Geodesic
On the Schr\"odinger maximal function in higher dimension
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Orthogonal series, approximation theory, and related problems, Tome 280 (2013), pp. 53-66.

Voir la notice de l'article provenant de la source Math-Net.Ru

New estimates on the maximal function associated to the linear Schrödinger equation are established. It is shown that the almost everywhere convergence property of $e^{it\Delta}f$ for $t\to0$ holds for $f\in H^s(\mathbb R^n)$, $s>\frac12-\frac1{4n}$, which is a new result for $n\geq3$. We also construct examples showing that $s\geq\frac12-\frac1n$ is certainly necessary when $n\geq4$. This is a further contribution to our understanding of how L. Carleson's result for $n=1$ generalizes in higher dimension. From the methodological point of view, crucial use is made of J. Bourgain and L. Guth's results and techniques that are based on the multi-linear oscillatory integral theory developed by J. Bennett, T. Carbery and T. Tao. 
@article{TM_2013_280_a3,
     author = {J. Bourgain},
     title = {On the {Schr\"odinger} maximal function in higher dimension},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {53--66},
     publisher = {mathdoc},
     volume = {280},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM_2013_280_a3/}
}
TY  - JOUR
AU  - J. Bourgain
TI  - On the Schr\"odinger maximal function in higher dimension
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2013
SP  - 53
EP  - 66
VL  - 280
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_2013_280_a3/
LA  - en
ID  - TM_2013_280_a3
ER  - 
%0 Journal Article
%A J. Bourgain
%T On the Schr\"odinger maximal function in higher dimension
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2013
%P 53-66
%V 280
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_2013_280_a3/
%G en
%F TM_2013_280_a3
J. Bourgain. On the Schr\"odinger maximal function in higher dimension. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Orthogonal series, approximation theory, and related problems, Tome 280 (2013), pp. 53-66. http://geodesic.mathdoc.fr/item/TM_2013_280_a3/

[1] Bennett J., Carbery T., Tao T., “On the multilinear restriction and Kakeya conjectures”, Acta Math., 196 (2006), 261–302 | DOI | MR | Zbl

[2] Bourgain J., “A remark on Schrödinger operators”, Isr. J. Math., 77 (1992), 1–16 | DOI | MR | Zbl

[3] Bourgain J., “Some new estimates on oscillatory integrals”, Essays on Fourier analysis in honor of Elias M. Stein, Proc. conf. (Princeton, NJ, 1991), Princeton Math. Ser., 42, Princeton Univ. Press, Princeton, NJ, 1995, 83–112 | MR | Zbl

[4] Bourgain J., Guth L., “Bounds on oscillatory integral operators based on multilinear estimates”, Geom. Funct. Anal., 21:6 (2011), 1239–1295 | DOI | MR | Zbl

[5] Carleson L., “Some analytic problems related to statistical mechanics”, Euclidean harmonic analysis, Proc. semin. (Univ. Maryland, 1979), Lect. Notes Math., 779, Springer, Berlin, 1980, 5–45 | DOI | MR

[6] Dahlberg B.E.J., Kenig C.E., “A note on the almost everywhere behavior of solutions to the Schrödinger equation”, Harmonic analysis, Proc. conf. (Minneapolis, 1981), Lect. Notes Math., 908, Springer, Berlin, 1982, 205–209 | DOI | MR

[7] Lee S., “On pointwise convergence of the solutions to Schrödinger equations in $\mathbb R^2$”, Int. Math. Res. Not., 2006 (2006), 32597 | MR | Zbl

[8] Moyua A., Vargas A., Vega L., “Schrödinger maximal function and restriction properties of the Fourier transform”, Int. Math. Res. Not., 1996:16 (1996), 793–815 | DOI | MR | Zbl

[9] Sjölin P., “Regularity of solutions to the Schrödinger equation”, Duke Math. J., 55:3 (1987), 699–715 | DOI | MR | Zbl

[10] Tao T., “A sharp bilinear restriction estimate for paraboloids”, Geom. Funct. Anal., 13:6 (2003), 1359–1384 | DOI | MR | Zbl

[11] Tao T., Vargas A., “A bilinear approach to cone multipliers. I: Restriction estimates”, Geom. Funct. Anal., 10:1 (2000), 185–215 | DOI | MR | Zbl

[12] Tao T., Vargas A., “A bilinear approach to cone multipliers. II: Applications”, Geom. Funct. Anal., 10:1 (2000), 216–258 | DOI | MR | Zbl

[13] Vega L., “Schrödinger equations: Pointwise convergence to the initial data”, Proc. Amer. Math. Soc., 102:4 (1988), 874–878 | MR | Zbl