This entry devotes the following fundamental question: if is Hölder continuous, then how about for some constant ? Throughout this entry, we work on which is not necessarily bounded.

Firstly, we have an elementary result

Proposition. If and are -Hölder continuous and bounded, so is .

*Proof*. The proof is simple, we just observe that

which yields

.

Consequently,

for any positive integer number and any -Hölder continuous and bounded function , function is also -Hölder continuous and bounded.

Let us assume , is -Hölder continuous and bounded, is a constant. Let . Since , we may assume is also bounded away from zero, that means there exist two constants such that

.

We now study the -Hölder continuity of . Observe that function

is sub-additive in the sense that

.

Indeed, by dividing both sides by we arrive at

.

Since

the above inequality comes from the Bernoulli inequality. Therefore if there are some then

holds for any . We now arrive at

.

We now make use the above estimate to show that is -Hölder continuous. Indeed, we have the following

.

Consequently, we get

.

Thus, from

we claim that is -Hölder continuous.

Let us now turn to the case . For the sake of simplicity, let us denote , i.e. the function under investigating is . Obviously

.

Thus is -Hölder continuous since

for any . Consequently,

for any positive integer number and any -Hölder continuous and bounded function , function is -Hölder continuous.

Similarly, we can prove

.

Thus

is -Hölder continuous. It is now easy to prove that is also -Hölder continuous. So far we have proved the following

Theorem. If is -Hölder continuous, positive and bounded (i.e. ). We assume in addition is bounded away from zero (i.e. ). Then is also -Hölder continuous for any constant .

The boundedness of plays an important role in our argument. In practice, if strictly positive function is locally -Hölder continuous on , we are the able to apply the theorem. We will discuss something related to this application later.

Hi Anh,

I found a Holder continuity estimate which is potentially useful for me.

However, I have problem finding the book talking about it. Do you happen to know any relevant reference?

Thanks,

Chao

Comment by Chao — August 18, 2010 @ 16:53

Hello Chao,

Where did you find such an estimate?

Comment by Ngô Quốc Anh — August 18, 2010 @ 17:50

Hi Anh,

The book is

“Aspects of Sobolev Type Inequalities”, by Laurent Saloff-Coste.

I found the inequality is actually proved in the book. I will try to understand it. I am cs background. So reading math textbook is always a pain.

Will keep you updated.

Chao

Comment by Chao — August 18, 2010 @ 19:31

Sorry the formula should be

Comment by Chao — August 18, 2010 @ 16:53

I realize this might be a combination of Holder inequality and Soblev inequality, in the case when p>n, and n the dimension of the Euclidean space.

Comment by Chao — August 18, 2010 @ 17:23

Hi Chao,

I just go through the proof on page 24 of the book. So what is your problem here? The proof is quite clear. What you need is prove the following

.

Comment by Ngô Quốc Anh — August 18, 2010 @ 19:52

Hi Anh,

Let me think more before asking questions. I will come back later.

Chao

Comment by Chao — August 18, 2010 @ 20:24