On the Schrödinger maximal function in higher dimension

By Jean Bourgain. 2012

135 citations recorded by the source corpus.

AbstractNew estimates on the maximal function associated to the linear Schrödinger equation are established. It is shown that the almost everywhere convergence property of eitΔf for t → 0 holds for f ∈ Hs(ℝn), $$s > \tfrac{1} {2} - \tfrac{1} {{4n}}$$, which is a new result for n ≥ 3. We also construct examples showing that $$s \geqslant \tfrac{1} {2} - \tfrac{1} {n}$$ is certainly necessary when n ≥ 4. 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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4f53d4c360f1e92398f0d29b9a64cdf6118969f8

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In Applied Mathematics

1. Ruixiang Zhang
2. Xiumin Du
3. Jonathan Bennett
4. Larry Guth
5. Jean Bourgain
6. Bochen Liu

In Mathematical Physics

1. Ruixiang Zhang
2. Xiumin Du
3. Jonathan Bennett
4. Larry Guth
5. Jean Bourgain

Citations in the corpus

• Aspects of Multilinear Harmonic Analysis Related to Transversality (2014, background): “1 and the Bourgain–Guth method, see [20], [21], [23], [52] and [36].”
• Sharp L2 estimates of the Schrödinger maximal function in higher dimensions (2018, methodology): “6, we will use a broad-narrow analysis, which has similar spirit as the techniques in the work of Bourgain-Guth [6], Bourgain [3], Bourgain-Demeter [5] and Guth [19].”
• A note on pointwise convergence for the Schrödinger equation (2017, unknown): “1) holds as long as s > 2n−1 4n ; see [12, 3].”