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

We consider the Cauchy problem for the inhomogeneous biharmonic nonlinear Schr\"{o}dinger (IBNLS) equation $$iu_{t} +\Delta^{2} u=\lambda |x|^{-b}|u|^{\sigma}u,\;u(0)=u_{0} \in H^{s} (\mathbb R^{d}),$$ where $\lambda\in \mathbb R$, $d\in \mathbb N$, $0\le s<\min\left\{2+\frac{d}{2},d\right\}$, $0<b<\min \left\{4,\; d-s,\; 2+\frac{d}{2}-s \right\}$ and $0<\sigma\le \sigma_{c}(s)$ with $\sigma<\infty$. Here $\sigma_{c}(s)=\frac{8-2b}{d-2s}$ if $s<\frac{d}{2}$, and $\sigma_{c}(s)=\infty$ 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 $(-\Delta)^{s/2}$ 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 $\sigma<\infty$.