ENTRY POTENTIAL THEORY 
Authors: Oliver Knill: jan 2003
Literature: "T. Ransford","Potential theory in the complex plane".


+------------------------------------------------------------
| Analytic
+------------------------------------------------------------

Analytic Let D subset C be an open set. A continuous function 
f: D arrow C is called analytic in D, if for
all z in D the complex partial derivative
 fracpartial f partial z := 
   lim_|h| arrow 0 frac1h (f(z+h) - f(z)) 
exists and is finite. 
Analytic functions are also called holomorphic. Properties:
the sum and the product of analytic functions are analytic.
If f_n is a sequence of analytic maps which converges uniformly on 
compact subsets of D to a function f, then f is analytic too. 


+------------------------------------------------------------
| complex partial derivative
+------------------------------------------------------------

Define the complex partial derivative of a complex function 
f(z)=f(x+iy) in the complex plane is defined as
fracpartial fpartial z = 
  frac12 (fracpartialpartial x-ifracpartialpartial y) f.


+------------------------------------------------------------
| conformal map
+------------------------------------------------------------

A conformal map is a differentiable map 
from the complex plane to the complex plane which 
preserves angles.

 every conformal map which has continuous 
      partial derivatives is analytic.
 An analytic function f is conformal at every 
      point where its derivative f'(z) is different from 0.
 


+------------------------------------------------------------
| Dirichlet problem
+------------------------------------------------------------

Solution of the Dirichlet problem.
If D is a regular domain in the complex plane
and f is a continous function on the boundary of D,
then there exists a unique harmonic function f
on D such that h(z)=f(z) for all boundary points
of D. 


+------------------------------------------------------------
| Dirichlet problem
+------------------------------------------------------------

Let K be a compact subset of the complex plane. 
Let P(K) the set of all Borel probability measure 
on K. A measure nu maximizing the potential theoretical energy 
in P(K) is called an equilibrium measure of K.
Properties:

 every compact K has an equilibrium measure. 
 if K is not polar then the equilibrium measure is 
      unique. 



+------------------------------------------------------------
| fine topology
+------------------------------------------------------------

The fine topology on the complex plane is defined as 
the coarsest topology on the plane which makes all 
subharmonic functions continuous. 


+------------------------------------------------------------
| Frostman's theorem
+------------------------------------------------------------

Frostman's theorem: If nu is the equilibrium measure
on a compact set K, then the potential p_nu  of nu
is bounded below by I(nu) everywhere on C. Furthermore, 
p_nu=I(nu) everywhere on K except on a F_sigma polar 
subset E of the boundary of K. 


+------------------------------------------------------------
| Frostman's theorem
+------------------------------------------------------------

A function h on the complex plane is called harmonic 
in a region D if it satisfies the mean value property
on every disc contained in D. 


+------------------------------------------------------------
| harmonic measure
+------------------------------------------------------------

A harmonic measure w_D on a domain D is a function from
D to the set of Borel probability measures on the boundary
of D. The measure for z is defined as the
functional g mapsto H_D(g)(z), where H_D(g) is the
Perron function of g on D.

 if the boundary of D is non-polar, there exists a
      unique harmonic measure for D.
 if D=Im(z)<0, then
      w_D(z,a,b) =  arg( (z-b)/(z-a) )/pi




+------------------------------------------------------------
| Harnack inequality
+------------------------------------------------------------

The Harnack inequality assures that for any positive harmonic
function h on the disc D(w,R) and for any r<R and 0<t<2Pi
  h(w) (R-r)/(R+r) leq h(w+r e^it leq h(w) (R+r)/(R-r) 


+------------------------------------------------------------
| extended Liouville theorem
+------------------------------------------------------------

The extended Liouville theorem: if f is subharmonic
on the complex plane C-E, where E is a closed polar 
set and f is bounded above then f is constant. 


+------------------------------------------------------------
| generalized Laplacian
+------------------------------------------------------------

The generalized Laplacian Delta(f) of a subharmonic function
f on a domain D is the Radon measure mu on D defined 
as the linear functional g mapsto int_D u Delta g  dA.
The Laplacian of a subharmonic function is also called the 
Riesz measure. 
The Laplacian is known to exist and is unique. If
p_mu is the potential associated to mu, then 
Delta p_mu = mu. 


+------------------------------------------------------------
| Hadamard's three circle theorem
+------------------------------------------------------------

Hadamard's three circle theorem assures that for any 
subharmonic function f on the annulus r<|z|<R the function
M(f,r)=sup_|z|=r f(z) is an increasing convex function of log(r). 


+------------------------------------------------------------
| Jensen formula
+------------------------------------------------------------

Jensen formula If f is holomorphic in the disc D=B(0,R), r<R and
a_1,...,a_n are the zeros of f in the closure of D counted with 
multiplicity, then 
int_0^2pi log|f(r exp(i t))|  dt = log|f(0)| + N log(r)- sum_j=1^n log|a_j|. 


+------------------------------------------------------------
| Jensen formula
+------------------------------------------------------------

If f is a subharmonic function in a neighborhood of a point z
in the complex plane, then the limit lim_r to 0 M(f,r)/log(r)
exists and is called the Lelong number of f at z. Here
M(f,r)=sup_|z|=r f(z).


+------------------------------------------------------------
| hyperbolic domain
+------------------------------------------------------------

An open set in the extended complex plane is a hyperbolic domain
if there is a subharmonic function on G that is bounded above and not
constant on any component of G. A domain which is not hyperbolic is
called a parabolic domain. Known facts:

 every bounded region is a hyperbolic domain (take f(z)=Re(z)). 
 an open not connected set is hyperbolic. 
 the complex plane is not a hyperbolic domain



+------------------------------------------------------------
| Perron function
+------------------------------------------------------------

The Perron function for a domain D is defined as the functional
assigning to a continous function g on the boundary of D
the value H_D(g), which is the supremum of all subharmonic functions
u satisfying sup_z to w u(z) leq g(w). 


