Pluripolarity of Sets with Small Hausdorff Measure

Research Report MRR99-012

Pluripolarity of Sets with Small Hausdorff Measure

Denis A. Labutin

Abstract: We show that any set $E\subset { C}^n$, $n\geq 2$, with the finite Hausdorff measure $\Lambda_{(\log {1\over r} )^{-n}}(E)<+\infty$ is pluripolar. The result is sharp with respect to the measuring function. The idea of the proof is to combine a construction from the potential theory for the real variational integral $\int_\Omega |\nabla u|^m$, $\Omega\subset{ R}^m$, with properties of the pluricomplex relative extremal function for the Bedford-Taylor capacity.

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