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

S. H.  Saker; R. R.  Mahmoud; M. H.  Hassan

doi:10.37863/umzh.v75i3.6788

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

DOI: https://doi.org/10.37863/umzh.v75i3.6788

Keywords: Gehring class, Muckenhoupt class, Hardy inequality, Carleman inequality.

Abstract

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 $\varphi.$

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