PomeExample 5.33.3.36A8480CE7DF On a disk in $\mathbb C$ centered at the origin, with coordinate $z$, let $\lambda \geq 0$ and put $u(z)=\lambda \log |z|^2$ for $\lambda >0$, while $u=0$ for $\lambda =0$. Then
\[ \mathcal I(u)_0=(z^{\lfloor \lambda \rfloor }). \]
Indeed, if a nonzero holomorphic germ has order $m$ at the origin, its squared modulus is comparable to $|z|^{2m}$ near the origin. In polar coordinates, its weighted integral over a disk of sufficiently small radius $\epsilon >0$ is finite precisely when
\[ \int _0^\epsilon r^{2m-2\lambda +1}\, \mathrm{d}r<\infty , \qquad \text{equivalently}\qquad m>\lambda -1. \]
The smallest nonnegative integer satisfying this strict inequality is $\lfloor \lambda \rfloor $. When $\lambda $ is an integer, the borderline exponent $-1$ occurs for $m=\lambda -1$ and gives a divergent integral; this is where the strictness of the inequality matters.
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