Quantitative dependence of some discrete limiting classes on the Muckenhoupt ${\cal A}_1(u)$ class

  • S. H. Saker Mansoura University and New Mansoura University, Egypt
  • R. R. Mahmoud Rustaq College of Education, Rustaq-Sultanate of Oman and Faculty of Science, Fayoum University, Egypt
  • M. H. Hassan Mansoura University, Egypt
Keywords: Gehring class, Muckenhoupt class, Hardy inequality, Carleman inequality.


UDC 517.5

We prove some relations between the discrete Gehring classes $\mathcal{G}_{q}$ and the discrete Muckenhoupt classes  $\mathcal{A}_{p}.$  Specifically, by using some known Hardy-type and Carleman-type inequalities, we study the relationship between $\mathcal{G}_{1},$ $\mathcal{A}_{\infty }$ and $\mathcal{A}_{1}$ for nonincreasing and nondecreasing weights.  Finally, we establish some general results by introducing the notions of $\mathcal{G}_{\varphi }$ classes defined for nonnegative convex function $\varphi.$


G. Bennett, K.-G. Grosse-Erdmann, Weighted Hardy inequalities for decreasing sequences and functions, Math. Ann., 334, 489–531 (2006). DOI: https://doi.org/10.1007/s00208-005-0678-7

A. Böttcher, M. Seybold, Wackelsatz and Stechkin's inequality for discrete Muckenhoupt weights, Preprint № 99-7, TU Chemnitz (1999).

G. H. Hardy, J. E. Littlewood, G. Polya, Inequalities, 2nd ed., Cambridge Univ. Press (1934).

N. Levinson, Generalizations of an inequality of Hardy, Duke Math. J., 31, № 3, 389–394 (1964). DOI: https://doi.org/10.1215/S0012-7094-64-03137-0

M. M. Iddrisu, C. A. Okpoti, Applications of Taylor series for Carleman's inequality through Hardy inequality, Korean J. Math., 23, № 4, 655–664 (2015). DOI: https://doi.org/10.11568/kjm.2015.23.4.655

N. A. Malaksiano, The exact inclusions of Gehring classes in Muckenhoupt classes} (in Russian), Mat. Zametki, 7, № 5, 742–750 (2001); English translation: Math. Notes, 70, № 5-6, 673–681 (2001).

N. A. Malaksiano, The precise embeddings of one-dimensional Muckenhoupt classes in Gehring classes, Acta Sci. Math., 68, 237–248 (2002).

B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165, 207–226 (1972). DOI: https://doi.org/10.1090/S0002-9947-1972-0293384-6

E. N. Nikolidakis, A sharp integral Hardy type inequality and applications to Muckenhoupt weights on $mathbb{R},$} Ann. Acad. Sci. Fenn. Math., 39, 887–896 (2014). DOI: https://doi.org/10.5186/aasfm.2014.3947

S. H. Saker, I. Kubiaczyk, Higher summability and discrete

weighted Muckenhoupt and Gehring type inequalities, Proc. Edinb. Math. Soc., 62, № 4, 949–973 (2019). DOI: https://doi.org/10.1017/S0013091519000014

S. H. Saker, R. R. Mahmoud, Boundedness of discrete Hardy–Littlewood maximal operator via Muchenhoupt weights, Rocky Mountain J. Math., 51, № 2, 733–746 (2021). DOI: https://doi.org/10.1216/rmj.2021.51.733

S. H. Saker, S. S. Rabie, Ghada AlNemer, M. Zakarya, On structure of discrete Muckenhoupt and discrete Gehring classes, J. Inequal. and Appl., 2020, № 1, 1–18 (2020). DOI: https://doi.org/10.1186/s13660-020-02497-4

S. H. Saker, S. S. Rabie, J. Alzabut, D. O'Regan, R. P. Agarwal, Some basic properties of discrete Muckenhoupt and Gehring classes and fundumental relations, Adv. Difference Equat., 2021, № 1, 1–22 (2021). DOI: https://doi.org/10.1186/s13662-020-03105-x

S. H. Saker, D. O'Regan, R. P. Agarwal, Self-improving properties of discrete Muckenhoupt weights, Analysis, 41, № 3, 169–178 (2021). DOI: https://doi.org/10.1515/anly-2020-0052

How to Cite
Saker, S. H., R. R. Mahmoud, and M. H. Hassan. “Quantitative Dependence of Some Discrete Limiting Classes on the Muckenhoupt ${\cal A}_1(u)$ Class”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, no. 3, Apr. 2023, pp. 397 -15, doi:10.37863/umzh.v75i3.6788.
Research articles