Let \(B_k=\{x\in \mathbb R^n\); \(|x|\leq 2^k\}\) and \(C_k=B_k \setminus B_{k-1}\) for \(k\in \mathbb Z\). Let \(\chi_k\) denote the characteristic function of the set \(C_k\). For a function \(f\) and a nonnegative weight function \(\omega(x)\), one writes \(|f|_{L_\omega^q(\mathbb R^n)}= \bigl(\int_{\mathbb R^n}|f(x)|^q\omega(x) dx\bigr)^{1/q}\). For \(\alpha\in \mathbb R\), \(0<p<\infty\), \(1<q<\infty\) and nonnegative weight functions \(\omega_1\), \(\omega_2\), the non-homogeneous weighted Herz space \(K_q^{\alpha, p}(\omega_1, \omega_2)\) is defined in terms of \[ |f|_{K_q^{\alpha, p}(\omega_1, \omega_2)}= |f|_{L_{\omega_2}^q(\mathbb R^n)}+\Biggl\{\sum_{k=-\infty}^\infty[\omega_1(B_k) ]^{\alpha p/n}|f\chi_{k}|_{L_{\omega_2}^q(\mathbb R^n)}^p\Biggr\}^{1/p} \] by letting \[ K_q^{\alpha, p}(\omega_1, \omega_2)=\{f\in L_{\text{loc}}^q(\mathbb R^n, \omega_2); |f|_{K_q^{\alpha, p}(\omega_1, \omega_2)}<\infty\}. \] \(K_q^{\alpha,p}(1,1)\) was first introduced by C. S. Herz [J. Math. Mech. 18, 283-323 (1968; Zbl 0177.15701)] with different but equivalent norms and notations. Let \(f\in \mathcal S'(\mathbb R^n)\) and \(G_N(f)\) be the grand maximal function of \(f\). That is, \[ G_N(f)(x)= \sup_{\varphi \in \mathcal A_N}\sup_{|x-y|\leq t}|f\ast \varphi_t(x)|, \] where \[ \mathcal A_N=\{\varphi\in \mathcal S(\mathbb R^n);\;\sup_{|\alpha|,|\beta|\leq N} |x^\alpha\partial^\beta \varphi(x)|\leq 1\}, \] \(N\in \mathbb N\) is sufficiently large, and \(\varphi_t(x)=t^{-n}\varphi(x/t)\) \((t>0)\). Homogeneous versions are also considered.
The authors give some characterizations of these Herz-type Hardy spaces in terms of radial and nontangential maximal functions for a single \(\varphi\), and atomic decompositions, like as in the classical Hardy spaces \(H^p(\mathbb R^n)\). Furthermore, using these characterizations they investigate some properties of these Hardy spaces and consider the boundedness of some general potential operators in connection with partial differential equations on these spaces.