ABSTRACT
For open subsets of ℂ n with smooth boundaries, there are additional useful alternative characterizations of pseudoconvexity and various other properties closely related to pseudoconvexity. Recall that an open subset D ⊆ ℝ m is said to have a smooth boundary of class CT if for each point A ∈ ∂D there are an open neighborhood UA of A in ℝ m and a CT diffeomorphism F A : U A → B ( 0 ; 1 ) ⊆ ℝ m https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750063/1087b710-7222-48dd-8094-26eaf896ae1b/content/eq2726.tif"/> such that F A ( U a ∩ D ) = { X = ( x 1 , … , x m ) ∈ B ( 0 ; 1 ) : x m < 0 } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750063/1087b710-7222-48dd-8094-26eaf896ae1b/content/eq2727.tif"/> . A CT diffeomorphism is a homeomorphism FA with the property that all the component functions of both FA and the inverse mapping F A − 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750063/1087b710-7222-48dd-8094-26eaf896ae1b/content/eq2728.tif"/> have continuous partial derivatives of all orders ≤ _ r https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750063/1087b710-7222-48dd-8094-26eaf896ae1b/content/eq2729.tif"/> . The condition that an open subset ∂ D ⊆ ℝ m https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750063/1087b710-7222-48dd-8094-26eaf896ae1b/content/eq2730.tif"/> has a smooth boundary of class CT will be indicated briefly by writing ∂ D ∈ C T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750063/1087b710-7222-48dd-8094-26eaf896ae1b/content/eq2731.tif"/> . Although this is a significant smoothness condition even for r = 0, the interest here lies more in subsets with boundaries of class CT for r ≥ _ 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750063/1087b710-7222-48dd-8094-26eaf896ae1b/content/eq2732.tif"/> — indeed, usually for r ≥ _ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750063/1087b710-7222-48dd-8094-26eaf896ae1b/content/eq2733.tif"/> .
