Here's a great work from the financial literature on Hilbert Transforms and option pricing:

https://papers.ssrn.com/sol3/papers.cfm?abstract_id=993244

Several relationships between the Hilbert transform and characteristic function (with its links to the Laplace transform) are detailed.

@Prof Justin Lars Kirkby, thanks so much for your contribution.

check this : Article The Hilbert Transform as an Iterated Laplace Transform

Thanks to Prof. DJILANI KOBIBI YOUCEF ISLAM.

Hilbert transform has a unit impulse response 1/s. Laplace transform to the unit impulse is a step.

It is important to note that Hibert transform is not causal.

@ Prof. Ziad Sobih, thanks for your answer.