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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

Citations:

Zbl 0546.46027
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References:

[1] D. R. Adams, On the existence of capacitary strong type estimates inR n ,Ark. Mat.,14 (1976), 125–140. · Zbl 0325.31008
[2] G.Bourdaud,Sur les opérateurs pseudo-différentiels à coefficients peu reguliers, Diss. Université de Paris-Sud (1983).
[3] B. E. J.Dahlberg, A note on Sobolev spaces,Proc. Symp. in Pure Math. 35, part. 1, 183–185.
[4] C. Fefferman andE. M. Stein, Some maximal inequalities,Amer. J. Math. 93 (1971), 107–115. · Zbl 0222.26019
[5] J. Franke, On the spacesF p,q s of Triebel – Lizorkin type: Pointwise multipliers and spaces on domains,Math. Nachr. 125 (1986), 29–68. · Zbl 0617.46036
[6] S. Fucik, J. Necas andV. Soucek,Einführung in die Variationsrechnung, Teubner (Leipzig, 1977). · Zbl 0384.49001
[7] B. Jawerth, Some observations on Besov and Lizorkin – Triebel spaces,Math. Scand.,40 (1977), 94–104. · Zbl 0358.46023
[8] G. A. Kaljabin, Criteria of the multiplication property and the embedding inC of spaces of Besov – Lizorkin – Triebel type (Russian),Mat. Zametki,30 (1981), 517–526.
[9] M. A. Krasnoselski,Topological methods in the theory of nonlinear integral equations, Macmillan (New York, 1964).
[10] Y.Meyer, Régularité des solutions des équations aux dérivées partielles non linéaires [d’après J.-M. Bony],Sem. Bourbaki, 32e année (1979–80), n 560.
[11] Y. Meyer, Remarques sur un théorème de J. M. Bony,Suppl. Rendiconti Circ. Mat. Palermo, Serie II, n1 (1981), 1–20.
[12] J. Peetre, Interpolation of Lipschitz operators and metric spaces,Matematica (Cluj)12 (35), (1970), 1–20. · Zbl 0217.44504
[13] T. Runst, Pseudo differential operators of the ”exotic” classL 1,1 0 in spaces of Besov and Triebel – Lizorkin type,Annals of Global Analysis and Geometry 3 (1) (1985), 13–28. Mapping properties of non-linear operators in spaces of Triebel-Lizorkin and Besov type · Zbl 0566.46020
[14] T. Runst, Para-differential operators in spaces of Triebel – Lizorkin and Besov type,Z. Anal. Anwendungen 4 (1985), 557–573. · Zbl 0592.35011
[15] G. Stampacchia,Equations elliptiques du second ordre à coefficients discontinus, Univ. Montreal Press (Quebec, 1966). · Zbl 0151.15501
[16] H. Triebel,Fourier analysis and function spaces, Teubner (Leipzig, 1977). · Zbl 0345.42003
[17] H. Triebel,Spaces of Besov-Hardy-Sobolev type, Teubner (Leipzig, 1978). · Zbl 0408.46024
[18] H. Triebel,Theory of function spaces, Birkhäuser (Boston, 1983). · Zbl 0546.46028
[19] H.Triebel, Mapping properties of non-linear operators generated by{\(\Phi\)}(u)=|u| and by holomorphic{\(\Phi\)}(u) in function spaces of Besov-Hardy-Sobolev type. Boundary value problems for elliptic differential equations of type {\(\Delta\)}u=f(x)+{\(\Phi\)}(u),Math. Nachr. 117 (1984).
[20] H. Triebel,Interpolation theory, function spaces, differential operators, North-Holland (Amsterdam-New York-Oxford, 1978).
[21] M. M. Vajnberg,Variational methods for the study of nonlinear operators, Holden-Day (San Francisco, 1964). · Zbl 0136.09601
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