The essential norm of a composition operator mapping into the [formula omitted]-space

Abstract

An asymptotic formula, in terms of a global integral condition, for the essential norm ‖ C φ ‖ e of the composition operator C φ ( f ) = f ○ φ mapping from the weighted Bergman space A α p , 1 < p ⩽ 2 , or the weighted Dirichlet space D α into the Möbius invariant Q s -space is established. Moreover, it is shown that if C φ is bounded from the classical Dirichlet space to BMOA, then ‖ C φ ‖ e 2 = lim sup | a | → 1 − sup b ∈ D ∫ D | φ a ′ ( z ) | 2 N φ ○ φ b ( z ) d A ( z ) = lim sup | z | → 1 − sup b ∈ D N φ ○ φ b ( z ) , where N φ ○ φ b ( z ) is the Nevanlinna counting function for φ ○ φ b , and φ b ( z ) = ( b − z ) / ( 1 − b ¯ z ) .