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• W3158577444 abstract "Let $0<sigma<n/2$ and $H=(-Delta)^sigma +V(x)$ be Schrodinger type operators on $mathbb R^n$ with a class of scaling-critical potentials $V(x)$, which include the Hardy potential $a|x|^{-2sigma}$ with a sharp coupling constant $age -C_{sigma,n}$ ($C_{sigma,n}$ 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 Schrodinger 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 $sigmage n/(n+1)$. These extend the same properties for the Schrodinger 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 $n/(2n-1)<sigmale1$, 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." @default.
• W3158577444 created "2021-05-10" @default.
• W3158577444 creator A5018640779 @default.
• W3158577444 creator A5026724002 @default.
• W3158577444 date "2021-09-28" @default.
• W3158577444 modified "2023-10-05" @default.
• W3158577444 title "Kato Smoothing, Strichartz and Uniform Sobolev Estimates for Fractional Operators With Sharp Hardy Potentials" @default.
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• W3158577444 doi "https://doi.org/10.1007/s00220-021-04229-1" @default.
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