Strong and weak estimates for some sublinear operators in Herz spaces with power weights at indices beyond critical index

In 1996, X. Li and D. Yang found the best possible range of index \(\alpha \) for the boundedness of some sublinear operators on Herz spaces \({\dot{K}}_q^{\alpha , p}({{\mathbb {R}}}^n)\) or \(K_q^{\alpha , p}({{\mathbb {R}}}^n)\), under a certain size condition. Also, in 1994 and 1995, S. Lu and F. Soria showed that concerning the boundedness of above sublinear operator T on \({\dot{K}}_q^{\alpha , p}({{\mathbb {R}}}^n)\) or \(K_q^{\alpha , p}({{\mathbb {R}}}^n)\) with critical index of \(\alpha \), T is bounded on the power-weighted Herz spaces \({\dot{K}}_q^{\alpha , p}(w)({{\mathbb {R}}}^n)\) or \(K_q^{\alpha , p}(w)({{\mathbb {R}}}^n)\). In this paper, we will prove that for the two-power-weighted Herz spaces \({\dot{K}}_{q_1}^{\alpha , p}(w_1,w_2)({{\mathbb {R}}}^n)\) or \(K_{q_2}^{\alpha , p}(w_1,w_2)({{\mathbb {R}}}^n)\) with indices beyond critical index of \(\alpha \), the above T is bounded on them. Further, we will extend this result to a sublinear operator satisfying another size condition and a pair of Herz spaces \(K_q^{\alpha , p}(w_{\beta _1},w_{\beta _2})({{\mathbb {R}}}^n)\) and \(K_q^{\alpha , p}(w_{\gamma _1},w_{\gamma _2})({{\mathbb {R}}}^n)\). Moreover, we will also show the result of weak version of the above boundedness.