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Functional Calculus in Hölder-Zygmund Spaces


Authors: G. Bourdaud and Massimo Lanza de Cristoforis
Journal: Trans. Amer. Math. Soc. 354 (2002), 4109-4129
MSC (2000): Primary 46E35, 47H30
DOI: https://doi.org/10.1090/S0002-9947-02-03000-3
Published electronically: June 4, 2002
MathSciNet review: 1926867
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Abstract: In this paper we characterize those functions $f$ of the real line to itself such that the nonlinear superposition operator $T_{f}$ defined by $T_{f}[ g]:= f\circ g$ maps the Hölder-Zygmund space ${\mathcal C}^{s}({\mathbf R}^{n})$ to itself, is continuous, and is $r$ times continuously differentiable. Our characterizations cover all cases in which $s$ is real and $s>0$, and seem to be novel when $s>0$ is an integer.


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  • 1. J. Appell and P.P. Zabreiko, Nonlinear Superposition Operators, Cambridge Tracts in Mathematics 95, Cambridge University Press, Cambridge (1990). MR 91k:47168
  • 2. C. Bennett and R. Sharpley, Interpolation of Operators, Pure and Applied Mathematics 129, Academic Press, Boston (1988). MR 89e:46001
  • 3. G. Bourdaud, Le calcul fonctionnel dans les espaces de Sobolev, Invent. Math., 104 (1991), pp. 435-446. MR 93b:46053
  • 4. G. Bourdaud, Fonctions qui opèrent sur les espaces de Besov et de Triebel, Ann. Inst. Henri Poincaré, Analyse non linéaire, 10 (1993), pp. 413-422. MR 94m:46055
  • 5. G. Bourdaud, Analyse fonctionnelle dans l'espace Euclidien, 2ième édition, Pub. Math. Univ. Paris 7 (1995). MR 89a:46001 (1st ed.)
  • 6. P. Drábek, Continuity of Nemyckij's operator in Hölder spaces, Comm. Math. Univ. Carolinae, 16 (1975), pp. 37-57. MR 52:1447
  • 7. G.B. Folland, Real analysis, modern techniques and their applications, John Wiley and Sons, New York (1984). MR 86k:28001
  • 8. Y. Katznelson, An introduction to Harmonic Analysis, Dover, New York (1976). MR 54:10976
  • 9. H. Koch and W. Sickel, Pointwise multipliers of Besov spaces of smoothness zero and spaces of continuous functions, Rev. Math. Iberoamericana (to appear).
  • 10. A. Kufner, O. John and S. Fucik, Function Spaces, Noordhoff International Publishing, Leyden (1977). MR 58:2189
  • 11. M. Lanza de Cristoforis, Higher order differentiability properties of the composition and of the inversion operator, Indag. Matem. N.S., 5 (1994), pp. 457-482. MR 95j:47081
  • 12. M. Lanza de Cristoforis, Differentiability properties of an abstract autonomous composition operator, J. London Math. Soc. 61 (2000), pp. 923-936. MR 2001c:47069
  • 13. A. Marchaud, Sur les dérivées et sur les différences des fonctions des variables réelles, J. Math. Pure et Appl. 6 (1927), pp. 337-425.
  • 14. J. Peetre, New Thoughts on Besov spaces, Duke University Mathematics Series, No. 1. Mathematics Department, Duke University, Durham, N.C. (1976). MR 57:1108
  • 15. T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, De Gruyter, Berlin (1996). MR 98a:47071
  • 16. E. P. Sobolevskij, The superposition operator in Hölder spaces, Voronezh, VINITI No. 3765-84 (1984), (in Russian).
  • 17. P.M. Tamrazov, Structural and approximational properties of functions in the complex domain, Linear Spaces and Approximation, Birkhäuser, Basel, (1978), pp. 503-514. MR 80a:30044
  • 18. P.M. Tamrazov, Divided differences and moduli of smoothness of functions, function superpositions and their application, Constructive theory of functions 1984, Sofia, 1984, pp. 840-851.
  • 19. H. Triebel, Theory of Function Spaces II, Birkhäuser, Basel (1992). MR 93f:46029
  • 20. S.E. Warschawski, On the higher derivatives at the boundary in conformal mapping, Trans. Amer. Math. Soc. 38 (1935), pp. 310-340.
  • 21. A. Zygmund, Trigonometric Series, Cambridge University Press, New York (1959). MR 21:6498

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Additional Information

G. Bourdaud
Affiliation: Institut de Mathématiques de Jussieu, Équipe d’Analyse Fonctionnelle, Case 186, 4 place Jussieu, 75252 Paris Cedex 05, France
Email: bourdaud@ccr.jussieu.fr

Massimo Lanza de Cristoforis
Affiliation: University of Padova, Dipartimento di Matematica Pura ed Applicata, Via Belzoni 7, 35131 Padova, Italia
Email: mldc@math.unipd.it

DOI: https://doi.org/10.1090/S0002-9947-02-03000-3
Keywords: H\"older-Zygmund spaces, composition operators
Received by editor(s): June 13, 2000
Received by editor(s) in revised form: December 21, 2001
Published electronically: June 4, 2002
Additional Notes: The authors thank Jean-Pierre Kahane and Winfried Sickel for their help in the preparation of this paper. Massimo Lanza de Cristoforis wishes to thank Gérard Bourdaud and the University of Paris VII for hospitality during the months of September 1998 and of February 2000.
Article copyright: © Copyright 2002 American Mathematical Society

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