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 \(C_k\). Suppose \(-\infty0\) such that \(\int_{|x|\leq |Q|^{1/n}}\Phi(x) dx|Q|^{-1}[\omega(Q)]^{1/q_2} [\omega_1(Q)]^{1-1/q_1}\leq C\) for all cubes \(Q\subset \mathbb R^n\). This result corresponds to a work of Pérez in the case of the weighted Lebesgue spaces \(L_\omega^p\) [\textit{C. Pérez}, Indiana Univ. Math. J. 43, No. 2, 663-683 (1994; Zbl 0809.42007)]. The authors apply their results to the usual Riesz fractional integral operator \(I_\beta\). They also treat the weighted weak boundedness case and the weighted Herz-type Hardy space case.