1. ON ABSOLUTE VALUES OF 𝓠KFUNCTIONS
- Author
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Guanlong Bao, Ruishen Qian, Hasi Wulan, and Zengjian Lou
- Subjects
General Mathematics ,010102 general mathematics ,Boundary (topology) ,Function (mathematics) ,Hardy space ,Space (mathematics) ,01 natural sciences ,Measure (mathematics) ,Unit disk ,010101 applied mathematics ,Combinatorics ,symbols.namesake ,symbols ,Always true ,0101 mathematics ,Complex plane ,Mathematics - Abstract
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)
- Published
- 2016