论文标题
Kato平滑，Strichartz和均匀的Sobolev估计具有锋利势头的分数操作员
Kato smoothing, Strichartz and uniform Sobolev estimates for fractional operators with sharp Hardy potentials
论文作者
论文摘要
令$ 0 <σ<n/2 $和$ h =（ - δ）^σ+v（x）$ beschrödinger类型运算符在$ \ mathbb r^n $上，带有一类缩放的comminital潜力$ v（x）$，其中包括硬质$ a | x | x | x | x | x |^{ - 2σ} $ quart $ a | （$ C_ {σ，N} $是Hardy不平等$σ$的最佳常数）。在本文中，我们考虑了分解的几项全球估计，以及与$ h $相关的时间依赖性schrödinger方程的解决方案。对于亚临界耦合常数$ a> -c_ {σ，n} $，我们首先证明{\ it统一的resolvent估计} kato-yajima类型的所有$ 0 <σ<n/2 $，这结果与{\ it is kato smooly估计}相当于熟的问题。然后，我们为$σ> 1/2 $建立{\ it strichartz估计}，{\ it {\ it均匀的sobolev估计} kenig--ruiz--sogge类型的$σ\ ge n/（n+1）$。这些扩展了具有相同的特性，该特性具有具有逆向潜力的高阶和分数案例的相反方面的特性。此外，如果$ 1 <σ<n/2 $ $ 1 <σ<n/2 $，并且对于径向对称的数据，我们还可以获得{\ it {\ it rachartz的估计，如果$ n/（2n-1）<σ\ le1 $，则可以将相应的结果扩展到具有艰苦电位的情况。这些论点可以进一步应用于大量的高阶不均匀的椭圆运算符，甚至应用于Laplace操作员的某些远程度量扰动。最后，在关键耦合常数情况下（即$ a = -c_ {σ，n} $），我们显示的结果与亚临界情况相同的结果仍然适用于函数与径向函数的正交。
Let $0<σ<n/2$ and $H=(-Δ)^σ+V(x)$ be Schrödinger type operators on $\mathbb R^n$ with a class of scaling-critical potentials $V(x)$, which include the Hardy potential $a|x|^{-2σ}$ with a sharp coupling constant $a\ge -C_{σ,n}$ ($C_{σ,n}$ is the best constant of Hardy's inequality of order $σ$). In the present paper we consider several sharp global estimates for the resolvent and the solution to the time-dependent Schrödinger equation associated with $H$. In the case of the subcritical coupling constant $a>-C_{σ,n}$, we first prove {\it uniform resolvent estimates} of Kato--Yajima type for all $0<σ<n/2$, which turn out to be equivalent to {\it Kato smoothing estimates} for the Cauchy problem. We then establish {\it Strichartz estimates} for $σ>1/2$ and {\it uniform Sobolev estimates} of Kenig--Ruiz--Sogge type for $σ\ge n/(n+1)$. These extend the same properties for the Schrödinger operator with the inverse-square potential to the higher-order and fractional cases. Moreover, we also obtain {\it improved Strichartz estimates with a gain of regularities} for general initial data if $1<σ<n/2$ and for radially symmetric data if $n/(2n-1)<σ\le1$, which extends the corresponding results for the free evolution to the case with Hardy potentials. These arguments can be further applied to a large class of higher-order inhomogeneous elliptic operators and even to certain long-range metric perturbations of the Laplace operator. Finally, in the critical coupling constant case (i.e. $a=-C_{σ,n}$), we show that the same results as in the subcritical case still hold for functions orthogonal to radial functions.
