A coherent theory of tent spaces and homogeneous Triebel-Lizorkin spaces

We introduce and systematically investigate a scale of tent spaces that characterizes homogeneous Triebel-Lizorkin spaces ${\dot{\mathrm{F}}}_{p,q}^{\beta }$. These spaces generalize the classical spaces of Coifman, Meyer, and Stein, and are shown to be equivalent to the weighted tent spaces with Whitney averages developed by Huang. We show that these tent spaces follow a functional analytic theory that mirrors that of Triebel-Lizorkin spaces, including duality, embeddings, discrete characterizations, John-Nirenberg-type properties, as well as real and complex interpolation. Furthermore, we provide a novel characterization of the endpoint spaces ${\dot{\mathrm{F}}}_{\mathrm{\infty },q}^{\beta }$, completing earlier work by Auscher, Bechtel, and the author.