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May 2019, 18(3): 1433-1446. doi: 10.3934/cpaa.2019069

Hilbert transforms along double variable fractional monomials

Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

* Corresponding author

Received  July 2018 Revised  August 2018 Published  November 2018

In this paper, we obtain the
$L^2(\mathbb{R}^2)$
boundedness and single annulus
$L^p(\mathbb{R}^2)$
estimate for the Hilbert transform
$H_{α,β}$
along double variable fractional monomial
$u_1(x_1)[t]^α+u_2(x_1)[t]^β$
$H_{α,β}f(x_1,x_2): = \mathit{\rm{p.\,v.}}∈\int_{ - \infty }^\infty {} f(x_1-t,x_2-u_1(x_1)[t]^α-u_2(x_1)[t]^β)\,\frac{\textrm{d}t}{t}$
with the bounds are independent of the measurable function
$u_1$
and
$u_2$
. At the same time, we also obtain the
$L^p(\mathbb{R})$
boundedness of the corresponding Carleson operator
$\mathcal{C}_{α,β}f(x):=\mathop {\sup }\limits_{{N_1},{N_2} \in \mathbb{R}} |{\rm{p.\,v.}}\int_{ - \infty }^\infty {} e^{iN_1[t]^α+iN_2[t]^β}f(x-t)\,\frac{\textrm{d}t}{t}|,$
where
$[t]^α$
stands for either
$|t|^α$
or
$\textrm{sgn}(t)|t|^α$
,
$[t]^β$
stands for either
$|t|^β$
or
$\textrm{sgn}(t)|t|^β$
and
$α,β,p∈ (1,∞)$
.
Citation: Haixia Yu. Hilbert transforms along double variable fractional monomials. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1433-1446. doi: 10.3934/cpaa.2019069
References:
[1]

M. Bateman, Single annulus $L^p$ estimates for Hilbert transforms along vector fields, Rev. Mat. Iberoam., 29 (2013), 1021-1069. doi: 10.4171/RMI/748.

[2]

A. CarberyM. ChristJ. VanceS. Wainger and D. Watson, Operators associated to flat plane curves: $L^p$ estimates via dilation methods, Duke Math. J., 59 (1989), 675-700. doi: 10.1215/S0012-7094-89-05930-9.

[3]

A. CarberyJ. VanceS. Wainger and D. Watson, The Hilbert transform and maximal function along flat curves, dilations, and differential equations, Amer. J. Math., 116 (1994), 1203-1239. doi: 10.2307/2374944.

[4]

L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math., 116 (1966), 135-157. doi: 10.1007/BF02392815.

[5]

H. CarlssonM. ChristA. CordobaJ. DuoandikoetxeaJ. L. Rubio de FranciaJ. VanceS. Wainger and D. Weinberg, $L^p$ estimates for maximal functions and Hilbert transforms along flat convex curves in $\mathbb{R}^{2}$, Bull. Amer. Math. Soc. (N.S.), 14 (1986), 263-267. doi: 10.1090/S0273-0979-1986-15433-9.

[6]

C. Fefferman, Pointwise convergence of Fourier series, Ann. of Math., 98 (1973), 551-571. doi: 10.2307/1970917.

[7]

S. Guo, Hilbert transform along measurable vector fields constant on Lipschitz curves: $L^2$ boundedness, Anal. PDE, 8 (2015), 1263-1288. doi: 10.2140/apde.2015.8.1263.

[8]

S. Guo, Oscillatory integrals related to Carleson's theorem: fractional monomials, Commun. Pure Appl. Anal., 15 (2016), 929-946. doi: 10.3934/cpaa.2016.15.929.

[9]

S. Guo, Hilbert transform along measurable vector fields constant on Lipschitz curves: $L^p$ boundedness, Trans. Amer. Math. Soc., 369 (2017), 2493-2519. doi: 10.1090/tran/6750.

[10]

S. GuoJ. HickmanV. Lie and J. Roos, Maximal operators and Hilbert transforms along variable non-flat homogeneous curves, Proc. Lond. Math. Soc., 115 (2017), 177-219. doi: 10.1112/plms.12037.

[11]

S. GuoL. B. PierceJ. Roos and P. Yung, Polynomial Carleson operators along monomial curves in the plane, J. Geom. Anal., 27 (2017), 2977-3012. doi: 10.1007/s12220-017-9790-7.

[12]

R. A. Hunt, On the convergence of Fourier series, in Orthogonal Expansions and their Continuous Analogues (Proc. Conf., Edwardsville, Ill., 1967), Southern Illinois Univ. Press, Carbondale, Ill. (1968), 235-255.

[13]

M. Lacey and X. Li, Maximal theorems for the directional Hilbert transform on the plane, Trans. Amer. Math. Soc., 358 (2006), 4099-4117. doi: 10.1090/S0002-9947-06-03869-4.

[14]

M. Lacey and C. Thiele, A proof of boundedness of the Carleson operator, Math. Res. Lett., 7 (2000), 361-370. doi: 10.4310/MRL.2000.v7.n4.a1.

[15]

A. NagelJ. VanceS. Wainger and D. Weinberg, Hilbert transforms for convex curves, Duke Math. J., 50 (1983), 735-744. doi: 10.1215/S0012-7094-83-05036-6.

[16]

D. H. Phong and E. M. Stein, Hilbert integrals, singular integrals, and Radon transforms. I, Acta Math., 157 (1986), 99-157. doi: 10.1007/BF02392592.

[17]

E. Prestini and P. Sjölin, A Littlewood-Paley inequality for the Carleson operator, J. Fourier Anal. Appl., 6 (2000), 457-466. doi: 10.1007/BF02511540.

