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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 potential operators in the generalized Morrey spacesPublication . Persson, Lars-Erik; Samko, NatashaWe study the weighted p -> q-boundedness of the multi-dimensional Hardy type operators in the generalized Morrey spaces L-p.phi(R-n, w) defined by an almost increasing function phi(r) and radial type weight w(vertical bar x vertical bar). We obtain sufficient conditions, in terms of some integral inequalities imposed on phi and w, for such a p -> q-boundedness. In some cases the obtained conditions are also necessary. These results are applied to derive a similar weighted p -> q-boundedness of the Riesz potential operator. (c) 2010 Elsevier Inc. All rights reserved.
- Fractional integrals and hypersingular integrals in variable order Holder spaces on homogeneous spacesPublication . Samko, Natasha; Samko, Stefan; Vakulov, BorisWe consider non-standard Holder spaces H(lambda(.))(X) of functions f on a metric measure space (X, d, mu), whose Holder exponent lambda(x) is variable, depending on x is an element of X. We establish theorems on mapping properties of potential operators of variable order alpha(x), from such a variable exponent Holder space with the exponent lambda(x) to another one with a "better" exponent lambda(x) + alpha(x), and similar mapping properties of hypersingular integrals of variable order alpha(x) from such a space into the space with the "worse" exponent lambda(x) - alpha(x) in the case alpha(x) < lambda(x). These theorems are derived from the Zygmund type estimates of the local continuity modulus of potential and hypersingular operators via such modulus of their densities. These estimates allow us to treat not only the case of the spaces H(lambda(.))(X), but also the generalized Holder spaces H(w(.,.))(X) of functions whose continuity modulus is dominated by a given function w(x, h), x is an element of X, h > 0. We admit variable complex valued orders alpha(x), where R alpha(x) may vanish at a set of measure zero. To cover this case, we consider the action of potential operators to weighted generalized Holder spaces with the weight alpha(x).
- 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.