open access

https://doi.org/10.14943/83806

https://hdl.handle.net/2115/69460

Nonnegative functions in weighted hardy spaces

英語

2004

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

653

1-9

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

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