Boundedness Criterions for the Hardy Operator in Weighted L^(p(.))(0,l) Space

Farman Mamedov; Firana M. Mammadova; Mushviq Aliyev

Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

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

Keywords

Hardy operator
Hardy type inequality
variable exponent
weighted inequality
necessary and sufficient condition

Mathematics Subject Classification

42A05, 42B25, 26D10; 35A23

Suggested citation

F. Mamedov, F. M. Mammadova, M. Aliyev. “Boundedness Criterions for the Hardy Operator in Weighted L^(p(.))(0,l) Space.” Journal of Convex Analysis 22 (2015), No. 2, 553–568.