Suppose is the fundamental solution of laplace equation.

We know it has property

We can define the Newtonianial potential of

is well defined. We have the following propery of

**Thm:** Suppose is a bounded domain, . Assume is bounded and local integrable in , then is and

**Proof:** Fix , let and .

Since is uniformly bounded in , by the Lebesgue dominating theorem, is differentiable and

Considering , we will prove if

where .

For any , . So

Similarly for , we also have

Considering , we have

Combing the fact about , we get

So

Applying the fact that is differentiable and the fact about

Let , we get exists and

Next we shall prove is continuous for , then

By the dominating theorem

where . So is continuous everywhere.

**Remark:** This revised version also have errors. It is a shame. What do you mean is differentiable? Don’t you notice that the integral domain of also has relation with ?

To prove this theorem rigorously and neatly, we need the following lemma from calculus

**Lemma:** Suppose is a sequence of differntiable function on . If converges to another function uniformly( is finite somewhere) and converges uniformly to in then is differentiable and .

**Proof:** Let and , , .

Define

Let Since

which is in . By the Lebesgue Differentiable theorem, is differntiable and

So converges to uniformly on any compact subset of . And it is easy to prove converges to uniformly on any compact subset of . So by the lemma, we have is differentiable and has