In this paper, we study the structure of the discrete Muckenhoupt class $\mathcal{A}^{p}(\mathcal{C})$ and the discrete Gehring class $\mathcal{G}^{q}(\mathcal{K})$ . In particular, we prove that the self-improving property of the Muckenhoupt class holds, i.e., we prove that if $u\in \mathcal{A}^{p}(\mathcal{C})$ then there exists $q< p$ such that $u\in \mathcal{A}^{q}(\mathcal{C}_{1})$ . Next, we prove that the power rule also holds, i.e., we prove that if $u\in \mathcal{A}^{p}$ then $u^{q}\in \mathcal{A}^{p}$ for some $q>1$ . The relation between the Muckenhoupt class $\mathcal{A}^{1}(\mathcal{C})$ and the Gehring class is also discussed. For illustrations, we give exact values of the norms of Muckenhoupt and Gehring classes for power-low sequences. The results are proved by some algebraic inequalities and some new inequalities designed and proved for this purpose.

中文翻译：

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离散Muchenhoupt和离散Gehring类的结构

在本文中，我们研究了离散Muckenhoupt类$ \ mathcal {A} ^ {p}（\ mathcal {C}）$和离散Gehring类$ \ mathcal {G} ^ {q}（\ math { K}）$。特别是，我们证明了Muckenhoupt类的自我完善属性成立，即，我们证明了如果$ u \ in \ mathcal {A} ^ {p}（\ mathcal {C}）$中存在$ q < p $，这样\ u \ in \ mathcal {A} ^ {q}（\ mathcal {C} _ {1}）$中的$ u \。接下来，我们证明幂定律也成立，即我们证明如果$ u \ in \ mathcal {A} ^ {p} $，则$ u ^ {q} \ in \ mathcal {A} ^ {p} $对于$ q> 1 $。还讨论了Muckenhoupt类$ \ mathcal {A} ^ {1}（\ mathcal {C}）$与Gehring类之间的关系。为了说明，我们给出了低功率序列的Muckenhoupt和Gehring类的范式的精确值。