Tent spaces at endpoints

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.