Square function characterization of Hardy-type spaces

Abstract

The well-known characterization of Hardy spaces ${\mathrm{H}}_p(\mathbb{D})$, $0<p<\infty$, in terms of the Littlewood-Paley g-function $$Sf(\hat{I}\P )={\left({\int }_0^1|f^{\prime }(r\hat{I}\P ){|}^2(1-r)\mathrm{d}r\right)}^{1/2}\in {\mathrm{L}}_p$$ is generalized to Hardy-type spaces $X_A$ corresponding to quasi-Banach lattices $X$ on the unit circle $\mathbb{T}$ under the assumption that the Hardy-Littlewood maximal operator $M$ is bounded in $(X^{\hat{I}}´{)}^{\prime }$ with some $\hat{I}´>0$. As an application to composition operators $C_{\ddot{I}}†$, we derive an exact criterion for the boundedness and compactness of $C_{\ddot{I}}†:{\mathcal{B}}^{\ddot{I}}‰\to X_A$, where ${\mathcal{B}}^{\ddot{I}}‰=\{f\mid \sup |f^{\prime }|/\ddot{I}‰<\infty \}$ is the weighted Bloch space with a log-convex radial weight $\ddot{I}‰$, generalizing recent results in the one-dimensional setting.

10 pages