Exact exponents for inclusion of discrete Muckenhoupt classes into Gehring classes and reverse

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In this paper, we establish some embedding relationships between Muckenhoupt and Gehring classes A p and G r by proving transition and inclusion relations. We also identify the exact range of > 1 for which w A p . Additionally, we show that the weights that satisfy the A p -condition also meet the A -condition. Next, we prove the Jones factorization property of A p weights in terms of two A 1 weights by employing the discrete Rubio De Francia iterated algorithm. Finally, we determine the specific ranges of indices (sharp exponents) for which w belongs to G r (A p ) if w belongs to A p (G r ) and the precise range of q < p for which w belongs to A q given that it belongs to A p .

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