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On the differentiability of the solutions of non-local Isaacs equations involving $\frac{1}{2}$-Laplacian
Oscillatory integrals related to Carleson's theorem: fractional monomials
1. | Endenicher Allee 60, 53115, Bonn, Germany |
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] |
M. Bateman and C. Thiele, $L^p$ estimates for the Hilbert transforms along a one-variable vector field, Anal. PDE, 6 (2013), 1577-1600.
doi: 10.2140/apde.2013.6.1577. |
[3] |
L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math., 116 (1966), 135-157. |
[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 $\mathbb{R}^2$, Bull. Amer. Math. Soc. (N.S.), 14 (1986), 263-267.
doi: 10.1090/S0273-0979-1986-15433-9. |
[5] |
Y. Ding and H. Liu, Weighted $L^p$ boundedness of Carleson type maximal operators, Proc. Amer. Math. Soc., 140 (2012), 2739-2751.
doi: 10.1090/S0002-9939-2011-11110-9. |
[6] |
C. Fefferman, Inequalities for strongly singular convolution operators, Acta Math., 124 (1970), 9-36. |
[7] |
C. Fefferman, Pointwise convergence of Fourier series, Ann. of Math., 98 (1973), 551-571. |
[8] |
M. Folch-Gabayet and J. Wright, An oscillatory integral estimate associated to rational phases, J. Geom. Anal., 13 (2003), 291-299.
doi: 10.1007/BF02930698. |
[9] |
M. Folch-Gabayet and J. Wright, Singular integral operators associated to curves with rational components, Trans. Amer. Math. Soc., 360 (2008), 1661-1679 (electronic).
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, Studia Math., 210 (2012), 57-76.
doi: 10.4064/sm210-1-4. |
[11] |
I. I. Hirschman, On multiplier transformations, Duke Mathematical Journal, 26 (1959), 221-242. |
[12] |
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. |
[13] |
V. Lie, The (weak-$L^2$) boundedness of the quadratic Carleson operator, Geom. Funct. Anal., 19 (2009), 457-497.
doi: 10.1007/s00039-009-0010-x. |
[14] | |
[15] |
A. Nagel, N. Riviere and S. Wainger, On Hilbert transforms along curves, Bull. Amer. Math. Soc., 80 (1974), 106-108. |
[16] |
A. Nagel, N. Riviere and S. Wainger, On Hilbert transforms along curves. $II$, Amer. J. Math., 98 (1976), 395-403. |
[17] |
A. Nagel, J. Vance, S. Wainger and D. Weinberg, Hilbert transforms for convex curves, Duke Math. J., 50 (1983), 735-744.
doi: 10.1215/S0012-7094-83-05036-6. |
[18] |
A. Nagel, J. Vance, S. Wainger and D. Weinberg, Maximal functions for convex curves, Duke Math. J., 52 (1985), 715-722.
doi: 10.1215/S0012-7094-85-05237-8. |
[19] |
E. Stein, Singular integrals, harmonic functions, and differentiability properties of functions of several variables, In Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) (pp. 316-335). |
[20] |
E. Stein, Harmonic analysis, real-variable methods, orthogonality, and oscillatory integrals, With the assistance of Timothy S. Murphy. Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993. xiv+695 pp. |
[21] |
E. 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. |
[22] |
S. Wainger, Special trigonometric series in $k$-dimensions, Mem. Amer. Math. Soc., 59 (1965), 102 pp. |
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] |
M. Bateman and C. Thiele, $L^p$ estimates for the Hilbert transforms along a one-variable vector field, Anal. PDE, 6 (2013), 1577-1600.
doi: 10.2140/apde.2013.6.1577. |
[3] |
L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math., 116 (1966), 135-157. |
[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 $\mathbb{R}^2$, Bull. Amer. Math. Soc. (N.S.), 14 (1986), 263-267.
doi: 10.1090/S0273-0979-1986-15433-9. |
[5] |
Y. Ding and H. Liu, Weighted $L^p$ boundedness of Carleson type maximal operators, Proc. Amer. Math. Soc., 140 (2012), 2739-2751.
doi: 10.1090/S0002-9939-2011-11110-9. |
[6] |
C. Fefferman, Inequalities for strongly singular convolution operators, Acta Math., 124 (1970), 9-36. |
[7] |
C. Fefferman, Pointwise convergence of Fourier series, Ann. of Math., 98 (1973), 551-571. |
[8] |
M. Folch-Gabayet and J. Wright, An oscillatory integral estimate associated to rational phases, J. Geom. Anal., 13 (2003), 291-299.
doi: 10.1007/BF02930698. |
[9] |
M. Folch-Gabayet and J. Wright, Singular integral operators associated to curves with rational components, Trans. Amer. Math. Soc., 360 (2008), 1661-1679 (electronic).
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, Studia Math., 210 (2012), 57-76.
doi: 10.4064/sm210-1-4. |
[11] |
I. I. Hirschman, On multiplier transformations, Duke Mathematical Journal, 26 (1959), 221-242. |
[12] |
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. |
[13] |
V. Lie, The (weak-$L^2$) boundedness of the quadratic Carleson operator, Geom. Funct. Anal., 19 (2009), 457-497.
doi: 10.1007/s00039-009-0010-x. |
[14] | |
[15] |
A. Nagel, N. Riviere and S. Wainger, On Hilbert transforms along curves, Bull. Amer. Math. Soc., 80 (1974), 106-108. |
[16] |
A. Nagel, N. Riviere and S. Wainger, On Hilbert transforms along curves. $II$, Amer. J. Math., 98 (1976), 395-403. |
[17] |
A. Nagel, J. Vance, S. Wainger and D. Weinberg, Hilbert transforms for convex curves, Duke Math. J., 50 (1983), 735-744.
doi: 10.1215/S0012-7094-83-05036-6. |
[18] |
A. Nagel, J. Vance, S. Wainger and D. Weinberg, Maximal functions for convex curves, Duke Math. J., 52 (1985), 715-722.
doi: 10.1215/S0012-7094-85-05237-8. |
[19] |
E. Stein, Singular integrals, harmonic functions, and differentiability properties of functions of several variables, In Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) (pp. 316-335). |
[20] |
E. Stein, Harmonic analysis, real-variable methods, orthogonality, and oscillatory integrals, With the assistance of Timothy S. Murphy. Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993. xiv+695 pp. |
[21] |
E. 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. |
[22] |
S. Wainger, Special trigonometric series in $k$-dimensions, Mem. Amer. Math. Soc., 59 (1965), 102 pp. |
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