Abstract
Let 𝔻kdenote the unit polydisk in ℂk, k ≥ 1, and H(𝔻k) the set of all holomorphic functions on 𝔻k. Now, let v : 𝔻k→ (0, ∞) be a weight and [Formula: see text] the Banach space of holomorphic functions f on 𝔻ksuch that supz ∈ 𝔻kv(z)|f(z)| < ∞. A holomorphic self-map ϕ : 𝔻k→ 𝔻kinduces the linear composition operator Cϕ: H(𝔻k) → H(𝔻k), f ↦ f ◦ ϕ. We investigate under which conditions on the symbol ϕ and the weight v the operator Cϕacting on [Formula: see text] is a bellwether for boundedness of composition operators, where Cϕbeing a bellwether means that Cϕacts boundedly on [Formula: see text] if and only if all composition operators act boundedly on [Formula: see text].
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