Hardy spaces and rearrangements

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.