Nonnegative functions in weighted hardy spaces

Let W be a nonnegative summable function whose logarithm is also summable with respect to the Lebesgue measure on the unit circle. For 0 < p < ∞ , Hp (W) denotes a weighted Hardy space on the unit circle. When W ≡ 1, H p(W) is the usual Hardy space Hp . We are interested in Hp ( W)+ the set of all nonnegative functions in Hp ( W). If p ≥ 1/2, Hp + consists of constant functions. However Hp ( W)+ contains a nonconstant nonnegative function for some weight W. In this paper, if p ≥ 1/2 we determine W and describe Hp ( W)+ when the linear span of Hp ( W)+ is of finite dimension. Moreover we show that the linear span of Hp (W)+ is of infinite dimension for arbitrary weight W when 0 < p < 1/2.