Let $G(\phi,\epsilon) = 2Real \left ( (e^{-B+i\phi} * P_z \right ) \cdot ([-i\epsilon \cdot e^{-B-i\phi}]*\overline{P_z})$ with $\phi, \epsilon, B \in L^2(\Omega,\mathbb{R})$ and $P_z \in L^2(\Omega,\mathbb{C})$, where $\Omega$ is the space of positions, a bounded subset of $\mathbb{R}^2 $. The symbol $ * $ stands for 2D convolution and $\overline{?}$ for the conjugate form.
I'm trying to calculate the adjoint operator of $ G $ denoted $ G^* $ which can be defined as follows: $\langle G(\phi, \epsilon), y \rangle = \langle \epsilon , G^*(\phi, y) \rangle$ with $y$ the image of $ G $. I can calculate this type of operator for classical functions but I'm blocked mainly because of the convolution.
To put this into context, $P_z$ corresponds to the Fresnel propagator $P_z = \frac{1}{iz\lambda} e^{i\frac{\pi}{z\lambda }\sqrt{x^2+y^2}}$, B corresponds to the amplitude of a wavefront at z=0 and $\phi$ to its phase. G correspond to the Fréchet-differential of the projected intensity operator $ I_z(B,\phi ) = \lvert e^{-B+i\phi}*P_z \rvert^2 $ at $\phi$ defined as follows : $I_D(B,\phi+\epsilon) = I_D(B,\phi) + \langle~\nabla_\phi I_D(B,\phi),\epsilon~\rangle +o(\epsilon) = I_D(B,\phi) + G(\phi,\epsilon) + o(\epsilon) $.
Knowing the adjoint operator will be useful when using iteratives phase retrival algorithms. It's already been done, but I want to demonstrate the result (and make sure there's no mistake) like here "Non-linear iterative phase retrieval based on Frechet derivative" V. Davidoiu in chapiter 3,2. Note that in this paper $ I_z(B,\phi ) = \lvert e^{-B-i\phi}*P_z \rvert^2 $ to obtain their results .
Can you help me?