International Journal of Mathematics and Mathematical Sciences
Volume 8 (1985), Issue 3, Pages 441-448
doi:10.1155/S0161171285000485

Norm-preserving L−L integral transformations

Abstract

In this paper we consider an L−L integral transformation G of the form F(x)=∫0∞G(x,y)f(y)dy, where G(x,y) is defined on D={(x,y):x≥0,y≥0} and f(y) is defined on [0,∞). The following results are proved: For an L−L integral transformation G to be norm-preserving, ∫0∞|G*(x,t)|dx=1 for almost all t≥0 is only a necessary condition, where G*(x,t)=limh→0inf1h∫tt+hG(x,y)dy for each x≥0. For certain G's. ∫0∞|G*(x,t)|dx=1 for almost all t≥0 is a necessary and sufficient condition for preserving the norm of certain f ϵ L. In this paper the analogous result for sum-preserving L−L integral transformation G is proved.