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

A. Gogatishvill1 and J. Lang2

 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

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′.