Strichartz estimates for Schrödinger equation with singular and time dependent potentials and application to NLS equations

We establish inhomogeneous Strichartz estimates for the Schrödinger equation with singular and time-dependent potentials for some non-admissible pairs. Our work extends the results of Vilela (Trans Am Math Soc 359:2123–2136, 2007) and Foschi (J Hyperbolic Differ Equ 2:1–24, 2005), where they proved the results in the absence of potential. It also extends the works of Pierfelice (Asymptot Anal 47:1–18, 2006) and Burq et al. (J Funct Anal 203:519–549, 2003), who proved the estimates for admissible pairs. We also extend the recent work of Mizutani et al. (J Funct Anal 278:108350, 2020), and as an application of it, we improve the stability result of Kenig–Merle (Invent Math 166:645–675, 2006), which in turn establishes a proof (alternative to Yang in Commun Pure Appl Anal 20:77, 2020) of the existence of scattering solution for the energy-critical focusing NLS with inverse-square potentials.