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

DOI:
https://doi.org/10.1090/S0002-9939-1992-1076570-5

MathSciNet review:
1076570

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Abstract | References | Similar Articles | Additional Information

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