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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
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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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Additional Information

Samir H. Saker
Affiliation: Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
Email: shsaker@mans.edu.eg

Mario Krnić
Affiliation: University of Zagreb, Faculty of Electrical Engineering and Computing, Unska 3, 10000 Zagreb, Croatia
Email: mario.krnic@fer.hr

DOI: https://doi.org/10.1090/proc/15180
Keywords: Discrete Gehring class, discrete Muckenhoupt class, self-improving property, harmonic analysis
Received by editor(s): March 26, 2020
Received by editor(s) in revised form: April 29, 2020
Published electronically: October 9, 2020
Communicated by: Mourad Ismail
Article copyright: © Copyright 2020 American Mathematical Society