On structure of discrete Muckenhoupt and discrete Gehring classes

[摘要] 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 $q1$ . 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.

[学科分类] 电力

[关键词] Hardy type inequality;Discrete Muchenhoupt’s class;Reverse Hölder’s inequality;Higher summability