The vector-valued tent spaces T^1 and T^\infty

TL;DR

This paper extends the theory of vector-valued tent spaces to the endpoint cases p=1 and p=, providing atomic decomposition and duality results crucial for advanced functional calculus applications.

Contribution

It introduces the first atomic decomposition for vector-valued tent spaces at p=1 and explores their duality, extending scalar results to the vector-valued setting.

Findings

Atomic decomposition for p=1 tent spaces established.

Duality of vector-valued tent spaces characterized.

Extension of endpoint cases to vector-valued functions achieved.

Abstract

Tent spaces of vector-valued functions were recently studied by Hyt\"onen, 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.