Levi condition

A condition, which can be effectively verified, for pseudo-convexity in the sense of Levi of domains in the complex space . It was proposed by E.E. Levi

and consists of the following. Suppose that a domain is specified in a neighbourhood of a boundary point by the condition

where the real function belongs to the class and . If is Levi pseudo-convex at , then the (complex) Hessian

(1)

is non-negative for all that are complex orthogonal to , that is, are such that

(2)

Conversely, if the condition

(3)

is satisfied at the point for all satisfying (2), then is Levi pseudo-convex at .

For the inequalities (1) and (3) given above can be replaced by the simpler equivalent inequalities and , respectively, where

is the determinant of the Levi function .

The Levi condition (1)–(3) has also been generalized to domains on complex manifolds (see [4]).

References

[1a]	E.E. Levi, "Studii sui punti singolari essenziali delle funzioni analiticke de due o più variabili complesse" Ann. Mat. Pura Appl. , 17 (1910) pp. 61–87
[1b]	E.E. Levi, "Sulle ipersurficie dello spazio a 4 dimensione che possono essere frontiera del campo di esistenza di una funzione analitica di due variabili complesse" Ann. Mat. Pura Appl. , 18 (1911) pp. 69–79
[2]	V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)
[3]	B.V. Shabat, "Introduction of complex analysis" , 2 , Moscow (1976) (In Russian)
[4]	R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965)

Comments

By definition, is Levi pseudo-convex at if (1) is satisfied for vectors that satisfy (2); is called strictly (Levi) pseudo-convex at if (3) is satisfied for vectors that satisfy (2).

The domain is called (Levi) pseudo-convex if it is Levi pseudo-convex at every boundary point.

For domains with boundary, Levi pseudo-convexity is equivalent with any of the following:

a) is plurisubharmonic on (i.e. is Hartogs pseudo-convex), where denotes the Euclidean distance of to the boundary of .

b) relatively compact in implies relatively compact in , where .

References

[a1]	S.G. Krantz, "Function theory of several complex variables" , Wiley (1982)