A note on Boussinesq maximal estimate

Dan Li; Xiang Li; Dan Li; Xiang Li

doi:10.3934/math.2024088

AIMS Mathematics

2024, Volume 9, Issue 1: 1819-1830. doi: 10.3934/math.2024088

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A note on Boussinesq maximal estimate

Dan Li ¹,
Xiang Li ^{2
,
,}

1.
School of Mathematics and Statistics, Beijing Technology and Business University, Beijing 100048, China
2.
School of Mathematics and Finance, Chuzhou University, Chuzhou, Anhui 239012, China

Received: 22 August 2023 Revised: 19 November 2023 Accepted: 30 November 2023 Published: 14 December 2023
MSC : 42B25, 42B37

We considered the Boussinesq maximal estimate when $ n\geq1 $. We obtained the Boussinesq maximal operator $ \mathcal{B}_E^\ast f $ is bounded from $ L^2(\mathbb{R}^n) $ to $ L^2(\mathbb{R}^n) $ when $ f\in L^2(\mathbb{R}^n) $ and $ \text{supp}\; \hat f\subset B(0, \lambda) $.
- Boussinesq operator,
- maximal estimate,
- bounded,
- frequency decomposition,
- linearization
Citation: Dan Li, Xiang Li. A note on Boussinesq maximal estimate[J]. AIMS Mathematics, 2024, 9(1): 1819-1830. doi: 10.3934/math.2024088

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Abstract

We considered the Boussinesq maximal estimate when $ n\geq1 $. We obtained the Boussinesq maximal operator $ \mathcal{B}_E^\ast f $ is bounded from $ L^2(\mathbb{R}^n) $ to $ L^2(\mathbb{R}^n) $ when $ f\in L^2(\mathbb{R}^n) $ and $ \text{supp}\; \hat f\subset B(0, \lambda) $.

