Parameter depending almost monotonic functions and their applications to dimensions in metric measures spaces
Let \(X\) be a metric space with a positive measure \(\mu \) and \(\Omega \) a bounded open set in \(X\) with \(l=\text{diam\,}\Omega ,0<l<\infty \). The author studies functions \(w(x,r)\) defined on \(\Omega \times \left[ 0,l\right] \) which are almost increasing in the variable \(r,\) thus \(w(x,r_{1})\leq C_{w}w(x,r_{2})\) for \(0\leq r_{1}\leq r_{2}\leq l,\) \(x\in \Omega \), with \(1\leq C_{w}<\infty \). The index numbers \(m(w,x),M(w,x)\) of such functions and some generalized Zygmund, Bary, Lozinskii and Stechkin conditions are considered. Conditions for the uniform belonging of functions \(w(.,r)\) to Zygmund-Bary-Stechkin classes are determined.
scientific article; zbMATH DE number 5568059