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

Let $0<\sigma-C_{\sigma,n}$, first prove {\it uniform estimates} Kato--Yajima all $0<\sigma1/2$ Sobolev Kenig--Ruiz--Sogge $\sigma\ge n/(n+1)$. These extend same properties operator inverse-square higher-order fractional cases. Moreover, also obtain improved gain regularities} general initial data if $1<\sigma<n/2$ radially symmetric $n/(2n-1)<\sigma\le1$, extends corresponding results free evolution potentials. arguments can further applied large inhomogeneous elliptic even certain long-range metric perturbations Laplace operator. Finally, in critical (i.e. $a=-C_{\sigma,n}$), show that as still hold functions orthogonal radial functions.