On structure of discrete Muchenhoupt and discrete Gehring classes

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 < p $q< p$ such that u ∈ A q ( C 1 ) $u\in \mathcal{A}^{q}(\mathcal{C}_{1})$ . Next, we prove that the power rule also holds, i.e., we prove that if u ∈ A p $u\in \mathcal{A}^{p}$ then u q ∈ A p $u^{q}\in \mathcal{A}^{p}$ for some 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.