## Hardy spaces and rearrangements

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- by Burgess Davis
- Trans. Amer. Math. Soc.
**261**(1980), 211-233 - DOI: https://doi.org/10.1090/S0002-9947-1980-0576872-9
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## Abstract:

Let*f*be an integrable valued function on the unit circle in the complex plane, and let

*g*be the rearrangement of

*f*satisfying $g({e^{i\theta }}) \geqslant g({e^{i\varphi }})$ if $0 \leqslant \theta < \varphi < 2\pi$. Define \[ G(\theta ) = \int _{ - \theta }^\theta {g({e^{i\varphi }})} d\varphi \] . It is shown that some rearrangement of

*f*is in $\operatorname {Re} {H^1}$, that is, the distribution of

*f*is the distribution of a function in $\operatorname {Re} {H^1}$, if and only if $\int _0^\pi {|G(\theta )/\theta |} d\theta < \infty$, and that, if any rearrangement of

*f*is in $\operatorname {Re} {H^1}$, then

*g*is. The existence and form of rearrangements minimizing the ${H^1}$ norm are investigated. It is proved that

*f*is in $\operatorname {Re} {H^1}$ if and only if some rotation of

*f*is in the space dyadic ${H^1}$ of martingales. These results are extended to other ${H^p}$ spaces.

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## Bibliographic Information

- © Copyright 1980 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**261**(1980), 211-233 - MSC: Primary 42A50; Secondary 30D55, 42A61, 42B30, 60G46, 60J65
- DOI: https://doi.org/10.1090/S0002-9947-1980-0576872-9
- MathSciNet review: 576872