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