Merging to harmonic function?

I think this page should be either extended or what I 'd prefer most intergrated in the 'subharmonic'-Entry. Hottiger 21:17, 8 March 2006 (UTC)[reply]

Good point about this article being rather stubby. As history has it, this article was created before the one on subharmonic function, and that may explain why it was not merged to start with.

Now, I would think it would be fine if it stays the way it is. This article is indeed rather small, but on the other hand the one at subharmonic function is big enough, and in my view, it would be easier to read the subharmonic function without having to even think of n complex dimensions, which may be intimidating. :)

I would be more than happy if some expert (I am not) would extend this one. Oleg Alexandrov (talk) 21:57, 8 March 2006 (UTC)[reply]

Sandbox for entirely revised entry

{tt}Note: This is only the very beginning of some reformulation.{/tt}

In mathematics, plurisubharmonic functions form an important class of functions used in complex analysis. They are the higher dimensional generalization of subharmonic functions. They can be defined in full generality on complex spaces.

Formal definition

Let $X$ is any complex space and $\Delta \subset {\mathbb {C} }$ denote the unit disk. Then an upper semi-continuous function

f\colon X\to {\mathbb {R} }\cup \{-\infty \}

is said plurisubharmonic if and only if for any holomorphic map $\varphi \colon \Delta \to X$ the function

f\circ \varphi \colon \Delta \to {\mathbb {R} }\cup \{-\infty \}

is subharmonic.

Reductions

A upper semi-continuous function is plurisubharmonic if and only if its restriction to holomorphic disks through each point in each tangent direction is subharmonic, formally:

If $f\circ \varphi$ is subharmonic for a set of functions $\varphi$ such that all subsets $d\varphi (T\Delta )\subset TX$ span $TX$ , $f$ is in fact plurisubharmonic, in particular an upper semi-continuous function function $f\colon G\subset {\mathbb {C} }^{n}\to {\mathbb {R} }\cup \{-\infty \}$ is plurisubharmonic if for every complex line

\{a+bz\mid z\in {\mathbb {C} }\}\subset {\mathbb {C} }^{n}

with

a,b\in {\mathbb {C} }^{n}

the function $z\mapsto f(a+bz)$ is a subharmonic function on the set

\{z\in {\mathbb {C} }\mid a+bz\in G\}.

If $f$ of (differentiability) class $C^{2}$ , then $f$ is plurisubharmonic, if

the hermitian matrix $L_{f}=(\lambda _{ij})$ , called Levi matrix, with entries

$\lambda _{ij}={\frac {\partial ^{2}}{\partial z_{i}\partial {\bar {z}}_{j}}}$

is positive definite.

Properties

The set of plurisubharmonic functions form a convex cone in the vector space of semicontinuous functions, i.e.

if $f$ is a plurisubharmonic function and $c>0$ a positiv real, then the function $c\cdot f$ is plurisubharmonic,
if $f_{1}$ and $f_{2}$ are plurisubharmonic functions, then the sum $f_{1}+f_{2}$ is a plurisubharmonic function.

Plurisubharmonicity is a local property, i.e. a function is plurisubharmonic if and only if

it is plurisubharmonic in a neighborhood of each point.

If $f$ is plurisubharmonic and $\varphi :\mathbb {R} \to \mathbb {R}$ a monotonically increasing, convex function then $\varphi \circ f$ is plurisubharmonic.
If $f_{1}$ and $f_{2}$ are plurisubharmonic functions, then the function $f(x):=\sup(f_{1}(x),f_{2}(x))$ is plurisubharmonic.
If $f_{1},f_{2},...$ is a monotonically decreasing sequence of plurisubharmonic functions

then so is $f(x):=\lim _{n\to \infty }f_{n}(x)$ .

The inequality in the usual semi-continuity condition holds as equality, i.e. if $f$ is plurisubharmonic then

$\limsup _{x\to x_{0}}f(x)=f(x_{0})$

(see limit superior and limit inferior for the definition of lim sup).

Plurisubharmonic functions satisfy the maximum principle, i.e. if $f$ is plurisubharmonic on the connected domain $D$ and

$\sup _{x\in D}f(x)=f(x_{0})$

for some point $x_{0}\in D$ then $f$ is constant.

Applications

In Complex analysis, plurisubharmonic functions are used to describe pseudoconvex domains, domains of holomorphy and Stein manifolds.

References

Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Robert C. Gunning. Introduction to Holomorphic Functions in Several Variables, Wadsworth & Brooks/Cole

Category:Potential theory Category:Complex analysis

Comments on Sandbox

Enjoy working on it. :) Oleg Alexandrov (talk) 18:06, 9 March 2006 (UTC)[reply]

This is a sketch of what I'd propose for a revised article. Some formulations and some layout does not please me up to now. And there can be said even some more but I still think this is an improval.?? --unsigned

I think it is good to first state the definition in C^n, as it is more elementary that way than starting with a complex space, whatever that means. Wonder what think. Oleg Alexandrov (talk) 19:10, 15 March 2006 (UTC)[reply]