Complex Analytic and Differential Geometry 2024, 46: Holomorphic Functions & Subharmonic Functions

For an open set in complex space, and some complex variable z, a once-differentiable function follows

for a compact set K and a Lebesgue measure on complex space dλ. For holomorphic f, this reduces to the usual Cauchy formula. The set of holomorphic functions on an open set is a ring.

An open polydisk D(z, R) centered on z with multiradius R is a product of the disks with quantities of matching index in the argument. The distinguished boundary of D is the product of the boundary circles, which is smaller than the topological boundary. For a closed polydisk D' contained in Ω with a holomorphic function f on Ω, then

Calculus on currents functions largely analogous to complex differential calculus. Spaces decompose like this

On an open set Ω in multi-dimensional complex with compact subset K so that Ω\K is connected, all holomorphic functions f on Ω\K extends into some holomorphic function on Ω. If Ω is a neighborhood of 0

d'' is hypoelliptic in bidegree.

The Green kernel of a smoothly bounded domain Ω is a function G so that

For all smooth functions u, v on a smoothly bounded domain, and the derivative along the outward normal unit vector over the euclidean area measure

A Borel function u on a closed ball, the mean values are

Define the subharmonic function by equivalent statements for an upper semicontinuous functions

Decreasing sequences of subharmonic functions tend toward a subharmonic function. Convex, functions that are non-decreasing for all coordinates that can be extended by continuity into a function [-∞, ∞[ᴾ → [-∞, ∞[ then it is part of Sh(Ω), the set of subharmonic functions. If Ω is connected and u subharmonic, then u = -∞ or it's part of L¹(Ω). All families (u) have countable subfamilies with an upper envelope v, so that v ≤ u ≤ u* = v*, where u*(z) = lim sup u ≥ u(z)