On a maximum principle for pseudocontinuable functions

Let Θ be an inner function and let α ∈ ℂ, |α| = 1. Denote by σα the nonnegative singular measure whose Poisson integral is equal to Re α+Θ/α-Θ. A theorem of Clark provides a natural unitary operator Uα that identifies H2 ⊖ ΘH2 with L 2(σα). The following fact is established. Assume that f ∈ H2 ⊖ ΘH2, 2 < p ≤ + ∞, α ≠ β. Then ∥f∥Hp ≤ C(α,β,p)(∥Uα f∥Lp(σα) + ∥Uβf∥Lp(σβ)). ©1997 Plenum Publishing Corporation.