On structure of discrete Muckenhoupt and discrete Gehring classes

Abstract

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