On the Stieltjes transform of generalized functions

Abstract

If f(t) belongs to L(0, R) for every positive R and is such that the integralconverges for x > 0, then F(s) exists for complex s(s ╪ 0) not lying on the negative real axis andfor any positive ξ at which f(ξ+) and f(ξ−) both exist.We define an operator Lk, t[F(x)]byUnder the above conditions on f(t), it is known that for all points t of the Lebesgue set for the function f(t),Let Ln, x denote the differentiation operatorSuppose thatconverges for some x¬ 0; then, if f(t) belongs to L(R−1, R) for every R>1,