Remarks on endpoint Strichartz estimates for Schrodinger equations with the critical inverse-square potential

The purpose of this paper is to study the validity of global-in-time Strichartz estimates for the Schrodinger equation on R-n, n >= 3, with the negative inverse-square potential -sigma vertical bar x vertical bar(-2) in the critical case sigma = (n - 2)(2)/4. It turns out that the situation is different from the subcritical case sigma < (n-2)(2)/4 in which the full range of Strichartz estimates is known to be hold. More precisely, splitting the solution into the radial and non-radial parts, we show that (i) the radial part satisfies a weak-type endpoint estimate, which can be regarded as an extension to higher dimensions of the endpoint Strichartz estimate with radial data for the two-dimensional free Schrodinger equation; (ii) other endpoint estimates in Lorentz spaces for the radial part fail in general; (iii) the non-radial part satisfies the full range of Strichartz estimates. (C) 2017 Elsevier Inc. All rights reserved.