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Sharp \(L^{p}\)-boundedness of oscillatory integral operators with polynomial phases

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Abstract

In this paper, we shall prove the \(L^{p}\) endpoint decay estimates of oscillatory integral operators with homogeneous polynomial phases S in \(\mathbb {R} \times \mathbb {R}\). As a consequence, sharp \(L^{p}\) decay estimates are also obtained when polynomial phases have the form \(S(x^{m_{1}},y^{m_{2}})\) with \(m_1\) and \(m_2\) being positive integers.

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References

  1. Bak, J.G.: An \(L^p-L^q\) estimate for Radon transforms associated with polynomials. Duke Math. J. 101, 259–269 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bak, J.G., Oberlin, D., Seeger, A.: Two endpoint bounds for generalized Radon transforms in the plane. Rev. Mat. Iberoam. 18, 231–247 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beckner, W.: Pitt’s inequality with sharp convolution estimates. Proc. Am. Math. Soc. 136, 1871–1885 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Carbery, A., Christ, M., Wright, J.: Multidimensional van der Corput and sublevel estimates. J. Am. Math. Soc. 12, 981–1015 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Christ, M.: Failure of an endpoint estimate for integrals along curves. In: Garcia-Cuerva, J., Hernandez, E., Soria, F., Torrea, J.L. (eds.) Fourier Analysis and Partial Differential Equations, pp. 163–168. CRC Press, Boca Raton (1995)

    Google Scholar 

  6. Fefferman, C., Stein, E.M.: \(H^{p}\) spaces of several variables. Acta Math. 129, 137–193 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  7. Greenblatt, M.: A direct resolution of singularities for functions of two variables with applications to analysis. J. Anal. Math. 92, 233–257 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Greenblatt, M.: Sharp \(L^2\) estimates for one-dimensional oscillatory integral operators with \(C^\infty \) phase. Am. J. Math. 127, 659–695 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Greenleaf, A., Seeger, A.: Oscillatory and Fourier integral operators with folding canonical relations. Stud. Math. 132, 125–139 (1999)

    MATH  Google Scholar 

  10. Greenleaf, A., Seeger, A.: Oscillatory and Fourier integral operators with degenerate canonical relations. Publ. Mat. 93, 141 (2002)

    MathSciNet  MATH  Google Scholar 

  11. Greenleaf, A., Pramanik, M., Tang, W.: Oscillatory integral operators with homogeneous polynomial phases in several variables. J. Funct. Anal. 244, 444–487 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hörmander, L.: Oscillatory integrals and multipliers on \(FL^p\). Ark. Mat. 11, 1–11 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  13. Jurkat, W.B., Sampson, G.: The complete solution to the \((L^p, L^q)\) mapping problem for a class of oscillating kernels. Indiana Univ. Math. J. 30, 403–413 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lee, S.: Endpoint \(L^p-L^q\) estimates for degenerate Radon transforms in \(\mathbb{R}^2\) associated with real analytic functions. Math. Z. 243, 817–841 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lu, S.Z., Zhang, Y.: Criterion on \(L^p\) boundedness for a class of oscillatory singular integrals with rough kernels. Rev. Mat. Iberoam. 8, 201–219 (1992)

    MathSciNet  MATH  Google Scholar 

  16. Pan, Y.: Hardy spaces and oscillatory singular integrals. Rev. Mat. Iberoam. 7, 55–64 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  17. Pan, Y., Sampson, G., Szeptycki, P.: \(L^{2}\) and \(L^{p}\) estimates for oscillatory integrals and their extended domains. Stud. Math. 122, 201–224 (1997)

    MathSciNet  MATH  Google Scholar 

  18. Phong, D.H., Stein, E.M.: Hilbert integrals, singular integrals, and Radon transforms \(I\). Acta Math. 157, 99–157 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  19. Phong, D.H., Stein, E.M.: Oscillatory integrals with polynomial phases. Invent. Math. 110, 39–62 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  20. Phong, D.H., Stein, E.M.: Models of degenerate Fourier integral operators and Radon transforms. Ann. Math. 140, 703–722 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  21. Phong, D.H., Stein, E.M.: The Newton polyhedron and oscillatory integral operators. Acta Math. 179, 105–152 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  22. Phong, D.H., Stein, E.M.: Damped oscillatory integral operators with analytic phases. Adv. Math. 134, 146–177 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  23. Pramanik, M., Yang, C.W.: \(L^{p}\) decay estimates for weighted oscillatory integral operators on \(\mathbb{R}\). Rev. Mat. Iberoam. 21, 1071–1095 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ricci, F., Stein, E.M.: Harmonic analysis on nilpotent groups and singular integrals \(I\): oscillatory integrals. J. Funct. Anal. 73, 179–194 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  25. Rychkov, V.S.: Sharp \(L^2\) bounds for oscillatory integral operators with \(C^\infty \) phases. Math. Z. 236, 461–489 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  26. Seeger, A.: Degenerate Fourier integral operators in the plane. Duke Math. J. 71, 685–745 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  27. Seeger, A.: Radon transforms and finite type conditions. J. Am. Math. Soc. 11, 869–897 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  28. Sjölin, P.: Convolution with oscillating kernels. Indiana Univ. Math. J. 30, 47–56 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  29. Stein, E.M.: Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)

    MATH  Google Scholar 

  30. Varchenko, A.: Newton polyhedra and estimations of oscillatory integrals. Funct. Anal. Appl. 18, 175–196 (1976)

    MATH  Google Scholar 

  31. Yang, C.W.: Sharp estimates for some oscillatory integral operators on \(\mathbb{R}^1\). Ill. J. Math. 48, 1093–1103 (2004)

    MathSciNet  MATH  Google Scholar 

  32. Yang, C.W.: \(L^p\) improving estimates for some classes of Radon transforms. Trans. Am. Math. Soc. 357, 3887–3903 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, Hunan Province, People’s Republic of China

    Zuoshunhua Shi

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, People’s Republic of China

    Zuoshunhua Shi

  3. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Dunyan Yan

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  1. Zuoshunhua Shi
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  2. Dunyan Yan
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Correspondence to Zuoshunhua Shi.

Additional information

The work of D. Yan was supported in part by National Natural Foundation of China under Grant Nos. 11471309 and 11561062.

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Shi, Z., Yan, D. Sharp \(L^{p}\)-boundedness of oscillatory integral operators with polynomial phases. Math. Z. 286, 1277–1302 (2017). https://doi.org/10.1007/s00209-016-1800-0

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  • DOI: https://doi.org/10.1007/s00209-016-1800-0

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