On structure of discrete Muchenhoupt and discrete Gehring classes

Abstract

Abstract In this paper, we study the structure of the discrete Muckenhoupt class A p ( C ) $\mathcal{A}^{p}(\mathcal{C})$ and the discrete Gehring class G q ( K ) $\mathcal{G}^{q}(\mathcal{K})$ . In particular, we prove that the self-improving property of the Muckenhoupt class holds, i.e., we prove that if u ∈ A p ( C ) $u\in \mathcal{A}^{p}(\mathcal{C})$ then there exists q 1 $q>1$ . The relation between the Muckenhoupt class A 1 ( C ) $\mathcal{A}^{1}(\mathcal{C})$ 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.

Keywords

• Hardy type inequality
• Discrete Muchenhoupt’s class
• Reverse Hölder’s inequality
• Higher summability