Abstract
We establish an \(L^\infty \times L^2 \rightarrow L^2\) norm estimate for a bilinear oscillatory integral operator along parabolas incorporating oscillatory factors \(e^{i|t|^{-\beta}}\).
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A. Carbery, M. Christ, J. Wright, Multidimensional van der Corput and sublevel set estimates, J. Amer. Math. Soc., 12 (1999), 981–1015.
S. Chandarana, Lp bounds for hypersingular integral operators along curves, Pacific J. Math., 175 (1996), 389–416.
J. Chen, D. Fan, M. Wang, X. Zhu, Lp bounds for hyper Hilbert transform along curves, Proc. Amer. Math Soc., 136 (2008), 3145–3153.
M. Christ, Hilbert transforms along curves, I: Nilpotent groups, Ann. Math., 122 (1985), 575–596.
M. Christ, Hilbert transforms along curves, II: A flat case, Duke Math. J. 52 (1985), 887–894.
M. Christ, X. Li, T. Tao, C. Thiele, On multilinear oscillatory integrals, singular and nonsingular, Duke Math. J., 130(2) (2005), 321–351.
M. Christ, A. Nagel, E. Stein, S. Wainger, Singular and maximal Radon transforms: analysis and geometry, Ann Math., 150 (1999), 489–577.
R.R. Coifman, Y. Meyer, Commutateurs d′ intégrales singuliéres at opérateuers multilinéaires, Ann. Inst. Fourier, Greenoble 28 (1978), 177–202.
R.R. Coifman, Y. Meyer, Au-delà des opérateurs pseudo-différentiels, Asterisque, 57 (1978).
J. Duoandikoetxea, J.L. Rubio de Francia, Maximal and singular integral operators via Fourier transform estimates, Invent. Math., 84 (1986), 541–561.
E.B. Fabes, N.M. Riviere, Singular integral with mixed homogeneity, Studia. Math., 27 (1966), 19–38.
C. Fefferman, E. Stein, Hpspaces of several variables, Acta Math., 229 (1972), 137–193.
H. Furstenberg, Nonconventional ergodic averages, Proceedings of Symposia in Pure Math., 50 (1990), 43–56
L. Grafakos, X. Li, The disc as a bilinear multiplier, Am. J. Math., 128 (2006), 91–119.
I. Hirschman Jr., On multiplier transformations, Duke Math. J., 26 (1959), 221–242.
L. Hörmander, Oscillatory integrals and multipliers on FLp, Ark. Mat., 11 (1973), 1–11.
B. Host, B. Kra, Nonconventional ergodic averages and nilmanifolds, Ann. Math. (2) 161(1) (2005), 397–488.
M. Lacey, C. Thiele, Lp estimates for the bilinear Hilbert tansform for\(2< p <\infty\), Ann. Math. 146(2) (1997), 693–724.
A. Nagel, I. Vance, S. Wainger, D. Weinberg, Hilbert transforms for convex curves, Duke. Math. J., 50 (1983), 735–744.
D.H. Phong, E.M. Stein, On a stopping process for oscillatory integrals, J. Geom. Anal., 4 (1994), 104–120.
C.D. Sogge, Fourier integrals in classical analysis, Cambridge University Press, (1993).
E. Stein, Harmonic analysis, real-variable methods, orthogonality, and oscillatory integrals, Princeton (1993).
E. Stein, Oscillatory integrals related to Radon-like transforms, J. Fourier Anal. Appl., Kahane special issue (1995), 535–551.
E. Stein, S. Wainger, Problems in harmonic analysis related to curvature, Bull. Am. Math. Soc., 84 (1978), 1239–1295.
C. Thiele, Singular integrals meet modulation invariance survey article, Proceedings of ICM 2002, Higher Education Press, Beijing, 2002, vol. II, 721–732.
S. Wainger, Special trigonometric series in k dimensions, Mem. Am. Math Soc., 59 (1965), AMS.
M. Zielinski, Highly oscillatory integrals along curves, Ph.D. Thesis, University of Wisconsin, Madison, (1985).
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The first author was partially supported by NSF grant of China grant 10671079. The second author was supported by NSF grant DMS-0456976.
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Fan, D., Li, X. A bilinear oscillatory integral along parabolas. Positivity 13, 339–366 (2009). https://doi.org/10.1007/s11117-008-2270-3
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DOI: https://doi.org/10.1007/s11117-008-2270-3