The essential norm of a composition operator mapping into the Qs-space

An asymptotic formula, in terms of a global integral condition, for the essential norm parallel to C-phi(l)parallel to of the composition operator C-phi(f) = f o phi mapping from the weighted Bergman space A(alpha)(p), 1 < p <= 2, or the weighted Dirichlet space D-phi into the Mbbius invariant Q(s)-space is established. Moreover, it is shown that if C-phi is bounded from the classical Dirichlet space to BMOA, then [GRAPHICS] where N-phi o phi l, (z) is the Nevanlinna counting function for phi o phi(b), and phi(b) (z) = (b-z)/(1-(b) over barz) (c) 2006.Elsevier Inc. All rights reserved.