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On Pointwise Convergence for Schrödinger Operator in a Convex Domain

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Abstract

In this paper, we prove that the maximal inequality

$$\begin{aligned} \big \Vert \sup _{|t|<1}|e^{it\Delta _D}f(x,y)|\big \Vert _{L^2_{\mathrm{loc}}(\Omega )}\le C\Vert f\Vert _{H^s_D(\Omega )},\quad \forall ~f\in H^s_D(\Omega ) \end{aligned}$$

holds for any \(s>\tfrac{1}{2}\) with \(\Omega =\{(x,y)\in \mathbb {R}^2\mid x>0\}\) and \(\Delta _D=\partial _x^2+(1+x)\partial _y^2\). As a direct application, we obtain the pointwise convergence for the free Schrödinger equation \(i\partial _tu+\Delta _D u=0\) with initial data \(u(0)=f\) inside strictly convex domain.

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Acknowledgements

The author would like to express his gratitude to the anonymous referees for their invaluable comments and suggestions. The author would like to thank Fabrice Planchon for his helpful discussions and encouragement. The author was also partly supported by the ANR-16-TERC-0006-01, ANADEL.

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Authors and Affiliations

  1. Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China

    Jiqiang Zheng

  2. LJAD, Université Côte d’Azur, Nice, France

    Jiqiang Zheng

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  1. Jiqiang Zheng
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Correspondence to Jiqiang Zheng.

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Communicated by Luis Vega.

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Zheng, J. On Pointwise Convergence for Schrödinger Operator in a Convex Domain. J Fourier Anal Appl 25, 2021–2036 (2019). https://doi.org/10.1007/s00041-018-09658-6

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  • DOI: https://doi.org/10.1007/s00041-018-09658-6

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