How to define a linear operator in Maple that commutes with derivatives?

Question

I would like to simplify an expression involving the Hilbert transform in Maple. The Hilbert transform is defined by $$ Hf(x) = \frac{1}{\pi} \ \mathrm{p.v.} \int_{-\infty}^{+\infty} \frac{f(z)}{z-x} \mathrm{d}x, $$ where $\mathrm{p.v.}$ denotes the Cauchy principal value. Formally, the derivative of the Hilbert transform is the Hilbert transform of the derivative, i.e. these two linear operators commute: $ \frac{\mathrm{d}}{\mathrm{d}x} Hf(x) = H \left( \frac{\mathrm{d}}{\mathrm{d}x} f(x) \right) $. However, for slightly complex functions $f(x)$, Maple is unable to recognize this equation. For example, Maple is unable to recognize the equality between $ \frac{\mathrm{d}}{\mathrm{d}x} Hf^2(x) $ and $ H \left( \frac{\mathrm{d}}{\mathrm{d}x} f^2(x) \right) $. Furthermore, the expression I seek to compute is very complicated, causing the computer to run for a long time without producing a result.

In fact, the simplification process does not involve applying the Hilbert transform to specific functions. Therefore, I would like to define an abstract operator that satisfies certain properties of the Hilbert transform, and use it for the simplification. Specifically, I need to define an operator $H$ that satisfies the following properties:

1. Linearity: $$ H \left( \sum_{i=1}^{N} a_i f_i(x,y,t) \right) = \sum_{i=1}^{N} a_i H\left( f_i(x,y,t)) \right), \quad a_i \in \mathbb{R}, \ N \in \mathbb{N} .$$
2. Differentiation: $$ \frac{\partial^m}{\partial x^m} \frac{\partial^n}{\partial y^n} \frac{\partial^k}{\partial t^k} H(f(x,y,t)) = H(\frac{\partial^m}{\partial x^m} \frac{\partial^n}{\partial y^n} \frac{\partial^k}{\partial t^k} f(x,y,t) ), \quad m,n,k \in \mathbb{N} .$$
3. Inverse transform: $$ H(H(f(x,y,t)))=-f(x,y,t) .$$
4. Identity: $$ H(fH(g)) + H(gH(f)) = (H(f))(H(g)) - fg .$$

Although all of these properties are useful in the simplification process, the first two properties are indispensable. If possible, please provide a definition for an operator that satisfies at least the first two properties. Additionally, it would be great if there is a solution for the commutativity of the Hilbert transform and derivatives.

I welcome both answers with a detailed code and answers that point me towards the necessary tools. Thank you!