Leray formula

Cauchy–Fantappié formula

A formula for the integral representation of holomorphic functions of several complex variables , , which generalizes the Cauchy integral formula (see Cauchy integral).

Let be a finite domain in the complex space with piecewise-smooth boundary and let be a smooth vector-valued function of with values in such that the scalar product

everywhere on for all . Then any function holomorphic in and continuous in the closed domain can be represented in the form

(*)

Formula (*) generalizes Cauchy's classical integral formula for analytic functions of one complex variable and is called the Leray formula. J. Leray, who obtained this formula (see [1]), called it the Cauchy–Fantappié formula. In this formula the differential forms and are constituted according to the laws:

and

where is the sign of exterior multiplication (see Exterior product). By varying the form of the function it is possible to obtain various integral representations from formula (*). One should bear in mind that, generally speaking, the Leray integral in (*) is not identically zero when is outside .

See also Bochner–Martinelli representation formula.

References

[1]	J. Leray, "Le calcul différentielle et intégrale sur une variété analytique complexe" Bull. Soc. Math. France , 87 (1959) pp. 81–180
[2]	B.V. Shabat, "Introduction of complex analysis" , 2 , Moscow (1976) (In Russian)

Comments

Often the Leray formula is understood to be a more general representation formula, valid for arbitrary sufficiently smooth (e.g., ) functions on a domain in . Let , and be as defined above, . Furthermore, define for , and :

Let denote the right-hand side of (*). It is well defined for measurable functions on . Define for a continuous -form on ,

meaning that the exterior derivative in the definition of has to be with respect to as well as . Next, for -forms defined on there holds

the Bochner–Martinelli operator.

Now let be a continuous function on such that is continuous there too. Then Leray's formula reads

(a1)

where .

If is holomorphic on , then (a1) reduces to (*). Of particular importance are instances where , and hence also , is holomorphic as a function of for fixed — this can only occur if is pseudo-convex; is then a holomorphic support function (i.e. for all there is a neighbourhood of such that is holomorphic in this neighbourhood and ), the existence of which is closely related to the existence of continuously varying holomorphic peaking functions. (A continuously varying holomorphic peaking function for is a function such that for each fixed : 1) is holomorphic on and continuous on ; and 2) and for all . If , is required to be for each fixed .) Then is holomorphic for every continuous on and the operator

solves the inhomogeneous Cauchy–Riemann equations

(a2)

for continuous -forms on . Formula (a1) can be generalized to give a representation formula for -forms as well (see [a2]).

Thus, the Leray formula has become an important tool for solving the Levi problem (work of G.M. Khenkin [a1] and of E. Ramirez de Arellano [a3]) and for obtaining estimates for solutions of (a2). In particular, the following sharp Hölder estimates hold on strictly pseudo-convex domains: There is a solution with , where depends on the domain only, denotes the Hölder -norm and denotes the sup-norm. Many analysts made contributions in this direction, notably Khenkin and A.V. Romanov; H. Grauert and I. Lieb; and N. Kerzman and R.M. Range.

References

[a1]	G.M. [G.M. Khenkin] Henkin, "Integral representations of functions holomorphic in strictly pseudoconvex domains and some applications" Math. USSR Sb. , 78 (1969) pp. 611–632 Mat. Sb. , 7 (1969) pp. 597–616
[a2]	J.L. Leiterer, "Theory of functions on complex manifolds" , Birkhäuser (1984)
[a3]	E. Ramirez de Arellano, "Ein Divisionsproblem und Randintegraldarstellungen in der komplexen Analysis" Math. Ann. , 184 (1970) pp. 172–187
[a4]	R.M. Range, "Holomorphic functions and integral representation in several complex variables" , Springer (1986) pp. Chapt. VI, Par. 6