We first prove some new weighted refinements of inequalities of Hardy’s type with negative powers. Next, we prove that any \(A_{\lambda }^{1}\) Muckenhoupt class with a weight \(\lambda \) belongs to some weighted Gehring class \(G_{\lambda }^{p}\) for \(p>1\). We also prove that the self-improving property of the weighted Muckenhoupt class \(A_{\lambda }^{q}\) holds. The main results give exact values of the limit exponents and the constants of the new classes. The self-improving property of the weighted Muckenhoupt class will then be applied to prove the self-improving property of the Gehring class with a sharp value on the exponents.

我们首先证明具有负幂的Hardy型不等式的一些新的加权细化。接下来，我们证明了任何\（A _ {\拉姆达} ^ {1} \） Muckenhoupt类与重\（\拉姆达\）属于一些加权Gehring集团类\（G _ {\拉姆达} ^ {P} \）为\（p> 1 \）。我们还证明了加权Muckenhoupt类\（A _ {\ lambda} ^ {q} \）的自我改进性质 成立。主要结果给出了极限指数和新类常数的精确值。然后，将使用加权的Muckenhoupt类的自我改进属性来证明Gehring类的自我改进属性，并对指数具有敏锐的价值。