Revistes Catalanes amb Accés Obert (RACO)

Mean growth of $H^p$ functions

D. Girela, Maria Auxiliadora Márquez


A classical result of Hardy and Littlewood asserts that if $0 < p < q < \infty $ and $f$ is a function which is analytic in the unit disc and belongs to the Hardy space $H^p$, then, if $\lambda \ge p$ and $\alpha = \frac{1}{p} - \frac{1}{q}$, we have
\int_0^1 (1-r)^{\lambda \alpha - 1} \left ( \frac{1}{2 \pi} \int_0^{2 \pi} \bigl \vert f (re^{i\theta}) \bigr \vert^q\, d\theta \right )^{\lambda / q}\, dr < \infty.
We prove that this result is sharp in a very strong sense. Indeed, we prove that if $p$, $q$, $\lambda$ and $\alpha$ are as above and $\varphi$ is a positive, continuous and increasing function defined in $[0, \infty )$ with $\frac{\varphi(x)}{x^q} \to \infty$, as $x \to \infty$, then there exists a function $f \in H^p$ such that
\int_0^1 (1-r)^{\lambda \alpha - 1} \left ( \int _I \varphi \left ( \bigl \vert f (re^{i \theta}) \bigr \vert \right ) \, d\theta \right )^{\lambda/q}\, dr = \infty,
for every non-degenerate interval $I\subset [0, 2\pi ]$. We also prove a result of the same kind concerning functions $f$ such that $f' \in H^p$, $0 < p < 1$.

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