Null-form estimates and nonlinear waves

Abstract

We present bilinear and trilinear as well as quadrilinear null-form estimates arising in connection with the wave-maps problem. These estimates fit into the program of S. Klainerman begun in papers such as [4] and [1]. While the latter used the framework of $X^{s,b}$ spaces, our estimates involve the Banach spaces introduced by D. Tataru in [13] and further developed by T. Tao in [12]. In this paper we attempt to give a somewhat systematic account of the basic properties of (a certain brand of) these spaces in $n=3$ space dimensions. In particular, we solve some cases of the "division and summation problem" for semilinear wave equations whose nonlinearity has null-form structure.