Journal of Inequalities and Applications
Volume 4 (1999), Issue 1, Pages 1-16
doi:10.1155/S1025583499000272
The generalized hardy operator with kernel and variable integral limits in banach function spaces
1Razmadze Mathematical Institute, Georgian Academy of Sciences, M. Aleksidze st., Tbilisi 380093, Georgia
2University of Missouri-Columbia, Department of Mathematics, 202 Mathematical Sciences Building, Columbia 65211, MO, USA
Received 18 December 1997; Revised 15 September 1998
Copyright © 1999 A. Gogatishvill and J. Lang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Let we have an integral operator
Kf(x): =v(x)∫a(x)b(x)k(x,y)u(y)f(y)dy
for x>0 where a and b are nondecreasing functions, u and v are non-negative and finite functions, and k(x,y)≥0 is nondecreasing in x, nonincreasing in y and k(x,z)≤D[k(x,b(y))+k(y,z)] for y≤x and a(x)≤z≤b(y). We show that the integral operator K:X→Y where X and Y are Banach functions spaces with l-condition is bounded if and only if A<∞. Where A:=A0+A1 and A0:=supx≤y,a(y)≤b(x)‖χ(x,y)(.)v(.)k(.,b(x))‖Y ‖χ(a(y),b(x))u‖x′A1:=supx≤y,a(y)≤b(x) ‖χ(x,y)v‖Y ‖χ(a(y),b(x))(.)K(x,.)u(.)‖x′.