Essential norm of weighted composition operators from H∞ to the Zygmund space

Let ϕ be an analytic self-map of the unit disk D and u ∈ H(D), the space of analytic functions on D. The weighted composition operator, denoted by uCϕ, is defined by (uCϕf )(z) = u(z)f (ϕ(z)), f ∈ H(D), z ∈ D. In this paper, we give three different estimates for the essential norm of the operator uCϕ from H∞ into the n Zygmund space, denoted by Z. In particular, we show that lluCϕlle,H∞→Z ≈ lim supn→∞ lluϕ^nllz.