Sobolev-Lorentz spaces with an application to the inhomogeneous biharmonic NLS equation

We consider the Cauchy problem for the inhomogeneous biharmonic nonlinear Schrödinger (IBNLS) equation $$i{u}_t+{\mathrm{\Delta }}^{2}u=\lambda \mathrm{\mid }x{\mathrm{\mid }}^{-b}\mathrm{\mid }u{\mathrm{\mid }}^{\sigma }u,u(0)=u_0\in H^s({\mathbb{R}}^d),$$ where $\lambda \in \mathbb{R}$, $d\in \mathbb{N}$, $0\le s<\min \{2+\frac{d}{2},d\}$, $0<b<\min \{4,d-s,2+\frac{d}{2}-s\}$ and $0<\sigma \le {\sigma }_c(s)$ with $\sigma <\mathrm{\infty }$. Here ${\sigma }_c(s)=\frac{8-2b}{d-2s}$ if $s<\frac{d}{2}$, and ${\sigma }_c(s)=\mathrm{\infty }$ if $s\ge \frac{d}{2}$. First, we give some remarks on Sobolev-Lorentz spaces and extend the chain rule under Lorentz norms for the fractional Laplacian $(-\mathrm{\Delta }{)}^{s\mathrm{/}2}$ with $s\in (0,1]$ established by [Discrete Contin. Dyn. Syst. 41 (2021) 5409-5437] to any $s>0$. Applying this estimate and the contraction mapping principle based on Strichartz estimates in Lorentz spaces, we then establish the local well-posedness in $H^s$ for the IBNLS equation in both of subcritical case $\sigma <{\sigma }_c(s)$ and critical case $\sigma ={\sigma }_c(s)$. We also prove that the IBNLS equation is globally well-posed in $H^s$, if the initial data is sufficiently small and $\frac{8-2b}{d}\le \sigma \le {\sigma }_c(s)$ with $\sigma <\mathrm{\infty }$.