Journal of Convex Analysis 22 (2015), No. 2, 553--568
Copyright Heldermann Verlag 2015
Boundedness Criterions for the Hardy Operator in Weighted Lp(.)(0,l) Space
Farman Mamedov
Mathematics and Mechanics Institute, National Academy of Sciences, B. Vahabzade 9, Baku 1141, Azerbaijan
farman-m@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
[Abstract-pdf]
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
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