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Strichartz estimates for Schrödinger equations in weighted $L^2$ spaces and their applications

  • * Corresponding author: Ihyeok Seo

    * Corresponding author: Ihyeok Seo

Y. Koh was supported by NRF Grant 2016R1D1A1B03932049 (Republic of Korea). I. Seo was supported by the TJ Park Science Fellowship of POSCO TJ Park Foundation and by the NRF grant funded by the Korea government(MSIP) (No. 2017R1C1B5017496)

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  • We obtain weighted $L^2$ Strichartz estimates for Schrödinger equations $i\partial_tu+(-\Delta)^{a/2}u=F(x, t)$, $u(x, 0)=f(x)$, of general orders $a>1$ with radial data $f, F$ with respect to the spatial variable $x$, whenever the weight is in a Morrey-Campanato type class. This is done by making use of a useful property of maximal functions of the weights together with frequency-localized estimates which follow from using bilinear interpolation and some estimates of Bessel functions. As consequences, we give an affirmative answer to a question posed in [1] concerning weighted homogeneous Strichartz estimates, and improve previously known Morawetz estimates. We also apply the weighted $L^2$ estimates to the well-posedness theory for the Schrödinger equations with time-dependent potentials in the class.

    Mathematics Subject Classification: Primary: 35B45, 35A01; Secondary: 35Q40, 42B35.


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  • Figure 1.  The region of $(s,1/p)$ for (7) particularly when $a=2$

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