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May  2016, 15(3): 929-946. doi: 10.3934/cpaa.2016.15.929

## Oscillatory integrals related to Carleson's theorem: fractional monomials

 1 Endenicher Allee 60, 53115, Bonn, Germany

Received  August 2015 Revised  November 2015 Published  February 2016

Stein and Wainger [21] proved the $L^p$ bounds of the polynomial Carleson operator for all integer-power polynomials without linear term. In the present paper, we partially generalise this result to all fractional monomials in dimension one. Moreover, the connections with Carleson's theorem and the Hilbert transform along vector fields or (variable) curves %and a polynomial Carleson operator along the paraboloid are also discussed in details.
Citation: Shaoming Guo. Oscillatory integrals related to Carleson's theorem: fractional monomials. Communications on Pure & Applied Analysis, 2016, 15 (3) : 929-946. doi: 10.3934/cpaa.2016.15.929
##### References:
 [1] M. Bateman, Single annulus $L^p$ estimates for Hilbert transforms along vector fields,, \emph{Rev. Mat. Iberoam.}, 29 (2013), 1021.  doi: 10.4171/RMI/748.  Google Scholar [2] M. Bateman and C. Thiele, $L^p$ estimates for the Hilbert transforms along a one-variable vector field,, \emph{Anal. PDE}, 6 (2013), 1577.  doi: 10.2140/apde.2013.6.1577.  Google Scholar [3] L. Carleson, On convergence and growth of partial sums of Fourier series,, \emph{Acta Math.}, 116 (1966), 135.   Google Scholar [4] H. Carlsson, M. Christ, A. Cordoba, J. Duoandikoetxea, J. L. Rubio de Francia, J. Vance, S. Wainger and D. Weinberg, $L^p$ estimates for maximal functions and Hilbert transforms along flat convex curves in $\R^2$,, \emph{Bull. Amer. Math. Soc. (N.S.)}, 14 (1986), 263.  doi: 10.1090/S0273-0979-1986-15433-9.  Google Scholar [5] Y. Ding and H. Liu, Weighted $L^p$ boundedness of Carleson type maximal operators,, \emph{Proc. Amer. Math. Soc.}, 140 (2012), 2739.  doi: 10.1090/S0002-9939-2011-11110-9.  Google Scholar [6] C. Fefferman, Inequalities for strongly singular convolution operators,, \emph{Acta Math.}, 124 (1970), 9.   Google Scholar [7] C. Fefferman, Pointwise convergence of Fourier series,, \emph{Ann. of Math.}, 98 (1973), 551.   Google Scholar [8] M. Folch-Gabayet and J. Wright, An oscillatory integral estimate associated to rational phases,, \emph{J. Geom. Anal.}, 13 (2003), 291.  doi: 10.1007/BF02930698.  Google Scholar [9] M. Folch-Gabayet and J. Wright, Singular integral operators associated to curves with rational components,, \emph{Trans. Amer. Math. Soc.}, 360 (2008), 1661.  doi: 10.1090/S0002-9947-07-04349-8.  Google Scholar [10] M. Folch-Gabayet and J. Wright, Weak type $(1, 1)$ bounds for oscillatory singular integrals with rational phases,, \emph{Studia Math.}, 210 (2012), 57.  doi: 10.4064/sm210-1-4.  Google Scholar [11] I. I. Hirschman, On multiplier transformations,, \emph{Duke Mathematical Journal}, 26 (1959), 221.   Google Scholar [12] M. Lacey and C. Thiele, A proof of boundedness of the Carleson operator,, \emph{Math. Res. Lett.}, 7 (2000), 361.  doi: 10.4310/MRL.2000.v7.n4.a1.  Google Scholar [13] V. Lie, The (weak-$L^2$) boundedness of the quadratic Carleson operator,, \emph{Geom. Funct. Anal.}, 19 (2009), 457.  doi: 10.1007/s00039-009-0010-x.  Google Scholar [14] V. Lie, The Polynomial Carleson Operator,, arXiv:1105.4504., ().   Google Scholar [15] A. Nagel, N. Riviere and S. Wainger, On Hilbert transforms along curves,, \emph{Bull. Amer. Math. Soc.}, 80 (1974), 106.   Google Scholar [16] A. Nagel, N. Riviere and S. Wainger, On Hilbert transforms along curves. $II$,, \emph{Amer. J. Math.}, 98 (1976), 395.   Google Scholar [17] A. Nagel, J. Vance, S. Wainger and D. Weinberg, Hilbert transforms for convex curves,, \emph{Duke Math. J.}, 50 (1983), 735.  doi: 10.1215/S0012-7094-83-05036-6.  Google Scholar [18] A. Nagel, J. Vance, S. Wainger and D. Weinberg, Maximal functions for convex curves,, \emph{Duke Math. J.}, 52 (1985), 715.  doi: 10.1215/S0012-7094-85-05237-8.  Google Scholar [19] E. Stein, Singular integrals, harmonic functions, and differentiability properties of functions of several variables,, In \emph{Singular Integrals} (Proc. Sympos. Pure Math., (1966), 316.   Google Scholar [20] E. Stein, Harmonic analysis, real-variable methods, orthogonality, and oscillatory integrals,, With the assistance of Timothy S. Murphy. Princeton Mathematical Series, (1993).   Google Scholar [21] E. Stein and S. Wainger, Oscillatory integrals related to Carleson's theorem,, \emph{Math. Res. Lett.}, 8 (2001), 789.  doi: 10.4310/MRL.2001.v8.n6.a9.  Google Scholar [22] S. Wainger, Special trigonometric series in $k$-dimensions,, \emph{Mem. Amer. Math. Soc.}, 59 (1965).   Google Scholar

