Journal of Inequalities and Applications ( IF 1.470 ) Pub Date : 2020-10-16 , DOI: 10.1186/s13660-020-02497-4
S. H. Saker; S. S. Rabie; Ghada AlNemer; M. Zakarya

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.

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