Functional Calculus in Hölder-Zygmund Spaces

Bourdaud, G.; Lanza de Cristoforis, Massimo

doi:10.1090/S0002-9947-02-03000-3

Functional Calculus in Hölder-Zygmund Spaces
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by G. Bourdaud and Massimo Lanza de Cristoforis PDF

Trans. Amer. Math. Soc. 354 (2002), 4109-4129 Request permission

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 $\mathcal {S}^s(\mathbf {R}^n)$ 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.

References

Jürgen Appell and Petr P. Zabrejko, Nonlinear superposition operators, Cambridge Tracts in Mathematics, vol. 95, Cambridge University Press, Cambridge, 1990. MR 1066204, DOI 10.1017/CBO9780511897450
Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR 928802
Gérard Bourdaud, Le calcul fonctionnel dans les espaces de Sobolev, Invent. Math. 104 (1991), no. 2, 435–446 (French). MR 1098617, DOI 10.1007/BF01245083
Gérard Bourdaud, Fonctions qui opèrent sur les espaces de Besov et de Triebel, Ann. Inst. H. Poincaré C Anal. Non Linéaire 10 (1993), no. 4, 413–422 (French, with English and French summaries). MR 1246460, DOI 10.1016/S0294-1449(16)30209-8
Gérard Bourdaud, Analyse fonctionnelle dans l’espace euclidien, Publications Mathématiques de l’Université Paris VII [Mathematical Publications of the University of Paris VII], vol. 23, Université de Paris VII, U.E.R. de Mathématiques, Paris, 1986 (French). MR 901939
Pavel Drábek, Continuity of Nĕmyckij’s operator in Hölder spaces, Comment. Math. Univ. Carolinae 16 (1975), 37–57. MR 380547
Gerald B. Folland, Real analysis, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. Modern techniques and their applications; A Wiley-Interscience Publication. MR 767633
Yitzhak Katznelson, An introduction to harmonic analysis, Second corrected edition, Dover Publications, Inc., New York, 1976. MR 0422992
H. Koch and W. Sickel, Pointwise multipliers of Besov spaces of smoothness zero and spaces of continuous functions, Rev. Math. Iberoamericana (to appear).
Alois Kufner, Oldřich John, and Svatopluk Fučík, Function spaces, Monographs and Textbooks on Mechanics of Solids and Fluids, Mechanics: Analysis, Noordhoff International Publishing, Leyden; Academia, Prague, 1977. MR 0482102
Massimo Lanza de Cristoforis, Higher order differentiability properties of the composition and of the inversion operator, Indag. Math. (N.S.) 5 (1994), no. 4, 457–482. MR 1307964, DOI 10.1016/0019-3577(94)90018-3
Massimo Lanza de Cristoforis, Differentiability properties of an abstract autonomous composition operator, J. London Math. Soc. (2) 61 (2000), no. 3, 923–936. MR 1766115, DOI 10.1112/S0024610700008802
A. Marchaud, Sur les dérivées et sur les différences des fonctions des variables réelles, J. Math. Pure et Appl. 6 (1927), pp. 337-425.
Jaak Peetre, New thoughts on Besov spaces, Duke University Mathematics Series, No. 1, Duke University, Mathematics Department, Durham, N.C., 1976. MR 0461123
Thomas Runst and Winfried Sickel, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, De Gruyter Series in Nonlinear Analysis and Applications, vol. 3, Walter de Gruyter & Co., Berlin, 1996. MR 1419319, DOI 10.1515/9783110812411
E. P. Sobolevskij, The superposition operator in Hölder spaces, Voronezh, VINITI No. 3765-84 (1984), (in Russian).
P. M. Tamrazov, Structural and approximational properties of functions in the complex domain, Linear spaces and approximation (Proc. Conf., Math. Res. Inst., Oberwolfach, 1977) Lecture Notes in Biomath., vol. 21, Springer, Berlin-New York, 1978, pp. 503–514. MR 501503
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.
Hans Triebel, Theory of function spaces. II, Monographs in Mathematics, vol. 84, Birkhäuser Verlag, Basel, 1992. MR 1163193, DOI 10.1007/978-3-0346-0419-2
S.E. Warschawski, On the higher derivatives at the boundary in conformal mapping, Trans. Amer. Math. Soc. 38 (1935), pp. 310-340.
A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776