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 $({\mathit{\phi }}_0,{\text{bA}}_0)\in H^s({\mathbb{R}}^2)\times H^{s-\frac{1}{2}}({\mathbb{R}}^2)$, $s>1$, where the critical regularity for ${\mathit{\phi }}_0$ is $s_c=1$. Our proof is based on identifying null forms within the system and utilizing bilinear estimates in wave-Sobolev space.