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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Composition operators on potential spaces

Authors: David R. Adams and Michael Frazier
Journal: Proc. Amer. Math. Soc. 114 (1992), 155-165
MSC: Primary 46E35; Secondary 47B38
MathSciNet review: 1076570
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Abstract: By a result of B. Dahlberg, the composition operators $ {T_H}f = H \circ f$ need not be bounded on some of the Sobolev spaces (or spaces of Bessel potentials) even for very smooth functions $ H = H\left( t \right),H\left( 0 \right) = 0$, unless of course, $ H\left( t \right) = ct$. In this note a natural domain is found for $ {T_H}$ that is, in a sense, maximal and on which the $ \left\{ {{T_H}} \right\}$ form an algebra of bounded operators. Here the functions $ H\left( t \right)$ need not be bounded though they are required to have a sufficient number of bounded derivatives.

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Keywords: Sobolev space, potential space, composition operator
Article copyright: © Copyright 1992 American Mathematical Society