Banach Journal of Mathematical Analysis

Tent spaces at endpoints

Yong Ding and Ting Mei

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Abstract

In 1985, Coifman, Meyer, and Stein gave the duality of the tent spaces; that is, $(T_{q}^{p}(\mathbb{R}^{n+1}_{+}))^{\ast}=T_{q'}^{p'}(\mathbb{R}^{n+1}_{+})$ for $1\lt p,q\lt \infty$, and $(T_{\infty}^{1}(\mathbb{R}^{n+1}_{+}))^{\ast}=\mathscr{C}(\mathbb{R}^{n+1}_{+})$, $(T_{q}^{1}(\mathbb{R}^{n+1}_{+}))^{\ast}=T_{q'}^{\infty}(\mathbb{R}^{n+1}_{+})$ for $1\lt q\lt \infty$, where $\mathscr{C}(\mathbb{R}^{n+1}_{+})$ denotes the Carleson measure space on $\mathbb{R}^{n+1}_{+}$. We prove that $(\mathscr{C}_{v}(\mathbb{R}^{n+1}_{+}))^{\ast}=T_{\infty}^{1}(\mathbb{R}^{n+1}_{+})$, which we introduced recently, where $\mathscr{C}_{v}(\mathbb{R}^{n+1}_{+})$ is the vanishing Carleson measure space on $\mathbb{R}^{n+1}_{+}$. We also give the characterizations of $T_{q}^{\infty}(\mathbb{R}^{n+1}_{+})$ by the boundedness of the Poisson integral. As application, we give the boundedness and compactness on $L^{q}(\mathbb{R}^{n})$ of the paraproduct $\pi_{F}$ associated with the tent space $T_{q}^{\infty}(\mathbb{R}^{n+1}_{+})$, and we extend partially an interesting result given by Coifman, Meyer, and Stein, which establishes a close connection between the tent spaces $T_{2}^{p}(\mathbb{R}^{n+1}_{+})$ $(1\le p\le\infty)$ and $L^{p}(\mathbb{R}^{n})$, $H^{p}(\mathbb{R}^{n})$ and $\mathit{BMO}(\mathbb{R}^{n})$ spaces.

Article information

Source
Banach J. Math. Anal. Volume 11, Number 4 (2017), 841-863.

Dates
Received: 20 June 2016
Accepted: 20 November 2016
First available in Project Euclid: 17 August 2017

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1502935413

Digital Object Identifier
doi:10.1215/17358787-2017-0020

Subjects
Primary: 42B35: Function spaces arising in harmonic analysis
Secondary: 42B99: None of the above, but in this section

Keywords
tent space vanishing Carleson measure vanishing tent space Poisson integral paraproduct

Citation

Ding, Yong; Mei, Ting. Tent spaces at endpoints. Banach J. Math. Anal. 11 (2017), no. 4, 841--863. doi:10.1215/17358787-2017-0020. https://projecteuclid.org/euclid.bjma/1502935413


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