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On boundedness of superposition operators in spaces of Triebel-Lizorkin type. (English) Zbl 0693.46039
This paper contains certain results announced in the previous note [W. Sickel, Seminar analysis 1987, Teubner Texte Math. 106, 319-326 (1988)]. Therefore the notations introduced there will here be employed with no explication. Letting \(0<p<\infty\), \(0<q\leq \infty\), \(\sigma_{\rho}+1<s<n/p\) and \(\rho >\sigma_{\rho}\), we observe first that \(T_ G\) is a bounded mapping \(\cup_{0<q\leq \infty}\tilde F^ s_{p,q}\to \cap_{0<r\leq \infty}F^{\rho}_{p,r}\) and that for any \(0<r\), a constant c is given such that \(\| G(f)| F^{\rho}_{p,r}\| \leq c(\| f| F^ s_{p,\infty}\| +\| f| F^ s_{p,\infty}\|^{\rho}),\) \(f\in \tilde F^ s_{p,\infty}\) (Theorem 1).
Particular interests are taken in the case where \(G(t)=t^{\mu}\). Then putting \(T_ G=T_{\mu}\), \(s_{\mu}=s-(\mu -1)((n/p)-s)\) and letting \(\mu =m=2,3,4,...\), \(n\cdot \max (0,(1/p)-(1/m))<s<n/p\), one observes that for any \(0<r\leq \infty\), there exists a constant c such that \(\| f^ m| F^{s_ m}_{p,r}\| \leq c\| f| F^ s_{p,q}\|^ m\), \(f\in F^ s_{p,q}\) (Theorem 4).
In case \(\mu >1\) is not necessarily an integer and \(0<s<n/p\), \(\sigma_{\rho}<s_{\mu}<\mu\), it is seen that \(T_{\mu}\) is a bounded mapping \(\cup_{0<q\leq \infty}\tilde F^ s_{p,q}\to \cap_{0<r\leq \infty}F^{s_{\mu}}_{p,r}\) and for any \(0<r\leq \infty\) the inequality \(\| f^{\mu}| F^{s_{\mu}}_{p,r}\| \leq c\| f| F^ s_{p,\infty}\|^{\mu},\) \(f\in \tilde F^ s_{p,\infty}\), holds true for some constant c. If, in particular, \(\max (1,\sigma_{\rho})<\mu <n/p\) and \(s=1+((\mu -1)/\mu)(n/p)\), then \(T_{\mu}\) is a bounded mapping \(\cup_{0<q\leq \infty}\tilde F^ s_{p,q}\to F^{\mu}_{p,\infty}\) and there exists a constant c such that \(\| f^{\mu}| F^{\mu}_{p,\infty}\| \leq c\| f| F^ s_{p,\infty}\|^{\mu}\), \(f\in \tilde F^ s_{p,\infty}\) (Theorem 5).
As for weakening the differentiability of G, there are given particular studies, e.g. Theorem 2, 3. Furthermore, we are given 28 remarks, most of which are devoted to the examination of various inequalities that are assumed among parameters in each theorem.
Reviewer: K.Yoshinaga

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47B38 Linear operators on function spaces (general)
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References:
[1] D. R. Adams: Qn the existence of capacitary strong type estimates in \(R_n\). Ark. Mat. 14 (1976), 125-140. · Zbl 0325.31008 · doi:10.1007/BF02385830
[2] G. Bourdaud: Sur les opérateurs peudo-différentiels à coefficients peu réguliers. Diss. Univ. de Paris-Sud, 1983.
[3] B. E. J. Dahlberg: A note on Sobolev spaces. Proc. Symp. Pure Math. 35, 1979, part I, 183-185. · Zbl 0421.46027
[4] D. E. Edmunds, H. Triebel: Remarks on nonlinear elliptic equations of the type \(\Delta u + u = |u|^p + f\) bounded domains. J. London Math. Soc. (2) 31 (1985), 331-339.
