Counterexamples for the fractal Schrödinger convergence problem with an Intermediate Space Trick

Daniel Eceizabarrena; Felipe Ponce-Vanegas

doi:10.3934/cpaa.2022122

Article Contents

2022, Volume 21, Issue 11: 3777-3812. Doi: 10.3934/cpaa.2022122

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Counterexamples for the fractal Schrödinger convergence problem with an Intermediate Space Trick

Daniel Eceizabarrena^1, and
Felipe Ponce-Vanegas^2, ,

1.
Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA 01003, USA
2.
BCAM – Basque Center for Applied Mathematics, Bilbao, 48009, Basque Country – Spain

^*Corresponding author: Felipe Ponce-Vanegas
^*Corresponding author: Felipe Ponce-Vanegas

Received: January 2022

Revised: June 2022

Early access: September 2022

Published: November 2022

Daniel Eceizabarrena is supported by the Simons Foundation Collaboration Grant on Wave Turbulence (Nahmod's Award ID 651469), and by the National Science Foundation under Grant No. DMS-1929284 while he was in residence at ICERM - Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Hamiltonian Methods in Dispersive and Wave Evolution Equations program. Felipe Ponce-Vanegas is funded by the Basque Government (BERC 2018-2021), and by the Spanish State Research Agency (SEV-2017-0718), (PGC2018-094528-B-I00 - IHAIP) and (FJC2019-039804-I).

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Abstract

We construct counterexamples for the fractal Schrödinger convergence problem by combining a fractal extension of Bourgain's counterexample and the intermediate space trick of Du–Kim–Wang–Zhang. We confirm that the same regularity as Du's counterexamples for weighted $ L^2 $ restriction estimates is achieved for the convergence problem. To do so, we need to construct the set of divergence explicitly and compute its Hausdorff dimension, for which we use the Mass Transference Principle, a technique originated from Diophantine approximation.

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Mathematics Subject Classification: Primary 35J10; Secondary 42B37 35E15.

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Figure 1. Representation of Theorem 1.1 for $ n = 15 $, where we show the improvement with respect to the former lower bound (1.3). The positive result refers to Theorem 2.3 in [13]

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Figure 2. Arrangement of the slabs of $ F_k $

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Figure 3. In blue, the unit cell $ \widetilde{X}_R^{a_1,a_2} $. On the right, $ \Omega_R^{a_1,a_2} = T(\widetilde{X}_R^{a_1,a_2}) $. In black on the right, the image by $ T $ of the original slabs, and in yellow, the image of the slabs dilated by $ a_1,a_2 $. To apply the Mass Transference Principle, we must prove that $ \Omega_k^{\boldsymbol{{a}}} $ covers a positive portion of the unit cell

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Figure 4. Restrictions on $ \boldsymbol{{u}} = (u_1, u_2, u_3) $

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Figure 5. For fixed $ \alpha $, the maximum regularity is attained on the blurred, green line

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Figure 6. For fixed $ \alpha $, the maximum regularity is attained on the blurred, green line

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Figure 7. For fixed $ \alpha $, the maximum regularity is attained on the blurred, green zone

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Figure 8. Comparison between $ s_m $ and $ s_{m-1} $; see Proposition 5(ⅰ)

