**Theorem** Suppose is a subsolution of , . Then for all , ,

**Remark 1** Without loss of generality one can assume . Since if is a subsolution, is also a subsolution. Since the result is scaling invariant, one can assume . Further more one can assume , if we have , for any , any , we can take a point such that . Then

Let , we can derive the inequality for any .

**Lemma** Let be a nonnegative bounded function on , if there exist , nonnegative constant such that

for all . Then

*Proof:* Set , for some to be chosen later. Then . So

After iterating we will have

Now we can choose such that . Let ,

**Remark 2** We only need to prove for case , and use the above lemma to recover the cases when . That is because for any , ,

Set and . Apply the above lemma we will have

*Proof of the theorem:* By the above remarks, it suffices to prove the case when . Since

for all . Take , where is a cut off function. Then we have

which implies

By ellipticity, we have

By Holder inequality we will have

Since , we can derive , which implies

By Sobolev inequality, we have

Now let , so and as . Take , , and on . Then . Therefore

Take , Then

Therefore

After iteration, we will have

Since we can take and get

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