Oscillatory integrals related to Carleson's theorem: fractional monomials

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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

Abstract
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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.

Keywords: uniform estimates, Carleson's theorem, Singular integrals, oscillatory integrals, Hilbert transforms along variable curves..

Mathematics Subject Classification: Primary: 42A50; Secondary: 42A2.

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.
[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.
[3]	L. Carleson, On convergence and growth of partial sums of Fourier series,, \emph{Acta Math.}, 116 (1966), 135.
[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.
[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.
[6]	C. Fefferman, Inequalities for strongly singular convolution operators,, \emph{Acta Math.}, 124 (1970), 9.
[7]	C. Fefferman, Pointwise convergence of Fourier series,, \emph{Ann. of Math.}, 98 (1973), 551.
[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.
[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.
[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.
[11]	I. I. Hirschman, On multiplier transformations,, \emph{Duke Mathematical Journal}, 26 (1959), 221.
[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.
[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.
[14]	V. Lie, The Polynomial Carleson Operator,, arXiv:1105.4504., ().
[15]	A. Nagel, N. Riviere and S. Wainger, On Hilbert transforms along curves,, \emph{Bull. Amer. Math. Soc.}, 80 (1974), 106.
[16]	A. Nagel, N. Riviere and S. Wainger, On Hilbert transforms along curves. $II$,, \emph{Amer. J. Math.}, 98 (1976), 395.
[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.
[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.
[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.
[20]	E. Stein, Harmonic analysis, real-variable methods, orthogonality, and oscillatory integrals,, With the assistance of Timothy S. Murphy. Princeton Mathematical Series, (1993).
[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.
[22]	S. Wainger, Special trigonometric series in $k$-dimensions,, \emph{Mem. Amer. Math. Soc.}, 59 (1965).

show all references

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.
[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.
[3]	L. Carleson, On convergence and growth of partial sums of Fourier series,, \emph{Acta Math.}, 116 (1966), 135.
[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.
[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.
[6]	C. Fefferman, Inequalities for strongly singular convolution operators,, \emph{Acta Math.}, 124 (1970), 9.
[7]	C. Fefferman, Pointwise convergence of Fourier series,, \emph{Ann. of Math.}, 98 (1973), 551.
[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.
[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.
[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.
[11]	I. I. Hirschman, On multiplier transformations,, \emph{Duke Mathematical Journal}, 26 (1959), 221.
[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.
[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.
[14]	V. Lie, The Polynomial Carleson Operator,, arXiv:1105.4504., ().
[15]	A. Nagel, N. Riviere and S. Wainger, On Hilbert transforms along curves,, \emph{Bull. Amer. Math. Soc.}, 80 (1974), 106.
[16]	A. Nagel, N. Riviere and S. Wainger, On Hilbert transforms along curves. $II$,, \emph{Amer. J. Math.}, 98 (1976), 395.
[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.
[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.
[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.
[20]	E. Stein, Harmonic analysis, real-variable methods, orthogonality, and oscillatory integrals,, With the assistance of Timothy S. Murphy. Princeton Mathematical Series, (1993).
[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.
[22]	S. Wainger, Special trigonometric series in $k$-dimensions,, \emph{Mem. Amer. Math. Soc.}, 59 (1965).

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