to study the action of operators, such as the Hardy-Littlewood maximal operator, Calderón-Zygmund operators and Riesz potentials, on tent spaces.

More precisely, recall that the *tent space $T^q_r(\rr_+^{n+1})$*, with $q,\,r\in(0,\fz)$, consists of all measurable functions $F$ on $\rr_+^{n+1}:=\rn\times(0,\fz)$ such that $$\|F\|_{T^q_r(\rr_+^{n+1})}:=\lf\|\lf[\int_0^\fz\int_{B(\cdot,t)}|F(y,t)|^r
\,\frac{dy\,dt}{t^{n+1}}\r]^{1/r}\r\|_{L^q(\rn)}<\fz,$$ here and hereafter, for any $(x,t)\in\rr_+^{n+1}$, $B(x,t):=\{y\in\rn:\ |y-x|<t\}$. It is known (see [@cms]) that $T^q_r(\rr_+^{n+1})$ can be represented as $\sum_{i=1}^\fz\lz_i A_i$ with $\{\lz_i\}_{i\in\nn}\in \ell^q$ and $\{A_i\}_{i\in\nn}$ being $T^q_r(\rr_+^{n+1})$-atoms, that is, for any $i\in\nn$, there exists a ball $B_i\subset \rn$ such that $$\supp (A_i)\subset \widehat{B}_i:=\{(x,t)\in\rr_+^{n+1}:\ d(x, \rn\setminus B_i)\ge t\}$$ and $\iint_{\widehat{B}_i}|A_i(x,t)|^r\,\frac{dx\,dt}{t}\le |B_i|^{1-\frac{r}{q}}$, where $d(x, \rn\setminus B_i):=\inf\{|x-y|:\ y\in\rn\setminus B_i\}$. As a subspace of $T^q_r(\rr_+^{n+1})$, Auscher and Prisuelos-Arribas [@ap] introduced the space $\mathfrak{T}^q_r(\rr_+^{n+1})$ consisting of all functions $F\in T^q_r(\rr_+^{n+1})$ which can be represented as $\sum_{i=1}^\fz\lz_i A_i$ with $\{\lz_i\}_{i\in\nn}\in \ell^q$ and $T^q_r(\rr_+^{n+1})$-atoms $\{A_i\}_{i\in\mathbb{N}}$ satisfying the additional moment condition $\int_{\rr_+^{n+1}}A_i(x,t)\,dx=0$ for almost every $t\in (0,\fz)$ and any $i\in\mathbb{N}$. In [@ap], Auscher and Prisuelos-Arribas studied the behaviors of the Hardy-Littlewood maximal operator, Calderón-Zygmund operators and Riesz potentials on $T^q_r(\rr_+^{n+1})$ and $\mathfrak{T}^q_r(\rr_+^{n+1})$. As Auscher and Prisuelos-Arribas mentioned in [@ap], “it would be interesting to explore further these spaces (interpolation, etc) and their applications".

One key tool used in [@ap] is the slice space which is 