On the Behaviour of the Distributional Stieltjes Transformation at the Origin

The sets of real and natural numbers are denoted by 91 and91, respectively. 0() = and °'(R) = T' denote the spaces of rapidly decreasing functions and tempered distributions, respectivel5. The space 7'(r), r E R \ (-1) is defined in [3] as a subspace of = E à°'(R); supp / c [0, co)) consisting of all / of the form/ = DmF, for some m E 9? = IR u {0}, FE L' bc, supp F c[O, oo), D is the distributional derivative, such that