[18]

E. M. Stein and S. Wainger, Oscillatory integrals related to Carleson's theorem, Math. Res. Lett., 8 (2001), 789-800. doi: 10.4310/MRL.2001.v8.n6.a9.

[19]

J. VanceS. Wainger and J. Wright, The Hilbert transform and maximal function along nonconvex curves in the plane, Rev. Mat. Iberoam., 10 (1994), 93-121. doi: 10.4171/RMI/146.

[20]

J. Wright, $L^p$ estimates for operators associated to oscillating plane curves, Duke Math. J., 67 (1992), 101-157. doi: 10.1215/S0012-7094-92-06705-6.

[21]

H. Yu and J. Li, $L^p$ Boundedness of Hilbert Transforms Associated with Variable Plane Curves, preprint, arXiv: 1806.08589.

show all references

References:
[1]

M. Bateman, Single annulus $L^p$ estimates for Hilbert transforms along vector fields, Rev. Mat. Iberoam., 29 (2013), 1021-1069. doi: 10.4171/RMI/748.

[2]

A. CarberyM. ChristJ. VanceS. Wainger and D. Watson, Operators associated to flat plane curves: $L^p$ estimates via dilation methods, Duke Math. J., 59 (1989), 675-700. doi: 10.1215/S0012-7094-89-05930-9.

[3]

A. CarberyJ. VanceS. Wainger and D. Watson, The Hilbert transform and maximal function along flat curves, dilations, and differential equations, Amer. J. Math., 116 (1994), 1203-1239. doi: 10.2307/2374944.

[4]

L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math., 116 (1966), 135-157. doi: 10.1007/BF02392815.

[5]

H. CarlssonM. ChristA. CordobaJ. DuoandikoetxeaJ. L. Rubio de FranciaJ. VanceS. Wainger and D. Weinberg, $L^p$ estimates for maximal functions and Hilbert transforms along flat convex curves in $\mathbb{R}^{2}$, Bull. Amer. Math. Soc. (N.S.), 14 (1986), 263-267. doi: 10.1090/S0273-0979-1986-15433-9.

[6]

C. Fefferman, Pointwise convergence of Fourier series, Ann. of Math., 98 (1973), 551-571. doi: 10.2307/1970917.

[7]

S. Guo, Hilbert transform along measurable vector fields constant on Lipschitz curves: $L^2$ boundedness, Anal. PDE, 8 (2015), 1263-1288. doi: 10.2140/apde.2015.8.1263.

[8]

S. Guo, Oscillatory integrals related to Carleson's theorem: fractional monomials, Commun. Pure Appl. Anal., 15 (2016), 929-946. doi: 10.3934/cpaa.2016.15.929.

[9]

S. Guo, Hilbert transform along measurable vector fields constant on Lipschitz curves: $L^p$ boundedness, Trans. Amer. Math. Soc., 369 (2017), 2493-2519. doi: 10.1090/tran/6750.

[10]

S. GuoJ. HickmanV. Lie and J. Roos, Maximal operators and Hilbert transforms along variable non-flat homogeneous curves, Proc. Lond. Math. Soc., 115 (2017), 177-219. doi: 10.1112/plms.12037.

[11]

S. GuoL. B. PierceJ. Roos and P. Yung, Polynomial Carleson operators along monomial curves in the plane, J. Geom. Anal., 27 (2017), 2977-3012. doi: 10.1007/s12220-017-9790-7.

[12]

R. A. Hunt, On the convergence of Fourier series, in Orthogonal Expansions and their Continuous Analogues (Proc. Conf., Edwardsville, Ill., 1967), Southern Illinois Univ. Press, Carbondale, Ill. (1968), 235-255.

[13]

M. Lacey and X. Li, Maximal theorems for the directional Hilbert transform on the plane, Trans. Amer. Math. Soc., 358 (2006), 4099-4117. doi: 10.1090/S0002-9947-06-03869-4.

[14]

M. Lacey and C. Thiele, A proof of boundedness of the Carleson operator, Math. Res. Lett., 7 (2000), 361-370. doi: 10.4310/MRL.2000.v7.n4.a1.

[15]

A. NagelJ. VanceS. Wainger and D. Weinberg, Hilbert transforms for convex curves, Duke Math. J., 50 (1983), 735-744. doi: 10.1215/S0012-7094-83-05036-6.

[16]

D. H. Phong and E. M. Stein, Hilbert integrals, singular integrals, and Radon transforms. I, Acta Math., 157 (1986), 99-157. doi: 10.1007/BF02392592.

[17]

E. Prestini and P. Sjölin, A Littlewood-Paley inequality for the Carleson operator, J. Fourier Anal. Appl., 6 (2000), 457-466. doi: 10.1007/BF02511540.

[18]

E. M. Stein and S. Wainger, Oscillatory integrals related to Carleson's theorem, Math. Res. Lett., 8 (2001), 789-800. doi: 10.4310/MRL.2001.v8.n6.a9.

[19]

J. VanceS. Wainger and J. Wright, The Hilbert transform and maximal function along nonconvex curves in the plane, Rev. Mat. Iberoam., 10 (1994), 93-121. doi: 10.4171/RMI/146.

[20]

J. Wright, $L^p$ estimates for operators associated to oscillating plane curves, Duke Math. J., 67 (1992), 101-157. doi: 10.1215/S0012-7094-92-06705-6.

[21]

H. Yu and J. Li, $L^p$ Boundedness of Hilbert Transforms Associated with Variable Plane Curves, preprint, arXiv: 1806.08589.

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