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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Hardy spaces and rearrangements


Author: Burgess Davis
Journal: Trans. Amer. Math. Soc. 261 (1980), 211-233
MSC: Primary 42A50; Secondary 30D55, 42A61, 42B30, 60G46, 60J65
MathSciNet review: 576872
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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

$\displaystyle 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 {\vert G(\theta )/\theta \vert} \,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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1980-0576872-9
PII: S 0002-9947(1980)0576872-9
Keywords: Hardy spaces, conjugate function, rearrangements, Brownian motion, martingale
Article copyright: © Copyright 1980 American Mathematical Society