Hilbert transform of the integral of a function

Question

Given that f and g are Hilbert transform pair $$Hf(x) = g(x)$$ Although the derivative will maintain the transform pair relation $$Hf'(x) = g'(x)$$ Does the Hilbert transform pair relation apply to its integral/(anti-derivative)? I speculate that $$H\int_{0}^{x}f(t) \, dt = \int_{0}^{x}g(t) \, dt + C$$ I could not find such a relationship in the references I have! I am trying to use the above relationship for the integral of the Airy Ai function and the integral of the Scorer Gi function. $$HAi_1 (x) = Gi_1(x)$$

Answer 1

In any situation where everything makes sense and works as desired:

The Hilbert transform commutes with differentiation (as you observe). That is, differentiating the Hilbert transform of the integral gives the Hilbert transform of the derivative $f$ of the integral. And this is $g$, by assumption. Since the only functions (or distributions!) annihilated by differentiation are constants, this gives your anticipated assertion.