Kato smoothing, Strichartz and uniform Sobolev estimates for fractional operators with sharp Hardy potentials

Abstract

Let $0<\sigma -C_{\sigma,n}$, we first prove {\it uniform resolvent estimates} of Kato--Yajima type for all $0<\sigma 1/2$ 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.