Fractional derivatives of composite functions and the Cauchy problem for the nonlinear half wave equation

Abstract: 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 $s>1/2$.