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

Abstract

The purpose of this paper is to study the validity of global-in-time Strichartz estimates for the Schrödinger equation on Rn, n≥3, with the negative inverse-square potential −σ|x|−2 in the critical case σ=(n−2)2/4. It turns out that the situation is different from the subcritical case σ<(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 Schrödinger 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.