Local well-posedness for Chern-Simons gauged $O(3)$ sigma equations under the Lorenz gauge

In this paper, we study the Cauchy problem for the Chern-Simons gauged $O(3)$ sigma model under the Lorenz gauge condition. We prove the local well-posedness of solutions if the initial matter field and gauge field satisfy $(\bm{\phi}_0, \bA_0) \in H^s(\R^2)\times H^{s-\frac12}(\R^2)$, $s>1$, where the critical regularity for $\bm{\phi}_0$ is $s_c=1$. Our proof is based on identifying null forms within the system and utilizing bilinear estimates in wave-Sobolev space.