Applications of Potential Theory to the Analysis of Property (P_(q))

Abstract

In the dissertation, we apply classical potential theory to study Property (P_(q)) and its relation with the compactness of the ∂ ̅-Neumann operator N_(q).

The main results in the dissertation consist of four parts. In the first part, we discuss the invariance property of Property (P_(q)) under holomorphic maps on any compact subset K in ℂ^(n).

In the second part, we show that if a compact subset K ⊂ ℂ^(n) has Property (P_(q)) (q ≥ 1), then for any q-dimensional affine subspace E in ℂ^(n), K ∩ E has empty interior with respect to the fine topology in ℂ^(q). We also discuss a special case of the converse statement on a smooth pseudoconvex domain when q = 1.

In the third part, we give two concrete examples of smooth complete Hartogs domains in ℂ^(3) regarding the smallness of the set of weakly pseudoconvex points on the boundary. Both examples conclude that if the Hausdorff 4-dimensional measure of the set of weakly pseudoconvex points is zero then the boundary has Property (P_(2)).

In the fourth part, we introduce a variant of Property (P_(n-1)) on smooth pseudoconvex domains in ℂ^(n) (n > 2) which implies the compactness of the ∂ ̅-Neumann operator N_(n-1).