  We show new results of wellposedness for the Cauchy problem for the half wave
equation with power-type nonlinear terms. For the purpose, we propose two
approaches on the basis of the contraction-mapping argument. One of them relies
upon the $L_t^q L_x^\infty$ Strichartz-type estimate together with the chain
rule of fairly general fractional orders. This chain rule has a significance of
its own. Furthermore, in addition to the weighted fractional chain rule
established in Hidano, Jiang, Lee, and Wang (arXiv:1605.06748v1 [math.AP]), the
other approach uses weighted space-time $L^2$ estimates for the inhomogeneous
equation which are recovered from those for the second-order wave equation. In
particular, by the latter approach we settle the problem left open in
Bellazzini, Georgiev, and Visciglia (arXiv:1611.04823v1 [math.AP]) concerning
the local wellposedness in $H^{s}_{{\rm rad}}({\mathbb R}^n)$ with $s>1/2$.
