Suppose is hölder continuous, where . It is equivalent to say for any ball centered at we have

where is independent from and the radius of the ball.

“necessity” If is hölder continuous, then

here is the radius of .

“sufficiency” Let is the ball centered at with radius . Denote then .

Since ,

Integrating this over , we get

So .

Since , from lebesgue differentiation theorem a.e. Let , we know a.e. .

So when , .

Set , , then

holds almost everywhere.

Since we have ,

Integrating this on , we get

So a.e. So is a hölder continuous.

Fanghua Lin, Qing Han. *Elliptic partial differential equations.*