**Thm:** Suppose is the upper half plane. is non-negative harmonic function in and on . Then for some . Note that there is no requirement on the growth of at infinity.

**Lemma:** Let , be positive solutions to in continuously vanishing on with

Then, in ,

and

**Remark:** Refer to Caffarelli’s book, *A Geometric Approach to Free Boundary Problems*, this lemma is related to boundary harnack inequality or Carleson estimate.

**Proof:** For , denote , then . Fix , define , for . Apply the lemma, we can get

where . These are equivalent to

Suppose

(i) If , then , large enough such that implies

This means in . Hence on .

(ii) If , then , large enough such that implies that

Since are arbitrary, then

Define

Then is non-negative and harmonic in , however

Applying the above lemma and repeating the above procedure, we get

So we construct

Then is non-negative and harmonic in with . Inductively, we construct

Then is non-negative and harmonic in with

From (i), we know , that is .

**Remark: **This problem was presented to me by Tianling Jin. As he said, there should have a very elementary proof, which do not require such advanced theorem.