Composition operators on potential spaces

Source abstract

By a result of B. Dahlberg, the composition operators T H f = H ∘ f {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 ( t ) , H ( 0 ) = 0 H = H\left ( t \right ),H\left ( 0 \right ) = 0 , unless of course, H ( t ) = c t H\left ( t \right ) = ct . In this note a natural domain is found for T H {T_H} that is, in a sense, maximal and on which the { T H } \left \{ {{T_H}} \right \} form an algebra of bounded operators. Here the functions H ( t ) H\left ( t \right ) need not be bounded though they are required to have a sufficient number of bounded derivatives.