# 1.1. Newtonian Potentials

By | January 16, 2021

The goal of this note is to study solvability of second-order elliptic and parabolic equations in Hölder spaces. The prototype of elliptic equation is the Poisson equation

To understand the property of solution , one of the easiest ways is to use Newtonian potential. In Section 1.1, we derive Newtonian potential and study some basic properties on this object. Also, we study a maximum principle for Poisson equation in Section 1.2. Such property plays a crucial role when we study second-order elliptic and parabolic equations.  Next, we introduce Hölder spaces which illustrate some smoothness of solutions in Section 1.3. After introducing Hölder spaces, in Section 1.4, we estimate a Hessian of Newtonian potential and a Hessian of solution for Poisson equation in Hölder space. This will lead us to think our main object of this note, so called Schauder estimate

To study properties of solution of the Poisson equation, we first seek a solution satisfying

A natural candidate for such solution is to assume that

i.e. radial solution. Write . In spherical coordinate, the Laplacian is written as

where is the spherical Laplacian (or Laplace-Beltrami operator on unit sphere). Since we seek a radial solution, this implies that should satisfy

Suppose that . Then

and hence for some constant . Hence if , then

where and are constants. This motivates us to introduce following one.

Definition 1.1. The function

defined for , is the fundamental solution of Laplacian. Here  denotes the volume of the unit ball in

We wil use the following computation:

and

From this, we see that  is harmonic in
For and , we define the convolution of and by

Theorem 1.1. Let  and define

Then

Proof. For simplicity, we prove the theorem when . The case can be similarly proved. We first show that is well-defined.

Since , there exists such that . A change of variable into polar coordinate gives

Hence is well-defined.
Fix . For , we have

since . So

By mean value property, it follows that

Since

it follows from dominated convergence theorem that

By induction, one can show that . It remains to show that
Fix . Then

Then

Since has compact support, integration by part gives

A change of variable gives that

Integration by part again gives

From this calculation, we have

(1)

Since is harmonic in , it follows that

Since

it follows that

Since is continuous and , it follows that

Hence letting in (1), we conclude that

This completes the proof.

Definition 1.2. Let . The function defined by

is called the Newtonian potential of .

Remark. Here we obtain another calculation related to some fundamental studies on Poisson equation. Following the above argument, we have

For simplicity, we set . Choose a radial function such that near at the origin. Then

where

We write

Note that

Since has compact support, integration by part gives

For any , we have

A change of variable gives

Hence it suffices to show that

If , the integral is zero since is antisymmetric. If , then a change of variable and harmonicity of give

for . Integration by part gives

If , the quantity is zero due to antisymmetric. If , then a change of coordinate and harmonicity of give

Letting , we conclude that

This shows that

(2)

for .
Now define

Note that the kernel satisfies

We call such kernel as Calderón-Zygmund singular kernel. Then  it follows from the well-known theory on singular integral operator that for , there exists a constant such that

for all . Hence by (2), we conclude that

We call such estimate as Calderón-Zygmund type estimate. This estimate plays a crucial roles when we study the Sobolev space theory for partial differential equations. Due to our purpose of this note, we will not go to any futher. We end this section by introducing several textbooks. When one has some interests on general theory on singular integrals, see Stein (1970). Modern theory of this theory can be found in Stein (1993). An application for Calderón-Zygmund estimate can be found in Gilbarg-Trudinger (1998) and Krylov (2008)

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