Kato Smoothing, Strichartz and Uniform Sobolev Estimates for Fractional Operators With Sharp Hardy Potentials
Kato Smoothing, Strichartz and Uniform Sobolev Estimates for Fractional Operators With Sharp Hardy Potentials
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DOI：
10.1007/s00220-021-04229-1
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发表时间：
2020-02
影响因子：
2.4
通讯作者：
H. Mizutani;X. Yao
中科院分区：
文献类型：
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作者：
H. Mizutani;X. Yao
用一类标度临界势（v (x)）对Schrödinger类型算子进行运算，其中包括具有尖锐耦合常数的Hardy势（是Hardy序不等式的最佳常数）。在本文中，我们考虑了与h相关的时间依赖Schrödinger方程的解的几个尖锐的全局估计。在次临界耦合常数的情况下，我们首先证明了所有问题的一致解估计的Kato-Yajima型，它与Cauchy问题的kato平滑估计等效。然后，我们建立了对于Kenig-Ruiz-Sogge型的strichartz估计和一致Sobolev估计。这些将具有平方反比势的Schrödinger运算符的相同性质扩展到高阶和分数阶情况。此外，我们还获得了一般初始数据if和径向对称数据if的改进的正则性Strichartz估计，将自由演化的相应结果推广到Hardy势的情况。这些论证可以进一步应用于一类大的高阶非齐次椭圆算子，甚至应用于拉普拉斯算子的某些远程度量摄动。最后，在临界耦合常数情况下（即），我们证明了与径向函数正交的函数在亚临界情况下的相同结果仍然成立。
Letandbe Schrödinger type operators onwith a class of scaling-critical potentialsV(x), which include the Hardy potentialwith a sharp coupling constant(is the best constant of Hardy’s inequality of order). In the present paper we consider several sharp global estimates for the resolvent and the solution to the time-dependent Schrödinger equation associated withH. In the case of the subcritical coupling constant, we first proveuniform resolvent estimatesof Kato–Yajima type for all, which turn out to be equivalent toKato smoothing estimatesfor the Cauchy problem. We then establishStrichartz estimatesforanduniform Sobolev estimatesof Kenig–Ruiz–Sogge type for. These extend the same properties for the Schrödinger operator with the inverse-square potential to the higher-order and fractional cases. Moreover, we also obtainimproved Strichartz estimates with a gain of regularitiesfor general initial data ifand for radially symmetric data if, 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.,), we show that the same results as in the subcritical case still hold for functions orthogonal to radial functions.