Regular Strichartz estimates in Lorentz-type spaces with application to the $H^s$-critical inhomogeneous biharmonic NLS equation

In this paper, we investigate the Cauchy problem for the $H^s$-critical inhomogeneous biharmonic nonlinear Schrödinger (IBNLS) equation $$i{u}_t\pm {\mathrm{\Delta }}^{2}u=\lambda \mathrm{\mid }x{\mathrm{\mid }}^{-b}\mathrm{\mid }u{\mathrm{\mid }}^{\sigma }u,u(0)=u_0\in H^s({\mathbb{R}}^d),$$ where $\lambda \in \mathbb{C}$, $d\ge 3$, $1\le s<\frac{d}{2}$, $0<b<\min \{4,2+\frac{d}{2}-s\}$ and $\sigma =\frac{8-2b}{d-2s}$. First, we study the properties of Lorentz-type spaces such as Besov-Lorentz spaces and Triebel-Lizorkin-Lorentz spaces. We then derive the regular Strichartz estimates for the corresponding linear equation in Lorentz-type spaces. Using these estimates, we establish the local well-posedness as well as the small data global well-posedness and scattering in $H^s$ for the $H^s$-critical IBNLS equation under less regularity assumption on the nonlinear term than in the recent work \cite{AKR24}. This result also extends the ones of \cite{SP23,SG24} by extending the validity of $d$, $b$ and $s$. Finally, we give the well-posedness result in the homogeneous Sobolev spaces ${\dot{H}}^s$.