Let $X=\{x=y^{3}\}$ and $\eta(x,y)=\log|1+y|$, so $\eta\,|_{{}_{X}}\in{\mathcal{L}}_{1/3}(X)$.

Given $k\geq 1$, there is a polynomial $Q_{k}(x,y)$ of degree $k+1$ so that $Q_{k}(y^{3},y)=(y+1)^{3k}$. In particular, $\psi_{k}=\frac{1}{3k}\log|Q_{k}|\in{\mathcal{L}}_{(k+1)/3k}({\mathbb{C}}^{2})$ is an extension of $\eta\,|_{{}_{X}}$.

There exists no polynomial $Q(x,y)$ of degree $k$ so that $Q(y^{3},y)=(y+1)^{3k}$. However, $\eta\,|_{{}_{X}}$ has an extension in ${\mathcal{L}}_{1/3}({\mathbb{C}}^{2})$.