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Sharp Self-Improving property of Discrete Muckenhoupt and Gehring Weights

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 sharp 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 the constant of transition are also explicitly given.

weight, discrete Gehring class, discrete Muckenhoupt class, self-improving property

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