Remarks on fractional Schr\"odinger equation on $\mathbb{R}^d$

We prove the well-posed results in sub-critical and critical cases for the pure power-type nonlinear fractional Schrodinger equations on $\mathbb{R}^d$. These results extend the previous ones in \cite{HongSire} for $\sigma\geq 2$. This covers the well-known result for the Schrodinger equation $\sigma = 2$ given in \cite{CazenaveWeissler}. In the case $\sigma \in (0,2)\backslash \{1\}$, we give the local well-posedness in sub-critical case for all exponent $\nu > 1$ in contrast of ones in \cite{HongSire}. This also generalizes the ones of \cite{ChoHwangKwonLee} when $d = 1$ and of \cite{GuoHuo} when $d \geq 2$ where the authors considered the cubic fractional Schrodinger equation with $\sigma\in (1,2)$. We also give the global existence in energy space under some assumptions. We finally prove the local well-posedness in sub-critical and critical cases for the pure power-type nonlinear fractional wave equations.