The weighted discrete Gehring classes, Muckenhoupt classes and their basic properties

The main objective of this paper is a study of the structure and basic properties of the weighted discrete Gehring classes, as well as the study of the relationship between discrete Muckenhoupt and Gehring classes. First, we prove that the weighted discrete Muckenhoupt class $\mathcal {A}_{\lambda }^{1}(C)$, $C>1$, consisting of nonincreasing sequences, belongs to the weighted discrete Gehring class $\mathcal {G}_{\lambda }^{p}(A)$ by giving explicit values of exponent $p$ and constant $A$. Next, we prove the self-improving property of the weighted Gehring class $\mathcal {G}_{\lambda }^{p}({K)}$, $p>1$, $K>1$, consisting of nonincreasing sequences. The exponent and constant of transition are explicitly given. Finally, utilizing the self-improving property of the weighted Gehring class, we also derive the self-improving property of a discrete Muckenhoupt class $\mathcal {A}^{p}(C)$, $p>1$, $C>1$, with exact values of exponent and constant of transition.