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##### References:
 [1] M. Bateman, Single annulus $L^p$ estimates for Hilbert transforms along vector fields,, \emph{Rev. Mat. Iberoam.}, 29 (2013), 1021.  doi: 10.4171/RMI/748.  Google Scholar [2] M. Bateman and C. Thiele, $L^p$ estimates for the Hilbert transforms along a one-variable vector field,, \emph{Anal. PDE}, 6 (2013), 1577.  doi: 10.2140/apde.2013.6.1577.  Google Scholar [3] L. Carleson, On convergence and growth of partial sums of Fourier series,, \emph{Acta Math.}, 116 (1966), 135.   Google Scholar [4] H. Carlsson, M. Christ, A. Cordoba, J. Duoandikoetxea, J. L. Rubio de Francia, J. Vance, S. Wainger and D. Weinberg, $L^p$ estimates for maximal functions and Hilbert transforms along flat convex curves in $\R^2$,, \emph{Bull. Amer. Math. Soc. (N.S.)}, 14 (1986), 263.  doi: 10.1090/S0273-0979-1986-15433-9.  Google Scholar [5] Y. Ding and H. Liu, Weighted $L^p$ boundedness of Carleson type maximal operators,, \emph{Proc. Amer. Math. Soc.}, 140 (2012), 2739.  doi: 10.1090/S0002-9939-2011-11110-9.  Google Scholar [6] C. Fefferman, Inequalities for strongly singular convolution operators,, \emph{Acta Math.}, 124 (1970), 9.   Google Scholar [7] C. Fefferman, Pointwise convergence of Fourier series,, \emph{Ann. of Math.}, 98 (1973), 551.   Google Scholar [8] M. Folch-Gabayet and J. Wright, An oscillatory integral estimate associated to rational phases,, \emph{J. Geom. Anal.}, 13 (2003), 291.  doi: 10.1007/BF02930698.  Google Scholar [9] M. Folch-Gabayet and J. Wright, Singular integral operators associated to curves with rational components,, \emph{Trans. Amer. Math. Soc.}, 360 (2008), 1661.  doi: 10.1090/S0002-9947-07-04349-8.  Google Scholar [10] M. Folch-Gabayet and J. Wright, Weak type $(1, 1)$ bounds for oscillatory singular integrals with rational phases,, \emph{Studia Math.}, 210 (2012), 57.  doi: 10.4064/sm210-1-4.  Google Scholar [11] I. I. Hirschman, On multiplier transformations,, \emph{Duke Mathematical Journal}, 26 (1959), 221.   Google Scholar [12] M. Lacey and C. Thiele, A proof of boundedness of the Carleson operator,, \emph{Math. Res. Lett.}, 7 (2000), 361.  doi: 10.4310/MRL.2000.v7.n4.a1.  Google Scholar [13] V. Lie, The (weak-$L^2$) boundedness of the quadratic Carleson operator,, \emph{Geom. Funct. Anal.}, 19 (2009), 457.  doi: 10.1007/s00039-009-0010-x.  Google Scholar [14] V. Lie, The Polynomial Carleson Operator,, arXiv:1105.4504., ().   Google Scholar [15] A. Nagel, N. Riviere and S. Wainger, On Hilbert transforms along curves,, \emph{Bull. Amer. Math. Soc.}, 80 (1974), 106.   Google Scholar [16] A. Nagel, N. Riviere and S. Wainger, On Hilbert transforms along curves. $II$,, \emph{Amer. J. Math.}, 98 (1976), 395.   Google Scholar [17] A. Nagel, J. Vance, S. Wainger and D. Weinberg, Hilbert transforms for convex curves,, \emph{Duke Math. J.}, 50 (1983), 735.  doi: 10.1215/S0012-7094-83-05036-6.  Google Scholar [18] A. Nagel, J. Vance, S. Wainger and D. Weinberg, Maximal functions for convex curves,, \emph{Duke Math. J.}, 52 (1985), 715.  doi: 10.1215/S0012-7094-85-05237-8.  Google Scholar [19] E. Stein, Singular integrals, harmonic functions, and differentiability properties of functions of several variables,, In \emph{Singular Integrals} (Proc. Sympos. Pure Math., (1966), 316.   Google Scholar [20] E. Stein, Harmonic analysis, real-variable methods, orthogonality, and oscillatory integrals,, With the assistance of Timothy S. Murphy. Princeton Mathematical Series, (1993).   Google Scholar [21] E. Stein and S. Wainger, Oscillatory integrals related to Carleson's theorem,, \emph{Math. Res. Lett.}, 8 (2001), 789.  doi: 10.4310/MRL.2001.v8.n6.a9.  Google Scholar [22] S. Wainger, Special trigonometric series in $k$-dimensions,, \emph{Mem. Amer. Math. Soc.}, 59 (1965).   Google Scholar
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