## Abstract

In 1985, Coifman, Meyer, and Stein gave the duality of the tent spaces; that is, $\left({T}_{q}^{p}\right({\mathbb{R}}_{+}^{n+1}){)}^{*}={T}_{q\text{'}}^{p\text{'}}({\mathbb{R}}_{+}^{n+1})$ for $1<p,q<\infty $, and $\left({T}_{\infty}^{1}\right({\mathbb{R}}_{+}^{n+1}){)}^{*}=\mathcal{C}({\mathbb{R}}_{+}^{n+1})$, $\left({T}_{q}^{1}\right({\mathbb{R}}_{+}^{n+1}){)}^{*}={T}_{q\text{'}}^{\infty}({\mathbb{R}}_{+}^{n+1})$ for $1<q<\infty $, where $\mathcal{C}\left({\mathbb{R}}_{+}^{n+1}\right)$ denotes the Carleson measure space on ${\mathbb{R}}_{+}^{n+1}$. We prove that $\left({\mathcal{C}}_{v}\right({\mathbb{R}}_{+}^{n+1}){)}^{*}={T}_{\infty}^{1}({\mathbb{R}}_{+}^{n+1})$, which we introduced recently, where ${\mathcal{C}}_{v}\left({\mathbb{R}}_{+}^{n+1}\right)$ is the vanishing Carleson measure space on ${\mathbb{R}}_{+}^{n+1}$. We also give the characterizations of ${T}_{q}^{\infty}\left({\mathbb{R}}_{+}^{n+1}\right)$ by the boundedness of the Poisson integral. As application, we give the boundedness and compactness on ${L}^{q}\left({\mathbb{R}}^{n}\right)$ of the paraproduct ${\pi}_{F}$ associated with the tent space ${T}_{q}^{\infty}\left({\mathbb{R}}_{+}^{n+1}\right)$, and we extend partially an interesting result given by Coifman, Meyer, and Stein, which establishes a close connection between the tent spaces ${T}_{2}^{p}\left({\mathbb{R}}_{+}^{n+1}\right)$ $(1\le p\le \infty )$ and ${L}^{p}\left({\mathbb{R}}^{n}\right)$, ${H}^{p}\left({\mathbb{R}}^{n}\right)$ and $\mathit{BMO}\left({\mathbb{R}}^{n}\right)$ spaces.

## Citation

Yong Ding. Ting Mei. "Tent spaces at endpoints." Banach J. Math. Anal. 11 (4) 841 - 863, October 2017. https://doi.org/10.1215/17358787-2017-0020

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