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 ${\mathbb{R}}^n$, $n\ge 3$, with the negative inverse-square potential $-\sigma |x{|}^{-2}$ 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 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.

18 pages; revised version; to appear in JDE

References in corpus (1)

• Strichartz estimates for the Wave and Schrodinger Equations with the Inverse-Square Potential

Cited by in corpus (3)

• The $W^{s,p}$-boundedness of stationary wave operators for the Schrödinger operator with inverse-square potential
• Global Kato smoothing and Strichartz estimates for higher-order Schrödinger operators with rough decay potentials
• Resolvent Estimates for the 3D Schrödinger Operator with Inverse-Square Potential