Hardy's inequality in L2 ([0,1]) and principal values of Brownian local times

We present in a unified framework two examples of a random function phi (omega, s) on R+ such that (a) the integral integral(0)(infinity) (omega, s) g (s) ds is well defined and finite (at least, as a limit in probability) for every deterministic and square integrable function g, and (b) phi does not belong to L-2 ([0, infinity), ds) with probability one. In particular, the second example is related to the existence of principal values of Brownian local times. Our key tools are Hardy's inequality, some semimartingale representation results for Brownian local times due to Ray, Knight and Jeulin, and the reformulation of certain theorems of Jeulin-Yor (1979) and Cherny (2001) in terms of bounded L-2 operators. We also establish, in the last paragraph, several weak convergence results.