and $p\in(0,\min\{1,q\})$. In [@bf], Bonami and Feuto introduced the Hardy type spaces $H_*^\Phi(\rn)$ and $h_*^\Phi(\rn)$ with respect to the amalgam space $(L^\Phi,\ell^1)(\rn)$, where $\Phi(t):=\frac{t}{\log(e+t)}$ for any $t\in[0,\infty)$ is an Orlicz function, and applied these Hardy-type spaces to study the linear decomposition of the product of the Hardy space $H^1(\rn)$ and its dual space $\BMO(\rn)$ as well as the local Hardy space $h^1(\rn)$ and its dual space $\bmo(\rn)$. Moreover, very recently, Cao et al. [@cky1] applied $h_*^\Phi(\rn)$ to study the bilinear decomposition of the product of the local Hardy space $h^1(\rn)$ and its dual space $\bmo(\rn)$. Recall that both the Hardy type spaces $H_*^\Phi(\rn)$ and $h_*^\Phi(\rn)$ were defined in [@bf] via the (local) radial maximal functions, while $h_*^\Phi(\rn)$ in [@cky1] was defined via the local grand maximal function. Moreover, *no other real-variable characterizations* of both the Hardy type spaces $H_*^\Phi(\rn)$ and $h_*^\Phi(\rn)$ are known so far.

On the other hand, recently, to study the classification of weak solutions in the natural classes for the boundary value problems of a $t$-independent elliptic system in the upper plane, Auscher and Mourgoglou [@am] introduced the slice spaces $E^q_t(\rn)$. In [@ap], Auscher and Prisuelos-Arribas further introduced a more general slice space $(E^q_r)_t(\rn)$ and applied it 