Chinese Annals of Mathematics,Series B, Volume. 46, Issue 4, 611(2025)

A Note on the Convergence Along Tangential Curve Associated with Fractional Schrödinger Propagator and Boussinesq Operator

LI Dan and LI Junfeng
References(31)

[1] [1] Bourgain, J., Some new estimates on oscillatory integrals, Essays on Fourier analysis in Honor of Elias M. Stein (Princeton, NJ, 1991), Princeton Math. Ser.,42, Princeton University Press, New Jersey, 1995, 83–112.

[2] [2] Bourgain, J., On the Schrdinger maximal function in higher dimension,Proc. Steklov Inst. Math.,280(1), 2013, 46–60.

[3] [3] Bourgain, J., A note on the Schrdinger maximal function,J. Anal. Math.,130, 2016, 393–396.

[4] [4] Boussinesq, J., Thorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond,J. Math. Pures Appl.,17(2), 1872, 55–108.

[5] [5] Carbery, A., Radial Fourier multipliers and associated maximal functions,Recent Progress in Fourier Analysis,111, 1985, 49–56.

[6] [6] Carleson, L., Some analytic problems related to statistical mechanics, Euclidean harmonic analysis (Proc. Sem., Univ. Maryland., College Park, Md., 1979), Lecture Notes in Math.,779, Springer-Verlag, Berlin, 1980, 5–45.

[7] [7] Cho, C. H. and Ko, H., A note on maximal estimates of generalized Schrdinger equation, arXiv: 1809.03246v1.

[8] [8] Cho, C. H., Lee, S. and Vargas, A., Problems on pointwise convergence of solutions to the Schrdinger equation,J. Fourier Anal. Appl.,18(5), 2012, 972–994.

[9] [9] Cowling, M., Pointwise behavior of solutions to Schrdinger equations,Lect. Notes Math.,992, 1983, 83–90.

[10] [10] Dahlberg, B. E. J. and Kenig, C. E., A note on the almost everywhere behavior of solutions to the Schrdinger equation, Harmonic analysis (Minneapolis, Minn., 1981), Lecture Notes in Math.,908, Springer-Verlag, Berlin, New York, 1982, 205–209.

[11] [11] Ding, Y. and Niu Y., Weighted maximal estimates along curve associated with dispersive equations,Anal. Appl.(Singap.),15(2), 2017, 225–240.

[12] [12] Demeter, C. and Guo, S., Schrdinger maximal function estimates via the pseudoconformal transformation, arXiv: 1608.07640, 2016.

[13] [13] Du, X., Guth, L. and Li, X., A sharp Schrdinger maximal eatimate inℝ2,Ann.of Math.,186(2), 2017, 607–640.

[14] [14] Du, X., Guth, L., Li, X. and Zhang, R., Pointwise convergence of Schrdinger solutions and multilinear refined Strichartz estimates,Forum Math. Sigma.,6, 2018, 18 pp.

[15] [15] Du, X. and Zhang, R., SharpL2 estimate of Schrdinger maximal function in higher dimensions,Ann. Math.,189(3), 2018, 837–861.

[16] [16] Grafakos, L., Modern Fourier Analysis, World Scientific Publishing, New York, 2014.

[17] [17] Lee, S., On pointwise convergence of the solutions to Schrdinger equations inℝ2,Int. Math. Res. Not., Art. ID 32597, 2006, 21 pp.

[18] [18] Lee, S. and Rogers, K., The Schrdinger equation along curves and the quantum harmonic oscillator,Adv. Math.,229, 2012, 1359–1379.

[19] [19] Li, D., Li, J. and Xiao, J., An upbound of Hausdorff's dimension of the divergence set of the fractional Schrdinger operator onHs(ℝn),Acta Math. Sci. Ser. B(Engl. Ed.),41(4), 2021, 1223–1249.

[20] [20] Li, W. and Wang, H., On convergence properties for generalized Schrdinger operators along tangential curves, arXiv: 2111.09186v2.

[21] [21] Li, W., Wang, H. and Yan, D., A note on non-tangential convergence for Schrdinger operators,J. Fourier Anal. Appl.,27(4), 2021, 14 pp.

[22] [22] Lu, S., Ding, Y. and Yan, D., Singular Integrals and Related Topics, World Scientific Publishing, Singapore, 2007.

[23] [23] Luc, R. and Rogers, K., An improved necessary condition for the Schrdinger maximal estimate, arXiv: 1506.05325.

[24] [24] Miao, C., Yang, J. and Zheng, J., An improved maximal inequality for 2D fractional order Schrdinger operators,Studia Math.,230(2), 2015, 121–165.

[25] [25] Moyua, A., Vargas, A. and Vega, L., Schrdinger maximal function and restriction properties of the Fourier transform,Internat. Math. Res. Notices,16, 1996, 793–815.

[26] [26] Shiraki, S., Pointwise convergence along restricted directions for the fractional Schrdinger equation,J. Fourier Anal. Appl.,26(4), 2020, 12 pp.

[27] [27] Sjgren, P. and Sjlin, P., Convergence properties for the time-dependent Schrdinger equation,Ann. Acad. Sci. Fenn. Ser. A I Math.,14(1), 1989, 13–25.

[28] [28] Sjlin, P., Regularity of solutions to the Schrdinger equation,Duke Math. J.,55(3), 1987, 699–715.

[29] [29] Stein, E. M., Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.

[30] [30] Tao, T. and Vargas, A., A bilinear approach to cone multipliers, I, Restriction estimate,Geom. Funct. Anal.,10(1), 2000, 185–215.

[31] [31] Vega, L., Schrdinger equations: Pointwise convergence to the initial data,Proc. Amer. Math. Soc.,102(4), 1988, 874–878.

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LI Dan, LI Junfeng. A Note on the Convergence Along Tangential Curve Associated with Fractional Schrödinger Propagator and Boussinesq Operator[J]. Chinese Annals of Mathematics,Series B, 2025, 46(4): 611

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Paper Information

Received: Sep. 7, 2022

Accepted: Aug. 25, 2025

Published Online: Aug. 25, 2025

The Author Email:

DOI:10.1007/s11401-025-0031-2

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