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References

[1]	C. An, R. Chu and L. B. Pierce, Counterexamples for high-degree generalizations of the Schrödinger maximal operator, arXiv: 2103.15003.
[2]	J. A. Barceló, J. Bennett, A. Carbery and Keith M Rogers, On the dimension of divergence sets of dispersive equations, Math. Ann., 349 (2011), 599-622. doi: 10.1007/s00208-010-0529-z.
[3]	V. Beresnevich and S. Velani, A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures, Ann. Math., 164 (2006), 971-992. doi: 10.4007/annals.2006.164.971.
[4]	J. Bourgain, A note on the Schrödinger maximal function, J. Anal. Math., 130 (2016), 393-396. doi: 10.1007/s11854-016-0042-8.
[5]	L. Carleson, Some analytic problems related to statistical mechanics, in Euclidean Harmonic Analysis (Proc. Sem., Univ. Maryland, College Park, Md., Springer, Berlin, 1980.
[6]	C-H. Cho and S. Lee, Dimension of divergence sets for the pointwise convergence of the Schrödinger equation, J. Math. Anal. Appl., 411 (2014), 254-260. doi: 10.1016/j.jmaa.2013.09.008.
[7]	C-H. Cho, S. Lee and A. Vargas, Problems on pointwise convergence of solutions to the Schrödinger equation, J. Fourier Anal. Appl., 18 (2012), 972-994. doi: 10.1007/s00041-012-9229-2.
[8]	E. Compaan, R. Lucà and G. Staffilani, Pointwise convergence of the Schrödinger flow, Int. Math. Res. Not., 1 (2021), 599-650. doi: 10.1093/imrn/rnaa036.
[9]	B. E. J. Dahlberg and C. E. Kenig, A note on the almost everywhere behavior of solutions to the Schrödinger equation, Harmonic Analysis (Minneapolis, Minn., Springer, Berlin-New York, 1982.
[10]	X. Du, Upper bounds for Fourier decay rates of fractal measures, J. Lond. Math. Soc., 102 (2020), 1318-1336. doi: 10.1112/jlms.12364.
[11]	X. Du, L. Guth and X. Li, A sharp Schrödinger maximal estimate in $\Bbb R^2$, Ann. of Math., 186 (2017), 607-640. doi: 10.4007/annals.2017.186.2.5.
[12]	X. Du, J. Kim, H. Wang and R. Zhang, Lower bounds for estimates of the Schrödinger maximal function, Math. Res. Lett., 27 (2020), 687-692. doi: 10.4310/MRL.2020.v27.n3.a4.
[13]	X. Du and R. Zhang, Sharp $L^2$ estimates of the Schrödinger maximal function in higher dimensions, Ann. Math., 189 (2019), 837-861. doi: 10.4007/annals.2019.189.3.4.
[14]	D. Eceizabarrena and R. Lucà, Convergence over fractals for the periodic Schrödinger equation, arXiv: 2005.07581.
[15]	D. Eceizabarrena and F. Ponce-Vanegas, Pointwise convergence over fractals for dispersive equations with homogeneous symbol, J. Math. Anal. Appl., 515 (2022), 57 pp. doi: 10.1016/j.jmaa.2022.126385.
[16]	K. Falconer, Fractal Geometry, John Wiley & Sons, Inc., Hoboken, NJ, 2003. doi: 10.1002/0470013850.
[17]	G. H. Hardy and E. M. Wright, An introduction to the theory of numbers. Edited and revised by D. R. Heath-Brown and J. H. Silverman. With a foreword by Andrew Wiles. 6th ed, Oxford: Oxford University Press, 2008.
[18]	W. Li, H. Wang and D. Yan, A note on non-tangential convergence for Schrödinger operators, J. Fourier Anal. Appl., 27 (2021), 14 pp. doi: 10.1007/s00041-021-09862-x.
[19]	Z. Li, J. Zhao and T. Zhao, $L^2$ Schrödinger maximal estimates associated with finite type phases in $\mathbb{R}^2$, arXiv: 2111.00897.
[20]	R. Lucà and K. M. Rogers, Average decay of the Fourier transform of measures with applications, J. Eur. Math. Soc., 21 (2019), 465-506. doi: 10.4171/JEMS/842.
[21]	R. Lucà and K. M. Rogers, A note on pointwise convergence for the Schrödinger equation, Math. Proc. Cambridge Philos. Soc., 166 (2019), 209-218. doi: 10.1017/S0305004117000743.
[22]	R. Lucà and F. Ponce-Vanegas, Convergence over fractals for the Schrödinger equation, arXiv: 2101.02495.
[23]	L. B. Pierce, On Bourgain's counterexample for the Schrödinger maximal function, Q. J. Math., 71 (2020), 1309-1344. doi: 10.1093/qmath/haaa032.
[24]	P. Sjögren and P. Sjölin, Convergence properties for the time-dependent Schrödinger equation, Ann. Acad. Sci. Fenn. Ser. A I Math., 14 (1989), 13-25. doi: 10.5186/aasfm.1989.1428.
[25]	B. Wang and J. Wu, Mass transference principle from rectangles to rectangles in Diophantine approximation, Math. Ann., 381 (2021), 243–317. doi: 10.1007/s00208-021-02187-6.
[26]	X. Wang and C. Zhang, Pointwise convergence of solutions to the Schrödinger equation on manifolds, Canad. J. Math., 71 (2019), 983-995. doi: 10.4153/cjm-2018-001-4.
[27]	D. Žubrinić, Singular sets of Sobolev functions, C. R. Math. Acad. Sci. Paris, 334 (2002), 539-544. doi: 10.1016/S1631-073X(02)02316-6.