Let $f(x,z)$, $x\in\mathbb{R}^N$, $z\in \mathbb{C}^M$, 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 H\"ormander $B_{p,k}$ space in the sense of Bj\"orck~\cite{bj}. 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.
Functional composition in Bp,k spaces and applications
OLIARO, Alessandro
2006-01-01
Abstract
Let $f(x,z)$, $x\in\mathbb{R}^N$, $z\in \mathbb{C}^M$, 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 H\"ormander $B_{p,k}$ space in the sense of Bj\"orck~\cite{bj}. 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.
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