A remark on the Half wave Schr\"odinger equation in the energy space

We investigate the Cauchy problem for the half wave Schr\"odinger equation in the energy space. We derive the local well-posedness in the energy space for the odd power type nonlinearities under certain additional assumption for the initial data, namely $\hat{u}_0 \in L^1_{\xi, \eta}(\mathbb{R}^2)$.