Composition operators on potential spaces

Adams, David R.; Frazier, Michael

doi:10.1090/S0002-9939-1992-1076570-5

Composition operators on potential spaces
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by David R. Adams and Michael Frazier

Proc. Amer. Math. Soc. 114 (1992), 155-165

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

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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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