[5] C. Fefferman, E. M. Stein: Some maximal inequalities. Amer. J. Math. 93 (1971), 107-115. · Zbl 0222.26019 · doi:10.2307/2373450
[6] C. Fefferman, E. M. Stein: \(H^p\) spaces of several variables. Acta Math. 129 (1972), 137-193. · Zbl 0257.46078 · doi:10.1007/BF02392215
[7] M. Marcus, V. J. Mizel: Absolute continuity on tracks and mappings of Sobolev spaces. Rational Mech. Anal. 42 (1972), 294-320. · Zbl 0236.46033 · doi:10.1007/BF00251378
[8] M. Marcus, V. J. Mizel: Complete characterizations of functions which act via superposition on Sobolev spaces. Trans. Amer. Math. Soc. 251 (1979), 187-218. · Zbl 0417.46035 · doi:10.2307/1998689
[9] J. Marschall: Pseudo-differential operators with nonregular symbols. Thesis, FU Berlin (West), 1985. · Zbl 0695.47047
[10] Y. Meyer: Remarques sur un théorème de J. M. Bony. Suppl. Rendiconti Circ. Mat. Palermo Serie II, 1 (1981), 1-20. · Zbl 0473.35021
[11] S. Mizohata: Lectures on the Cauchy problem. Tata Institute, Bombay 1965. · Zbl 0176.08502
[12] J. Peetre: Interpolation of Lipschitz operators and metric spaces. Matematica (Cluj) 12 (35) (1970), 325-334. · Zbl 0217.44504
[13] J. Peetre: On spaces of Triebel-Lizorkin type. Ark. Mat. 13 (1975), 123-130. · Zbl 0302.46021 · doi:10.1007/BF02386201
[14] J. Rauch: An \(L^2\)-proof that \(H^s\) is invariant under nonlinear maps for \(s > \frac{n}{2}\). Global Analysis, Analysis on Manifolds, Teubner-Texte Math., 57, Teubner, Leipzig 1983.
[15] Th. Runst: Para-differential operators in spaces of Triebel-Lizorkin and Besov type. Z. Anal. Anwendungen 4 (1985), 557-573. · Zbl 0592.35011
[16] Th. Runst: Mapping properties of non-linear operators in spaces of Triebel-Lizorkin and Besov type. Anal. Math. 12 (1986), 313-346. · Zbl 0644.46022 · doi:10.1007/BF01909369
[17] W. Sickel: On pointwise multipliers in Besov-Triebel-Lizorkin spaces. Seminar Analysis 1986 (ed. by B.-W. Schulze and H. Triebel), Teubner-Texte Math., 96, Teubner, Leipzig 1987.
[18] W. Sickel: Superposition offunctions in spaces of Besov-Triebel-Lizorkin type. The critical case \(1 < s < \frac{n}{p}\). Seminar Analysis 1987 (ed. by B.-W. Schulze and H. Triebel), Teubner-Texte Math. 106, Teubner, Leipzig, 1988.
[19] G. Stampacchia: Equations elliptiques du second ordre à coefficients discontinues. Univ. Montreal Press, Quebec, 1966. · Zbl 0151.15501
[20] E. M. Stein: Singular integrals and differentiability properties of functions. Princeton Univ. Press, Princeton 1979.
[21] H. Triebel: Theory of function spaces. Akad. Verlagsges. Geest and Portig K. G., Leipzig and Birkhäuser Verlag, Basel, Boston, Stuttgart 1983. · Zbl 0546.46028
[22] H. Triebel: Mapping properties of non-linear operators generated by holomorphic \(\Phi(u)\) in function spaces of Besov-Sobolev-Hardy type. Boundary value problems for elliptic differential equations of type \(\Delta u = f(x) + \Phi(u)\). Math. Nachr. 117 (1984), 193-213. · Zbl 0573.35032 · doi:10.1002/mana.3211170115
[23] M. Yamazaki: A quasi-homogeneous version of paradifferential operators I. Boundedness on spaces of Besov type. J. Fac. Sci. Univ. Tokyo, IA33 (1986), 131-174. · Zbl 0608.47058
[24] M. Yamazaki: A quasi-homogeneous version of the microlocal analysis for nonlinear partial differential equations. Preprint. · Zbl 0701.35162
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