Sharp Estimates on Divergence Sets of Convergence Rate for Schrödinger Mean

Abstract

We study the Hausdorff dimension of divergence sets for the convergence rate of fractional Schrödinger operators $e^{{{it}(-\Delta)}^{m \over 2}} \, f$, where f ∈ H^s. All results are sharp except at the endpoints.

Keywords

Schrödinger mean / maximal functions / divergence sets / Hausdorff dimension / convergence rate / 42B25

References

[1] Barceló JA, Bennett J, Carbery A, Rogers KM. On the dimension of divergence sets of dispersive equations. Math. Ann., 2011, 349(3): 599-622.

[2] Bourgain J. Some new estimates on oscillatory integrals. Essays on Fourier Analysis in Honor of Elias M. Stein (Princeton, NJ, 1991), 1995. Princeton, NJ, Princeton Univ. Press: 83-112. 42

[3] Bourgain J. On the Schrödinger maximal function in higher dimension. Tr. Mat. Inst. Steklova, 2013, 280: 53-66

[4] Bourgain J. A note on the Schrödinger maximal function. J. Anal. Math., 2016, 130: 393-396.

[5] Cao Z, Fan D, Wang M. The rate of convergence on Schrödinger operator. Illinois J. Math., 2018, 62(1–4): 365-380

[6] Carleson L. Some analytic problems related to statistical mechanics. Euclidean Harmonic Analysis (Proc. Sem., Univ. Maryland, College Park, Md., 1979), 1980. Berlin, Springer: 5-45. 779

[7] Cho C., Shiraki S., Dimension of divergence sets of oscillatory integrals with concave phase. J. Geom. Anal., 2025, 35(1): Paper No. 21, 18 pp.

[8] Cowling M. Pointwise behavior of solutions to Schroödinger equations. Harmonic Analysis (Cortona, 1982), 1983. Berlin, Springer: 83-90. 992

[9] Dahlberg BEJ, Kenig CE. A note on the almost everywhere behavior of solutions to the Schrödinger equation. Harmonic Analysis (Minneapolis, Minn., 1981), 1982. Berlin–New York, Springer: 205-209. 908

[10] Du X., Guth L., Li X., Zhang R., Pointwise convergence of Schrödinger solutions and multilinear refined Strichartz estimates. Forum Math. Sigma, 2018, 6: Paper No. e14, 18 pp.

[11] Du X, Zhang R. Sharp L² estimates of the Schrödinger maximal function in higher dimensions. Ann. of Math. (2), 2019, 189(3): 837-861.

[12] Eceizabarrena D, Ponce-Vanegas F. Counterexamples for the fractal Schrödinger convergence problem with an intermediate space trick. Commun. Pure Appl. Anal., 2022, 21(11): 3777-3812.

[13] Kenig CE, Ponce G, Vega L. Oscillatory integrals and regularity of dispersive equations. Indiana Univ. Math. J., 1991, 40(1): 33-69.

[14] Lee S., On pointwise convergence of the solutions to Schrödinger equations in ℝ². Int. Math. Res. Not., 2006, 2006: Art. ID 32597, 21 pp.

[15] Li D, Li J, Xiao J. An upbound of Hausdorff’s dimension of the divergence set of the fractional Schrödinger operator on H^s(ℝⁿ). Acta Math. Sci. Ser. B (Engl. Ed.), 2021, 41(4): 1223-1249

[16] Lucà R, Ponce-Vanegas F. Convergence over fractals for the Schrödinger equation. Indiana Univ. Math. J., 2022, 71(6): 2283-2307.

[17] Lucà R, Rogers KM. Coherence on fractals versus pointwise convergence for the Schrödinger equation. Comm. Math. Phys., 2017, 351(1): 341-359.

[18] Lucà R, Rogers KM. Average decay of the Fourier transform of measures with applications. J. Eur. Math. Soc. (JEMS), 2019, 21(2): 465-506.

[19] Lucà R, Rogers KM. A note on pointwise convergence for the Schrödinger equation. Math. Proc. Cambridge Philos. Soc., 2019, 166(2): 209-218.

[20] Rogers KM, Villarroya P. Sharp estimates for maximal operators associated to the wave equation. Ark. Mat., 2008, 46(1): 143-151.

[21] Sjögren P, Sjölin P. Convergence properties for the time-dependent Schrödinger equation. Ann. Acad. Sci. Fenn. Ser. A I Math., 1989, 14(1): 13-25

[22] Sjölin P. Regularity of solutions to the Schrödinger equation. Duke Math. J., 1987, 55(3): 699-715.

[23] Vega L. Schrödinger equations: pointwise convergence to the initial data. Proc. Amer. Math. Soc., 1988, 102(4): 874-878

[24] Walther BG. Maximal estimates for oscillatory integrals with concave phase. Harmonic Analysis and Operator Theory (Caracas, 1994), 1995. Providence, RI, Amer. Math. Soc.: 485-495. 189

RIGHTS & PERMISSIONS

Peking University