## 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

### MSC:

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

### Keywords:

generalized functions of Besov-Hardy-Sobolev type

Zbl 0546.46027
Full Text:

### References:

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