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$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}T^1$ AND 
$T^{\infty }$
Tent spaces of vector-valued functions were recently studied by Hytönen, van Neerven and Portal with an eye on applications to 
$H^{\infty }$-functional calculi. This paper extends their results to the endpoint cases 
$p=1$ and 
$p=\infty $ along the lines of earlier work by Harboure, Torrea and Viviani in the scalar-valued case. The main result of the paper is an atomic decomposition in the case 
$p=1$, which relies on a new geometric argument for cones. A result on the duality of these spaces is also given.
