Suppose we are on $(\mathbb C, i dz \wedge d \bar z)$ the 2-dimensional plane equipped with the usual volume form. I wonder for what $k$ values will we be able to view $\log|z|^{k}$ as a tempered distribution?
If it is $L^{1}$ then we are good, so the problem can be reduced to show that $-\int_{B_1(0)} k \log|z| i dz \wedge d \bar z$ is integrable. I think $k$ does not really matter here and $-\int_{B_1(0)} \log |z| i dz \wedge d \bar z$ is always integrable, so for all values of $k$ it can be viewed as a distribution.
If that is true, is there a terminology for such a distribution?