Representation of holomorphic functions by boundary integrals

Weighted Integrals of Holomorphic Functions in the Unit Polydisc

Let f be a measurable function defined on the unit polydisc Un in Cn and let ωj(zj), j = 1, . . . ,n, be admissible weights on the unit disk U , with distortion functions ψj(zj), p,q ω,N (U n) = { f | ‖ f ‖ p,q ω,N < ∞}, where ‖ f ‖ q p,q ω,N = ∫[0,1)n Mp( f ,r)∏nj=1ωj(r j)drj , and p,q ω,N (U n) = p,q ω,N (Un)∩H(Un). We prove the following result: if p,q ∈ [1,∞) and for all j = 1, . . . ,n, ...

On the Representation of Arbitrary Functions by Definite Integrals.

Holomorphic Matrix Integrals

We study a class of holomorphic matrix models. The integrals are taken over middle dimensional cycles in the space of complex square matrices. As the size of the matrices tends to infinity, the distribution of eigenvalues is given by a measure with support on a collection of arcs in the complex planes. We show that the arcs are level sets of the imaginary part of a hyperelliptic integral connec...

Boundary Values of Bounded Holomorphic Functions of Several Variables

A typical result (consequence of Theorem 1) is that if such a function tends to a limit X as x—»x° from inside an open cone with vertex at x°, then it tends "on the average" to X as x—>x° from inside any open cone with vertex at x°. In particular, if such a function tends to limits in each of two open cones with a common vertex, these limits must be equal; for n = l (when the cones are half-lin...

On the Boundary Values of Holomorphic Functions in Wedge Domains

WD = {z => x + iy\ x Q R , y & D). The distinguished boundary of WD is the set {z = x+iy \ xÇzR, y = 0 } , i.e., just R. In this paper we consider the following problem. Let ƒ be a real function on R satisfying certain regularity conditions. What are the conditions for the existence of a holomorphic function F on WD such that ƒ is the limit of Re -Fas the variable approaches the distinguished b...