Functional composition in $B_{p,k}$ spaces and applications

Let $f(x,z)$, $x\in\mathsf{R}^N$, $z\in \mathsf{C}^M$, be a smooth function in the sense that its Fourier transform has a good behaviour. We study the composition $f(x,u(x))$, where $u$ is in a generalized Hörmander $B_{p,k}$ space in the sense of Björck [1]. As a consequence we obtain results of local solvability and hypoellipticity of semilinear equations of the type $P(D)u+f(x,Q_1(D)u,\ldots,Q_M(D)u)=g$, with $g\in B_{p,k}$, and fully nonlinear elliptic equations.