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Thank you@ Daniele Tampieri.

@IgorKhavkine. Thank you for your help.

Iin my case $G(z,w)=K_1(|z-w|^2)e^{i Imz.\overline{w}}$. So, $\mathscr G(x,y)= E(x-y) - G(x,y)=G(x,y/|y|)-G(x,y)$. Is it true. @DanieleTampieri

@DanieleTampieri. in the above case, how the Poisson Kernel is defined?

Thank you for your response@Daniele Tampieri. In above we have $LK_1(|z-w|^2)=0$, but the Green function is not $G(z,w)=K_1(|z-w|^2)$, the Green function is the kernel of the operator $L^{-1}$. In our case $G(z,w)=K_1(|z-w|^2)e^{i Imz.\overline{w}}$

Thank you for your helo@Peter Humphries

Thank you for your helo@Abdemalek Abdesselam

Thank you for your help, my question is it false?