In the investigations of the boundedness of some sublinear operators, which do not hold the strong estimates, the researchers treat the weak estimates. In this occasion for the Herz spaces \(\dot{K}_q^{\alpha,p}({\mathbb{R}}^n)\) , in order to obtain more precise estimates than the weak estimates, the author [40] introduced the new “weak” Herz spaces \(\widetilde{W}\dot{K}_q^{\alpha,p}({\mathbb{R}}^n)\) and showed the new “weak” boundedness on \(\dot{K}_q^{\alpha,p}({\mathbb{R}}^n)\) . In this paper, we will extend the above new “weak” estimates to the sublinear operators satisfying another size condition. Further, we will extend these results on the Herz spaces with constant exponents \(\dot{K }_q^{\alpha,p}({\mathbb{R}}^n)\) to one’s on the Herz spaces with variable exponent \(\dot{K}_{q(\cdot)}^{\alpha,p}({\mathbb{R}}^n)\) .
