Local existence with minimal regularity for nonlinear wave equations

In this paper, we consider the Cauchy problem of semilinear wave equations which satisfy the "null condition." We prove that the problem is locally well-posed in H^s+1 for s > ¾ in 1 + 3 dimensions and s > ¼ in 1 + 2 dimensions. Moreover, it is shown that the result in 1 + 2 dimensions is sharp.