Pluripolarity of sets with small Hausdorff measure (Q1576581)

Let \(H(r)= (\log\frac 1r)^{-n}\), \(n\geq 2\). The author proves that any set \(E\subset\mathbb{C}^n\), \(n\geq 2\), with finite Hausdorff measure \(\Lambda_{H}(E)\leq\infty\) is pluripolar. The result is sharp with respect to the measuring function, i.e., for any measuring function \(h\) such that \[ \liminf_{r\to 0}\frac{h(r)}{H(r)}=0 \] there exists a set \(E\subset\mathbb{C}^n\) such that \(\Lambda_{h}(E)=0\) but \(E\) is not pluripolar.