Geodesic
The composition operator in Sobolev Morrey spaces
Eurasian mathematical journal, Tome 7 (2016) no. 2, pp. 50-67.

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In this paper we prove sufficent conditions on a map $f$ from the real line to itself in order that the composite map $f \circ g$ belongs to a Sobolev Morrey space of real valued functions on a domain of the $n$-dimensional space for all functions $g$ in such a space. Then we prove sufficient conditions on f in order that the composition operator $T_f$ defined by $T_f [g] \equiv f\circ g$ for all functions $g$ in the Sobolev Morrey space is continuous, Lipschitz continuous and differentiable in the Sobolev Morrey space. We confine the attention to Sobolev Morrey spaces of order up to one. 
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N. Kydyrmina; M. Lanza de Cristoforis. The composition operator in Sobolev Morrey spaces. Eurasian mathematical journal, Tome 7 (2016) no. 2, pp. 50-67. http://geodesic.mathdoc.fr/item/EMJ_2016_7_2_a3/

[1] J. Appell, P. P. Zabrejko, Nonlinear superposition operators, Cambridge Tracts in Mathematics, 95, Cambridge University Press, 1990 | MR | Zbl

[2] G. Bourdaud, M. Lanza de Cristoforis, “Regularity of the symbolic calculus in Besov algebras”, Studia Mathematica, 184 (2008), 271–298 | DOI | MR | Zbl

[3] G. Bourdaud, W. Sickel, “Composition operators on function spaces with fractional order of smoothness”, RIMS Kokyuroku Bessatsu B, 26 (93–132), 2011 | MR

[4] V. I. Burenkov, Sobolev spaces on domains, B. G. Teubner Verlagsgesellschaft mbH, Stuttgart–Leipzig, 1998 | MR | Zbl

[5] S. Campanato, “Proprietà di inclusione per spazi di Morrey”, Ricerche Mat., 12 (1963), 67–86 | MR | Zbl

[6] Ch. J. de la Vallée-Poussin, “Sur l'intégrale de Lebesgue”, Trans. Amer. Math. Soc., 16 (1915), 435–501 | MR | Zbl

[7] R. M. Dudley, R. Norvaisa, Concrete functional calculus, Springer, 2011 | MR | Zbl

[8] H. Federer, Geometric measure theory, Springer-Verlag New York Inc., New York, 1969 | MR | Zbl

[9] G. B. Folland, Real analysis: modern techniques and their applications, United States of America: A Wiley-Interscience publication, 1999 | MR

[10] M. Lanza de Cristoforis, “Differentiability properties of a composition operator”, Rendiconti del Circolo Matematico di Palermo, Serie II, 56 (1998), 157–165 | MR | Zbl

[11] M. Lanza de Cristoforis, “Differentiability properties of an abstract autonomous composition operator”, J. London Math. Soc. (2), 61:3 (2000), 923–936 | DOI | MR | Zbl

[12] M. Marcus, J. Mizel, “Absolute continuity on tracks and mappings of Sobolev spaces”, Arch. Rat. Mech. Anal., 45 (1972), 294–320 | DOI | MR | Zbl

[13] M. Marcus, V. J. Mizel, “Complete characterization of functions which act, via superposition, on Sobolev spaces”, Trans. Amer. Math. Soc., 251 (1979), 187–218 | DOI | MR | Zbl

[14] V. Rohlin, D. Fuchs, Premier cours de topologie — chapitres géometriques, Éditions Mir, M.

[15] T. Runst, W. Sickel, Sobolev spaces of fractional order, Nemytskij operators and nonlinear partial differential equations, De Gruyter, Berlin, 1996 | MR | Zbl

[16] T. Valent, “A property of multiplication in Sobolev spaces. Some applications”, Rend. Sem. Mat. Univ. Padova, 74 (1985), 63–73 | MR | Zbl

[17] W. Yuan, W. Sickel, D. Yang, Morrey and Campanato meet Besov, Lizorkin and Triebel, Lecture Notes in Math., 2005, Springer, Berlin, 2010 | DOI | MR | Zbl

[18] J. L. Zolesio, “Multiplication dans les espaces de Besov”, Proceedings of the Royal Society of Edinburgh, Sect. A, 78:1–2 (1977/78), 113–117 | DOI | MR | Zbl