On fractional integrals of pure imaginary order in 𝐿𝑝

The purpose of this paper is to extend some results of Kober [4] from L2 to Lp, 1 <p < oo. Sahnovic [7], using a similar method, obtained a result similar to Kober's (in L2). His purpose was to show the similarity of the operators M and M+i V in L2(0, 1) where (Mf) (x) = xf(x) and (Vf) (x) = fo'f(y) dy; he found that the operator P implementing this similarity, that is, a bounded operator P with bounded inverse P-1 such that P-'MP = M+i V, is precisely P = P where Jr is the operator of fractional integration