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- Parameter depending almost monotonic functions and their applications to dimensions in metric measure spacesPublication . Samko, NatashaIn 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.
- Weighted hardy and singular operators in morrey spacesPublication . Samko, NatashaWe study the weighted boundedness of the Cauchy singular integral operator S-Gamma in Morrey spaces L-p,L-lambda '(Gamma) on Curves satisfying the arc-chord condition, for a class of "radial type" almost monotonic weights. The non-weighted boundedness is shown to hold on an arbitrary Carleson curve. We show that the weighted boundedness is reduced to the boundedness of weighted Hardy operators in Morrey spaces L-p,L-lambda(0. e), e > 0. We find conditions for weighted Hardy operators to be bounded in Morrey spaces. To cover the case of curves we also extend the boundedness of the Hardy-Littlewood maximal operator in Morrey spaces, known in the Euclidean setting, to the case of Carleson curves. (c) 2008 Elsevier Inc. All rights reserved.