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 Cϕ e of the composition operator Cϕ(f ) = f ◦ ϕ mapping from the weighted Bergman space A pα , 1 < p 2, or the weighted Dirichlet space Dα into the Möbius invariant Qs -space is established. Moreover, it is shown that if Cϕ is bounded from the classical Dirichlet space to BMOA, then Cϕ 2e = lim sup |a|→1− sup b∈D D ϕ a(z) 2 Nϕ◦ϕb (z) dA(z) = lim sup |z|→1− sup b∈D Nϕ◦ϕb (z), where Nϕ◦ϕb (z) is the Nevanlinna counting function for ϕ ◦ ϕb, and ϕb(z) = (b −z)/(1− ¯bz). © 2006 Elsevier Inc. All rights reserved.