Duality of holomorphic Hardy type tent spaces

Abstract

We study the holomorphic tent spaces $\mathcal{HT}^p_{q,\alpha}(\Bn)$, which are motivated by the area function description of the Hardy spaces on one hand, and the maximal function description of the Hardy spaces on the other. Characterizations for these spaces under general fractional differential operators are given. We describe the dual of $\mathcal{HT}^p_{q,\alpha}(\Bn)$ for the full range $0<p,q<\infty$ and $\alpha>-n-1$. Here the case $1<p<\infty$, $q\leq 1$ leads us to the Hardy-Bloch type spaces $\mathcal{BT}^{p}(\Bn)$ and $\mathcal{BT}^{p,0}(\Bn)$, the latter being the predual of $\mathcal{HT}^p_{1,\alpha}(\Bn)$. In an analogous fashion, the case $p=1<q<\infty$ gives rise to Hardy-Carleson type spaces $\mathcal{CT}_{q,\alpha}(\Bn)$ and $\mathcal{CT}^0_{q,\alpha}(\Bn)$. In the remaining cases, the duality can be described in terms of the classical Bloch space $\mathcal{B}(\Bn)$. We also study these spaces in detail, providing characterizations in terms of fractional derivatives. When $p,q\geq 1$, our approach to the question of duality is strongly related to the boundedness of a certain weighted Bergman projections acting on tent spaces. For the small exponents, we will also need to apply some other techniques, such as atomic decompositions and embedding theorems. The work presented gives a unified approach for the classical Hardy and Bergman dualities, also involving the Bloch and BMOA spaces as well as some new spaces. We end the paper with discussion of some further questions and related topics.