Kato Smoothing, Strichartz and Uniform Sobolev Estimates for Fractional Operators With Sharp Hardy Potentials

Abstract

Let $0<\sigma<n/2$ and $H=(-\Delta)^\sigma +V(x)$ be Schr\"odinger type operators on $\mathbb R^n$ with a class of scaling-critical potentials $V(x)$, which include the Hardy potential $a|x|^{-2\sigma}$ with a sharp coupling constant $a\ge -C_{\sigma,n}$ ($C_{\sigma,n}$ is the best constant of Hardy's inequality of order $\sigma$). In the present paper we consider several sharp global estimates for the resolvent and the solution to the time-dependent Schr\"odinger equation associated with $H$. In the case of the subcritical coupling constant $a>-C_{\sigma,n}$, we first prove {\it uniform resolvent estimates} of Kato--Yajima type for all $0<\sigma<n/2$, which turn out to be equivalent to {\it Kato smoothing estimates} for the Cauchy problem. We then establish {\it Strichartz estimates} for $\sigma>1/2$ and {\it uniform Sobolev estimates} of Kenig--Ruiz--Sogge type for $\sigma\ge n/(n+1)$. These extend the same properties for the Schr\"odinger 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 $1<\sigma<n/2$ and for radially symmetric data if $n/(2n-1)<\sigma\le1$, 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. $a=-C_{\sigma,n}$), we show that the same results as in the subcritical case still hold for functions orthogonal to radial functions.