
Journal of Convex Analysis 22 (2015), No. 2, 553568 Copyright Heldermann Verlag 2015 Boundedness Criterions for the Hardy Operator in Weighted L^{p(.)}(0,l) Space Farman Mamedov Mathematics and Mechanics Institute, National Academy of Sciences, B. Vahabzade 9, Baku 1141, Azerbaijan farmanm@mail.ru Firana M. Mammadova Mathematics and Mechanics Institute, National Academy of Sciences, B. Vahabzade 9, Baku 1141, Azerbaijan mamedovafira@yahoo.com Mushviq Aliyev Mathematics and Mechanics Institute, National Academy of Sciences, B. Vahabzade 9, Baku 1141, Azerbaijan a.mushfiq@rambler.ru [Abstractpdf] Equivalent conditions are proved for the Hardy type weighted inequality $$ \Big\Vert W(\cdot)^{1}\sigma(\cdot)^{\frac{1}{p(\cdot)}} \int_{0}^{x} f(t)dt \Big \Vert_{L^{p(\cdot)}(0,l)} \leq C \Big \Vert \omega(\cdot)^{ \frac{1}{p(\cdot)}} f \Big \Vert_{L^{p(\cdot)}(0,l)}, \; \; \; f \geq 0 $$ to be fulfilled in the norms of a Lebesgue space with variable exponent $L^{p(.)}(0,l)$. It is assumed that the function $p(.)$ is a monotone function. Keywords: Hardy operator, Hardy type inequality, variable exponent, weighted inequality, necessary and sufficient condition. MSC: 42A05, 42B25, 26D10; 35A23 [ Fulltextpdf (161 KB)] for subscribers only. 