We derive simple explicit formulas for the Fourier-Laplace transforms of the Hilbert transform and related functionals of the local time of a symmetric Levy process. These formulas generalize results of Biane and Yor for Brownian motion. The method of proof provides an explanation (of sorts) for the presence of the hyperbolic functions in such formulas.
"On the Distribution of the Hilbert Transform of the Local Time of a Symmetric Levy Process." Ann. Probab. 20 (3) 1484 - 1497, July, 1992. https://doi.org/10.1214/aop/1176989702