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January  1999, 5(1): 157-184. doi: 10.3934/dcds.1999.5.157

Regularity of the composition operator in spaces of Hölder functions

Rafael De La Llave 1, and  R. Obaya 2,

1. 

Department of Mathematics, 1 University Station C1200, University of Texas, Austin, TX 78712, United States
2. 

Departamento de Matemática Aplicada a la Ingeniería, ETSII, Universidad de Valladolid, Valladolid, Spain

Received  October 1997 Revised  August 1998 Published  October 1998

We study the regularity of the composition operator $((f, g)\to g \circ f)$ in spaces of Hölder differentiable functions. Depending on the smooth norms used to topologize $f, g$ and their composition, the operator has different differentiability properties. We give complete and sharp results for the classical Hölder spaces of functions defined on geometrically well behaved open sets in Banach spaces. We also provide examples that show that the regularity conclusions are sharp and also that if the geometric conditions fail, even in finite dimensions, many elements of the theory of functions (smoothing, interpolation, extensions) can have somewhat unexpected properties.
Keywords: Composition operator, Hölder spaces, differentiability properties..
Mathematics Subject Classification: 58D15, 58B10. .
Citation: Rafael De La Llave, R. Obaya. Regularity of the composition operator in spaces of Hölder functions. Discrete & Continuous Dynamical Systems, 1999, 5 (1) : 157-184. doi: 10.3934/dcds.1999.5.157
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