ON ABSOLUTE VALUES OF 𝓠KFUNCTIONS

In this paper, the effect of absolute values on the behaviorof functions f in the spaces Q K is investigated. It is clear that g ∈Q K (∂D) ⇒ |g| ∈ Q K (∂D), but the converse is not always true. For fin the Hardy space H 2 , we give a condition involving the modulus ofthe function only, such that the condition together with |f|∈Q K (∂D) isequivalent to f ∈Q K . As an application, a new criterion for inner-outerfactorisation of Q K spaces is given. These results are also new for Q p spaces. 1. IntroductionDenote by ∂D the boundary of the unit disk D in the complex plane C.Let H(D) be the space of functions analytic in D. Throughout this paper, weassume that K: [0,∞) → [0,∞) is a right-continuous and increasing function.A function f∈ H(D) belongs to the space Q K ifkfk 2Q K = sup a∈D Z D |f ′ (z)| 2 K(g(a,z))dA(z)