Mapping properties of nonlinear operators in spaces of Triebel-Lizorkin and Besov type. (English) Zbl 0644.46022

The spaces \(B^ s_{p,q}\), \(F^ s_{p,q}\) of generalized functions of Besov-Hardy-Sobolev type in \({\mathbb{R}}^ n\) are defined in H. Triebel [Theory of function spaces (1983; Zbl 0546.46027)] for \(0<p\), \(q\geq \infty\), \(s\in {\mathbb{R}}\). Here, \(E=E^ s_{p,q}\) is the space of real elements of one of the spaces \(B^ s_{p,q}\), \(F^ s_{p,q}\). Given a sufficiently smooth function G:\({\mathbb{R}}\to {\mathbb{R}}\) such that \(G(0)=0\), G defines a mapping which assigns a real function \(G(f)=G\circ f\) to every real function \(f:{\mathbb{R}}^ n\to {\mathbb{R}}\). This paper studies conditions so that G yields a mapping from the space E into itself. It extends recent results due to G. Bourdaud, B. E. J. Dahlberg, J. Franke, Y. Meyer, J. Peetre, H. Triebel, etc.
The first result is a necessary condition: Suppose that \(2\leq 1+p<n\) and \(1+1/p<s<n/p\), or that \(2<1+1/p<n\) and \(n/p-(n-1-1/p)<s<n/p\). Then every \(G\in C^ 2({\mathbb{R}})\) which maps E into itself is of the type \(G(t)=ct\), \(c\in {\mathbb{R}}.\)
Another result supposes that \(G'\in L^{\infty}\). Then GE\(\subset E\) in the following cases:
1) \(E=real\) elements of \(B^ s_{p,q}\), \(1\leq p\) and \(0<s<1\), or \(1<1/p<1+1/n\) and \(n(1/p-1)<s<1.\)
2) \(E=real\) elements of \(F^ s_{p,q}\), p,q\(\geq 1\), \(0<s<1\), or \(p\geq 1\), \(1<1/q<1+1/n\), \(n(1/q-1)<s<1\), or \(q\geq p\), \(1<1/p<1+1/n\), \(n(1/p- 1)<s<1\). The case when \(G\in Lip(\alpha)\), \(0<\alpha \leq 1\), is also studied.
After proving that \(E\cap L^{\infty}\) is a multiplication algebra for s large enough, the author shows that \(G(f)\in E\cap L^ p\) if \(f\in E\cap L^{\infty}\) under suitable assumptions.
The paper also gives conditions so that the mapping \(x\mapsto h(x,u_ 1(x),...,u_ m(x))\) is in E when h is defined over \({\mathbb{R}}^{n+m}\) and \(u_ 1,...,u_ m\) are in appropriate spaces of type E.
Reviewer: P.Jeanquartier


46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47F05 General theory of partial differential operators


Zbl 0546.46027
Full Text: DOI


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