Measures Which Are Convolution Exponentials

Let M(R) denote the measure algebra on the additive group of the reals. R. G. Douglas recently pointed out to us the importance of the following question in the study of Wiener-Hopf integral equations: if fxÇ:M(R) is invertible, then under what conditions does jit = exp(j>) for some vGM(R)? The relevance of the above question in integral equations stems from the fact that if JJLÇÏM(R) is invertible, then // is an exponential if and only if /x has a factorization of the form M=MI * M2, where jUi and ju2 are invertible elements of M[0, 00) and M(— <*>, O] respectively. In fact, if/x = exp(j>) and^i = ^| [0,00), 2̂ =v\ (-00,0), thenjUi = exp(j>i) and M2 = exp(^2) yields such a factorization. Now if Wp is the Wiener-Hopf operator on L [0, 00 ) (p ̂ > 1) given by