \(C^\infty \) phase. Am. J. Math. 127, 659–695 (2005)
Gressman, P.T.: Uniform geometric estimates of sublevel sets. J. Anal. Math. 115, 251–272 (2011)
Gressman, P.T.: Damping oscillatory integrals by the Hessian determinant via Schrödinger. Math. Res. Lett. 23, 405–430 (2016)
Gressman, P.T., Xiao, L.: Maximal decay inequalities for trilinear oscillatory integrals of convolution type. J. Funct. Anal. 271, 3695–3726 (2016)
Hörmander, L.: Oscillatory integrals and multipliers on \(FL^p\). Ark. Mat. 11, 1–11 (1973)
Li, X.: Bilinear Hilbert transforms along curves I: the monomial case. Anal. PDE 6, 197–220 (2013)
Phong, D.H., Stein, E.M.: The Newton polyhedron and oscillatory integral operators. Acta Math. 179, 105–152 (1997)
Phong, D.H., Stein, E.M., Sturm, J.A.: Multilinear level set operators, oscillatory integral operators, and Newton polyhedra. Math. Ann. 319, 573–596 (2001)
Ricci, F., Stein, E.M.: Harmonic analysis on nilpotent groups and singular integrals \(I\): oscillatory integrals. J. Funct. Anal. 73, 179–194 (1987)
Rychkov, V.S.: Sharp \(L^2\) bounds for oscillatory integral operators with \(C^\infty \) phases. Math. Z. 236, 461–489 (2001)
Shi, Z.S.H., Xu, S.Z., Yan, D.Y.: Damping estimates for oscillatory integral operators with real-analytic phases and its applications. Forum Math. 31, 843–865 (2019)
Shi, Z.S.H., Yan, D.Y.: Sharp \(L^{p}\)-boundedness of oscillatory integral operators with polynomial phases. Math. Z. 286, 1277–1302 (2017)
Stein, E.M.: Harmonic Analysis: Real variable methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993).. (MR 95c:42002)
Varchenko, A.: Newton polyhedra and estimations of oscillatory integrals. Funct. Anal. Appl. 18, 175–196 (1976)
Xiao, L.: Sharp estimates for trilinear oscillatory integrals and an algorithm of two-dimensional resolution of singularities. Rev. Mat. Iberoam. 33, 67–116 (2017)
Acknowledgements
We would like to thank Dr. Lechao Xiao for his useful comments and nice suggestions. The second author was supported in part by the Natural Science Foundation of China under Grant No. 11701573, and the Natural Science Foundation of Hunan Province under Grant No. 2020JJ5683.
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Deng, Y., Shi, Z. & Yan, D. \(L^2\) Estimates of Trilinear Oscillatory Integrals of Convolution Type on \({\mathbb {R}}^2\). J Geom Anal 32, 190 (2022). https://doi.org/10.1007/s12220-022-00926-y
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DOI: https://doi.org/10.1007/s12220-022-00926-y