Abstract
In connection with application to various problems of operator theory, we study almost monotonic functions w(x, r) depending on a parameter x which runs a metric measure space X, and the so called index numbers m(w, x), M(w, x) of such functions, and consider some generalized Zygmund, Bary, Lozinskii and Stechkin conditions. The main results contain necessary and sufficient conditions, in terms of lower and upper bounds of indices m(w, x) and M(w, x) , for the uniform belongness of functions w(·, r) to Zygmund-Bary-Stechkin classes. We give also applications to local dimensions in metric measure spaces and characterization of some integral inequalities involving radial weights and measures of balls in such spaces.