The vector-valued tent spaces \(T^1\) and \(T^{\infty }\) (Q2877691)

In this article, the author introduces the vector-valued tent spaces \(T^1\) and \(T^\infty\), which extend the UMD spaces-valued tent spaces \(T^p\) of \textit{T. Hytönen} et al. [J. Anal. Math. 106, 317--351 (2008; Zbl 1165.46015)] to the endpoint cases \(p=1\) and \(p=\infty\). As the main result, by establishing a covering lemma on cones, the author obtains an atomic decomposition of the tent space \(T^1\). The duality between the spaces \(T^1\) and \(T^\infty\), and the embeddings from \(T^1\) into certain vector-valued Hardy spaces, and \(T^\infty\) into certain BMO spaces are also obtained.