论文标题
$ l^p $ eigenfunction范围用于分数schrödinger操作员
$L^p$ eigenfunction bounds for fractional Schrödinger operators on manifolds
论文作者
论文摘要
本文专门针对schödinger-type运算符$（ - Δ_g）^{α/2} +v $的$ l^p $界限。运算符$（-Δ_G）^{α/2} $是根据$-Δ_G$的特征函数来定义的。我们还获得了准示象和光谱簇估计。作为一个应用程序，我们得出了strichartz估计分数波方程$（\ partial_t^2+（-δ_g）^{α/2}+v）u = 0 $。 Bourgain-Shao-Sogge-Yao和Shao-Yao最近开发的Wave内核技术在本文中起着关键作用。我们使用几个本地运营商和一些良好的错误术语来构建一个新的复制操作员。此外，我们将证明这些本地运营商满足了Kenig-Ruiz-Sogge的“统一Sobolev估计”的某些可变系数版本。这些使我们能够处理至关重要的电位$ v $并证明了Quasimode的估计值。
This paper is dedicated to $L^p$ bounds on eigenfunctions of a Schödinger-type operator $(-Δ_g)^{α/2} +V$ on closed Riemannian manifolds for critically singular potentials $V$. The operator $(-Δ_g)^{α/2}$ is defined spectrally in terms of the eigenfunctions of $-Δ_g$. We obtain also quasimodes and spectral clusters estimates. As an application, we derive Strichartz estimates for the fractional wave equation $(\partial_t^2+(-Δ_g)^{α/2}+V)u=0$. The wave kernel techniques recently developed by Bourgain-Shao-Sogge-Yao and Shao-Yao play a key role in this paper. We construct a new reproducing operator with several local operators and some good error terms. Moreover, we shall prove that these local operators satisfy certain variable coefficient versions of the "uniform Sobolev estimates" by Kenig-Ruiz-Sogge. These enable us to handle the critically singular potentials $V$ and prove the quasimode estimates.
