There is some gap, in fact I can improve half of the argument of Discrete harmonic function , the pdf version is Discrete harmonic function in Z^n, but I still have some gap to deal with the residue half…

**1. The statement of result **

First of all, we give the definition of discrete harmonic function.

Definition 1 (Discrete harmonic function)We say a function is a discrete harmonic function on if and only if for any , we have:

In dimension 2, the definition reduce to:

Definition 2 (Discrete harmonic function in )We say a function is a discrete harmonic function on if and only if for any , we have:

The result establish in \cite{paper} is following:

Theorem 3 (Liouville theorem for discrete harmonic functions in )Given . There exists a constant related to such that, given a discrete harmonic function in satisfied for any ball with radius , there is portion of points satisfied . then is a constant function in .

Remark 1This type of result contradict to the intuition, at least there is no such result in . For example. the existence of poisson kernel and the example given in \cite{paper} explain the issue.

Remark 2There are reasons to explain why there could not have a result in but in ,

- The first reason is due to every radius there is only lattices in in so the mass could not concentrate very much in this setting.
- The second one is due to there do not have infinite scale in but in .
- The third one is the function in is automatically locally integrable.

The generation is following:

Theorem 4 (Liouville theorem for discrete harmonic functions in )Given . There exists a constant related to such that, given a discrete harmonic function in satisfied for any ball with radius , there is portion of points satisfied . then is a constant function in .

In this note, I give a proof of 4, and explicit calculate a constant satisfied the condition in 3, this way could also calculate a constant satisfied 4. and point the constant calculate in this way is not optimal both in high dimension and 2 dimension.

**2. some element properties with discrete harmonic function **

We warm up with some naive property with discrete harmonic function. The behaviour of bad points could be controlled, just by isoperimetric inequality and maximum principle we have following result.

Definition 5 (Bad points)We divide points of into good part and bad part, good part is combine by all point such that , and is the residue one. So .For all , we define for convenient.

Theorem 6 (The distribution of bad points)For all bad points in , they will divide into several connected part, i.e.

and every part satisfied .

Remark 3We say is connected in iff there is a path in connected .

Remark 4the meaning that every point So the behaviour of bad points are just like a tree structure given in the gragh.

*Proof:* A very naive observation is that for all is a connected compact domain, then there is a function

This could be proved by induction on the diameter if . Then, if there is a connected component of such that contradict to theorem 6 for simplify assume the connected component is just , then use the formula 5we know

The last line is due to consider around . But this lead to: which is contradict to the definition of . So we get the proof.

Now we begin another observation, that is the freedom of extension of discrete harmonic function in is limited.

Theorem 7we can say something about the structure of harmonic function space of , the cube, you will see, if add one value, then you get every value, i.e. we know the generation space of

*Proof:* For two dimension case, the proof is directly induce by the graph. The case of dimensional is similar.

Remark 5The generation space is well controlled. In fact is just like n orthogonal direction line in n dimensional case.

**3. sktech of the proof for \ref **

}

The proof is following, by looking at the following two different lemmas establish by two different ways, and get a contradiction.

\paragraph{First lemma}

Lemma 8 (Discrete poisson kernel)the poisson kernel in . We point out there is a discrete poisson kernel in , this is given by:

And the following properties is true:

- , .

Remark 6The proof could establish by central limit theorem, brown motion, see the material in the book of Stein \cite{stein}. The key point why this lemma 8 will be useful for the proof is due to this identity always true , So we will gain a lots of identity, These identity carry information which is contract by another argument.

\paragraph{Second lemma} The exponent decrease of mass.

Lemma 9The mass decrease at least for exponent rate.

Remark 7the proof reduce to a random walk result and a careful look at level set, reduce to the worst case by brunn-minkowski inequality or isoperimetry inequality.

\paragraph{Final argument} By looking at lemma 1 and lemma 2, we will get a contradiction by following way, first the value of on increasing too fast, exponent increasing by lemma2, but on the other hand, it lie in the integral expresion involve with poisson kernel, but the pertubation of poisson kernel is slow, polynomial rate in fact…

\newpage

{99} \bibitem{paper} A DISCRETE HARMONIC FUNCTION BOUNDED ON A LARGE PORTION OF Z2 IS CONSTANT

\bibitem{stein} Functional analysis