Abstract
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 is an element of A(p)(C) then there exists q < p such that u is an element of A(q)(C-1). Next, we prove that the power rule also holds, i.e., we prove that if u is an element of A(p) then u(q) is an element of 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.