On structure of discrete Muchenhoupt and discrete Gehring classes.

In this paper, we study the structure of the discrete Muckenhoupt class A p (C) and the discrete Gehring class G q (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) then there exists q < p such that u ∈ A q (C 1) . Next, we prove that the power rule also holds, i.e., we prove that if u ∈ A p then u q ∈ A p for some q > 1 . The relation between the Muckenhoupt class 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. [ABSTRACT FROM AUTHOR]