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

Saker, Samir; Krnić, Mario

doi:10.1090/proc/15180

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

Authors: Samir H. Saker and Mario Krnić
Journal: Proc. Amer. Math. Soc. 149 (2021), 231-243
MSC (2010): Primary 40D05, 40D25; Secondary 42C10, 43A55, 46B15
DOI: https://doi.org/10.1090/proc/15180
Published electronically: October 9, 2020
MathSciNet review: 4172600
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Abstract: 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.

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