currently I am trying to solve the following question: Let $f \in C^{\infty}_0(\mathbb C)$ and consider the following two integral operators. First we have the Cauchy integral operator given by $$ C[f](z) = -1/\pi \int \int_{\mathbb C \times \mathbb C} \frac{f(\zeta)}{z-\zeta} d\zeta^2, $$ and secondly we have $$ T[f](z) = \int \int_{\mathbb C \times \mathbb C} \frac{f(\zeta)}{\log(|z-\zeta|)} d\zeta^2. $$ The goal is now to show that if $f \in C^{\infty}_0(\mathbb C)$ then $T[f](z) \in C^{\infty}$. The idea would be to take the derivative of the second operator and relate it to the Cauchy integral one but this has been without much success. Basically just taking the derivative yields that $$ d/dz T[f](z) = \int \int_{\mathbb C \times \mathbb C} \frac{f(\zeta)}{\log(|z-\zeta|)^2} \frac{z-\zeta}{|z-\zeta|} d\zeta^2, $$ however I am not sure how to deal with the logarithmic nature of the kernel and if this is even the correct approach? Any suggestions?
Thanks in advance!