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A note on pointwise convergence for the Schrödinger equation

Published online by Cambridge University Press:  06 November 2017

RENATO LUCÀ  and
KEITH M. ROGERS
RENATO LUCÀ
Affiliation:
Departement Matematik und Informatik, Speigelgacse, Universität Basel, 4051, Switzerland. e-mail: renato.luca@unibas.ch
KEITH M. ROGERS
Affiliation:
Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Calle Nicolás Cabrera 13-15, Madrid, 28049, Spain. e-mail: keith.rogers@icmat.es
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Abstract

We consider Carleson's problem regarding pointwise convergence for the Schrödinger equation. Bourgain proved that there is initial data, in Hs(ℝn) with $s<\frac{n}{2(n+1)}$, for which the solution diverges on a set of nonzero Lebesgue measure. We provide a different example enabling the generalisation to fractional Hausdorff measure.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 166 , Issue 2 , March 2019 , pp. 209 - 218
Copyright
Copyright © Cambridge Philosophical Society 2017 

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References

REFERENCES

[1] Barceló, J. A., Bennett, J., Carbery, A. and Rogers, K. M. On the dimension of diverence sets of dispersive equations. Math. Ann. 349 (2011), 599–622.Google Scholar
[2] Barceló, J. A., Bennett, J., Carbery, A., Ruiz, A. and Vilela, M. C. Some special solutions of the Schrödinger equation. Indiana Univ. Math. J. 56 (2007), 1581–1593.Google Scholar
[3] Bourgain, J. Some new estimates on oscillatory integrals. In Essays on Fourier Analysis in Honor of Elias M. Stein. Princeton Math. Ser. 42 (Princeton, NJ, 1991), 83–112.Google Scholar
[4] Bourgain, J. A remark on Schrödinger operators. Israel J. Math. 77 (1992), 1–16.Google Scholar
[5] Bourgain, J. On the Schrödinger maximal function in higher dimension. Tr. Mat. Inst. Steklova 280 (2013), 53–66.Google Scholar
[6] Bourgain, J. A note on the Schrödinger maximal function. J. Anal. Math. 130 (2016), 393–396.Google Scholar
[7] Carbery, A. Radial Fourier multipliers and associated maximal functions. In Recent Progress in Fourier Analysis (El Escorial, 1983). North–Holland Math. Stud. 11 (North–Holland, Amsterdam), 49–56.Google Scholar
[8] Carleson, L. Some analytic problems related to statistical mechanics. In Euclidean Harmonic Analysis (Proc. Sem., Univ. Maryland, College Park, Md., 1979). Lecture Notes in Math. 779 (Springer, Berlin), 5–45.Google Scholar
[9] Cowling, M. Pointwise behaviour of solutions to Schrödinger equations. In Harmonic Analysis (Cortona, 1982). Lecture Notes in Math. 992 (Springer, Berlin), 83–90.Google Scholar
[10] Dahlberg, B. E. J. and Kenig, C. E. A note on the almost everywhere behavior of solutions to the Schrödinger equation. In Harmonic Analysis (Minneapolis, Minn., 1981). Lecture Notes in Math. 908 Springer, Berlin, 205–209.Google Scholar
[11] Demeter, C. and Guo, S. Schrödinger maximal function estimates via the pseudoconformal transformation. arXiv:1608.07640, (2016).Google Scholar
[12] Du, X., Guth, L. and Li, X. A sharp Schrödinger maximal estimate in ℝ2. Ann. of Math. 186 (2017), 607–640.Google Scholar
[13] Falconer, K. Classes of sets with large intersection. Mathematika 32 (1985), 191–205.Google Scholar
[14] Falconer, K. Fractal Geometry : Mathematical Foundations and Applications (Wiley, 2003).Google Scholar
[15] Lee, S. On pointwise convergence of the solutions to the Schrödinger equations in ℝ2. Int. Math. Res. Not. (2006), 1–21.Google Scholar
[16] Lucà, R. and Rogers, K. M. Coherence on fractals versus pointwise convergence for the Schrödinger equation. Comm. Math. Phys. 351 (2017), 341–359.Google Scholar
[17] Lucà, R. and Rogers, K. M. Average decay for the Fourier transform of measures with applications. arXiv:1503.00105 (2015), J. Eur. Math. Soc., to appear.Google Scholar
[18] Mattila, P. Fourier Analysis and Hausdorff Dimension. Cambridge Stud. Adv. Math. 150 (Cambridge University Press, Cambridge, 2015).Google Scholar
[19] Moyua, A., Vargas, A. and Vega, L. Schrödinger maximal function and restriction properties of the Fourier transform. Int. Math. Res. Not. 16 (1996), 793–815.Google Scholar
[20] Moyua, A., Vargas, A. and Vega, L. Restriction theorems and maximal operators related to oscillatory integrals in ℝ3. Duke Math. J. 96 (1999), 547–574.Google Scholar
[21] Nikišin, E. M. A resonance theorem and series in eigenfunctions of the Laplace operator. Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 795–813.Google Scholar
[22] Sjögren, P. and Sjölin, P. Convergence properties for the time-dependent Schrödinger equation. Ann. Acad. Sci. Fenn. 14 (1989), 13–25.Google Scholar
[23] Sjölin, P. Regularity of solutions to the Schrödinger equation. Duke Math. J. 55 (1987), 699–715.Google Scholar
[24] Stein, E. M. On limits of sequences of operators. Ann. of Math. 74 (1961), 140–170.Google Scholar
[25] Tao, T. A sharp bilinear restrictions estimate for paraboloids. Geom. Funct. Anal. 13 (2003), 1359–1384.Google Scholar
[26] Tao, T. and Vargas, A. A bilinear approach to cone multipliers II. Applications. Geom. Funct. Anal. 10 (2000), 185–258.Google Scholar
[27] Vega, L. Schrödinger equations: pointwise convergence to the initial data. Proc. Amer. Math. Soc. 102 (1988), 874–878.Google Scholar