References

[1]	J. Bourgain, Some new estimates on oscillatory integrals, In: Essays on Fourier analysis in honor of Elias M. Stein (PMS-42), Princeton: Princeton University Press, 1995. https://doi.org/10.1515/9781400852949.83
[2]	J. Bourgain, On the Schrödinger maximal function in higher dimension, Proc. Steklov Inst. Math., 280 (2013), 46–60. https://doi.org/10.1134/S0081543813010045 doi: 10.1134/S0081543813010045
[3]	J. Bourgain, A note on the Schrödinger maximal function, J. Anal. Math., 130 (2016), 393–396. https://doi.org/10.1007/s11854-016-0042-8 doi: 10.1007/s11854-016-0042-8
[4]	J. Boussinesq, Théorie 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 (1872), 55–108.
[5]	A. Carbery, Radial Fourier multipliers and associated maximal functions, North Holland Math. Stud., 111 (1985), 49–56. https://doi.org/10.1016/S0304-0208(08)70279-2 doi: 10.1016/S0304-0208(08)70279-2
[6]	L. Carleson, Some analytic problems related to statistical mechanics, In: Euclidean harmonic analysis, Berlin, Heidelberg: Springer, 1980, 5–45. https://doi.org/10.1007/BFb0087666
[7]	C. Cho, H. Ko, A note on maximal estimates of generalized Schrödinger equation, 2018, arXiv: 1809.03246.
[8]	M. G. Cowling, Pointwise behavior of solutions to Schrödinger equations, In: Harmonic analysis, Berlin, Heidelberg: Springer, 1982, 83–90. https://doi.org/10.1007/BFb0069152
[9]	B. E. J. Dahlberg, C. E. Kenig, A note on the almost everywhere behavior of solutions to the Schrödinger equation, In: Harmonic analysis, Berlin, Heidelberg: Springer, 1982,205–209. https://doi.org/10.1007/BFb0093289
[10]	C. Demeter, S. Guo, Schrödinger maximal function estimates via the pseudoconformal transformation, 2016, arXiv: 1608.07640.
[11]	Y. Ding, Y. M. Niu, Weighted maximal estimates along curve associated with dispersive equations, Anal. Appl., 15 (2017), 225–240. https://doi.org/10.1142/S021953051550027X doi: 10.1142/S021953051550027X
[12]	X. M. Du, L. Guth, X. C. Li, A sharp Schrödinger maximal eatimate in $\mathbb{R}^2$, Ann. Math., 186 (2017), 607–640. https://doi.org/10.4007/annals.2017.186.2.5 doi: 10.4007/annals.2017.186.2.5
[13]	X. M. Du, L. Guth, X. C. Li, R. X. Zhang, Pointwise convergence of Schrödinger solutions and multilinear refined Strichartz estimates, Forum Math. Sigma, 6 (2018), e14. https://doi.org/10.1017/fms.2018.11 doi: 10.1017/fms.2018.11
[14]	X. M. Du, R. X. Zhang, Sharp $L^2$ estimate of Schrödinger maximal function in higher dimensions, Ann. Math., 189 (2019), 837–861. https://doi.org/10.4007/annals.2019.189.3.4 doi: 10.4007/annals.2019.189.3.4
[15]	S. Lee, On pointwise convergence of the solutions to Schrödinger equations in $\mathbb{R}^2$, Int. Math. Res. Not., 2006 (2006), 32597. https://doi.org/10.1155/IMRN/2006/32597 doi: 10.1155/IMRN/2006/32597
[16]	S. Lee, K. M. Rogers, The Schrödinger equation along curves and the quantum harmonic oscillator, Adv. Math., 229 (2012), 1359–1379. https://doi.org/10.1016/j.aim.2011.10.023 doi: 10.1016/j.aim.2011.10.023
[17]	D. Li, J. F. Li, A Carleson problem for the Boussinesq operator, Acta Math. Sin. Engl. Ser., 39 (2023), 119–148. https://doi.org/10.1007/s10114-022-1221-4 doi: 10.1007/s10114-022-1221-4
[18]	D. Li, J. F. Li, J. Xiao, An upbound of Hausdorff's dimension of the divergence set of the fractional Schrödinger operator on $H^s(\mathbb{R}^n)$, Acta Math. Sci., 41 (2021), 1223–1249. https://doi.org/10.1007/s10473-021-0412-x doi: 10.1007/s10473-021-0412-x
[19]	W. J. Li, H. J. Wang, A study on a class of generalized Schrödinger operators, J. Funct. Anal., 281 (2021), 109203. https://doi.org/10.1016/j.jfa.2021.109203 doi: 10.1016/j.jfa.2021.109203
[20]	R. Lucà, K. Rogers, An improved necessary condition for the Schrödinger maximal estimate, 2015, arXiv: 1506.05325.
[21]	C. X. Miao, J. W. Yang, J. Q. Zheng, An improved maximal inequality for 2D fractional order Schrödinger operators, Stud. Math., 230 (2015), 121–165. https://doi.org/10.4064/sm8190-12-2015 doi: 10.4064/sm8190-12-2015
[22]	A. Moyua, A. Vargas, L. Vega, Schrödinger maximal function and restriction properties of the Fourier transform, Int. Math. Res. Not., 1996 (1996), 793–815. https://doi.org/10.1155/S1073792896000499 doi: 10.1155/S1073792896000499
[23]	P. Sjölin, Regularity of solutions to the Schrödinger equation, Duke Math. J., 55 (1987), 699–715. https://doi.org/10.1215/S0012-7094-87-05535-9 doi: 10.1215/S0012-7094-87-05535-9
[24]	P. Sjölin, J. O. Strömberg, Schrödinger means in higher dimensions, J. Math. Anal. Appl., 504 (2021), 125353. https://doi.org/10.1016/j.jmaa.2021.125353 doi: 10.1016/j.jmaa.2021.125353
[25]	P. Sjölin, J. O. Strömberg, Analysis of Schrödinger means, Ann. Fenn. Math., 46 (2021), 389–394. https://doi.org/10.5186/aasfm.2021.4616 doi: 10.5186/aasfm.2021.4616
[26]	E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton: Princeton University Press, 1993.
[27]	T. Tao, A. Vargas, A bilinear approach to cone multipliers Ⅰ. Restriction estimates, Geom. Funct. Anal., 10 (2000), 185–215. https://doi.org/10.1007/s000390050006 doi: 10.1007/s000390050006
[28]	L. Vega, Schrödinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc., 102 (1988), 874–878. https://doi.org/10.2307/2047326 doi: 10.2307/2047326

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