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»ã±¨ÌáÒª£ºLet 0<\sigma<n/2 and H=(-\Delta)^\sigma +V(x) be Schr\"odinger type operators with certain scaling-critical potentials V(x), which include the Hardy potential C|x|^{-2\sigma} with a subcritical coupling constant C as typical examples. In this talk, we consider several global estimates for the resolvent and the solution to the time-dependent Schr\"odinger equation associated with H. We first prove the uniform resolvent estimates of Kato-Yajima type for all 0<\sigma<n/2 using a version of Mourre's theory, which turn out to be equivalent to Kato smoothing estimates for the Cauchy problem. Using these estimates, we then obtain Strichartz estimates for \sigma>1/2 and uniform Sobolev estimates of Kenig-Ruiz-Sogge type for \sigma\ge n/(n+1). These completely extend the same properties for the Schr\"odinger operator with the inverse-square potential to the higher-order and fractional cases. Moreover, we point out that 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.

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