On structure of discrete Muckenhoupt and discrete Gehring classes

Journal of inequalities and applications

article

S. H. Saker¹  S. S. Rabie²  Ghada AlNemer³  M. Zakarya⁴ 

[1] Mathematics Division, Faculty of Advanced Basic Sciences, Galala University;Department of Mathematics, Faculty of Science, Mansoura University;Department of Mathematical Science, College of Science, Princess Nourah bint Abdulrahman University;Department of Mathematics, College of Science, King Khalid University;Department of Mathematics, Faculty of Science, Al-Azhar University

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

DOI  :  10.1186/s13660-020-02497-4

学科分类：电力

来源: SpringerOpen

【 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.

【 授权许可】

CC BY