+------------------------------------------------------------
| potential
+------------------------------------------------------------

A subharmonic function f is called a potential if 
f=p_mu, where mu= Delta f is the Laplacian of f and
p_mu(z) = -int_D log|z-w| dmu(w)/(2pi) is the potential
defined by mu. 


+------------------------------------------------------------
| logarithmic capacity
+------------------------------------------------------------

The logarithmic capacity of a subset E of the complex plane
is defined as c(E) = sup_mu exp(-I(mu)), where I(mu) is the 
potential theoretical energy of mu
and the supremum is taken over all Borel probability measures
mu on C whose support is a compact subset of E. Known facts:

 c(E)=0 if and only if E is polar. 
 a disc of radius r has capacity r
 a line segment of length h has capacity h/4. 
 if K has diameter d, then c(K) leq d/2. 
 if K has area A, then c(A) geq (A/pi)^1/2. 



+------------------------------------------------------------
| mean value property
+------------------------------------------------------------

The mean value property tells that if h is a harmonic function
in the disc D(w,R) and 0<r<R, then 
h(w) = int_0^2pi h(w+r e^it) dt/(2 pi).  


+------------------------------------------------------------
| polar set
+------------------------------------------------------------

A subset S of the complex plane is called a polar set
if the potential theoretical energy I(mu) is
-infinity for every finite Borel measure mu
with compact support  supp(mu) in S. 
Properties of polar sets:

 every countable union of polar sets is polar.
 every polar set has Lebesgue measure zero.



+------------------------------------------------------------
| Poisson integral formula
+------------------------------------------------------------

The Poisson integral formula: if h is harmonic
on the disk D(w,R'), then for all 0<r<R<R' and 0<t<2 pi, 
h(w+r e^it)=int_0^2pi h(w+R e^is) (R^2-r^2)/(R^2-2Rr cos(s-t)+r^2) ds/(2pi)


+------------------------------------------------------------
| potential theoretical energy
+------------------------------------------------------------

The potential theoretical energy I(mu) of a 
finite Borel measure mu of compact support 
on the complex plane is defined as 
 I(mu) = int_C int_C int int log|z-w| dmu(w) dmu(z)  .  


+------------------------------------------------------------
| potential theoretical energy
+------------------------------------------------------------

A function f on an open subset U of the complex 
plane is called subharmonic if it is 
upper semicontinous and satisfies the 
local submean inequality. Examples: 


 if g is holomorphic then 
      f=log|g| is subharmonic
 if mu is a Borel measure of compact
      support, then f(z) = int log|z-w|  dmu(w) 
      is subharmonic. 
 any harmonic function is subharmonic.
 if g is subharmonic, then exp(g) is subharmonic.



+------------------------------------------------------------
| regular
+------------------------------------------------------------

A boundary point w of a domain D is called regular
if there exists a barrier at w. A barrier is a 
subharmonic function f defined in a neighborhood N 
of w which is negative on D cap N and such that
lim_z to w f(z)=0. 
It is known that z is a regular boundary point if and 
only if the complement of D is non-thin at z. 


+------------------------------------------------------------
| irregular
+------------------------------------------------------------

A boundary point w of a domain D is called irregular 
if it is not regular. 
It is known that if z has a neighborhood N such 
that N intersected with the boundary of D is polar, 
then z is irregular.


+------------------------------------------------------------
| irregular
+------------------------------------------------------------

A domain D for which every point is regular is called
a regular domain.
For example, a simply connected domain D such that the 
complement of D in the Riemann sphere contains at least 
two points, is regular. 


+------------------------------------------------------------
| Riemann mapping Theorem
+------------------------------------------------------------

The Riemann mapping Theorem: if D is a simply connected
proper subdomain of the complex plane, there exists a 
conformal map of D onto the unit disc.


+------------------------------------------------------------
| Riesz decomposition theorem
+------------------------------------------------------------

The Riesz decomposition theorem tells that every 
subharmonic function f can be written as f=p_mu + h,
where mu=Delta f is the Laplacian of f, 2 pi p_mu 
is the potential of mu and where h is harmonic. 


+------------------------------------------------------------
| submean inequality
+------------------------------------------------------------

The local submean inequality for a function
in the complex plane tells that there exists R>0 
such that for all 0<r<R one has
 f(w) leq int_0^2pi f(w+r e^it)  dt/(2pi)  . 


+------------------------------------------------------------
| submean inequality
+------------------------------------------------------------

Let f be a subharmonic function on a domain D.
The maximum principle says that if f attains a
global maximum in the interior of D then f is constant.


+------------------------------------------------------------
| thin set
+------------------------------------------------------------

A subset S of the complex plane is called a thin set 
if for all w in the closure of S-w and all 
subharmonic functions f, 
limsup_z to w f(z) = f(w). Examples:


 every single point in the interiour of S is thin.
 F_sigma polar sets S are thin at every point. 
 connected sets of cardinality larger than 1 are 
      non-thin at every point of their closure
 A domain S is thin at a point z in S if and 
      only if z is regular. 



+------------------------------------------------------------
| Wiener criterion
+------------------------------------------------------------

The Wiener criterion gives a necessary and sufficient condition 
for a set S to be thin at a point w. 
Let S be a F_sigma subset of C and let w be in S. 
Let a<1 and define S_n =  z in S, a^n < |z-w| < a^(n-1)  .
The criterion says that S is thin at w if and only if 
sum_n geq 1 n/log(2/c(S_n)) < infinity, where 

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