We deal only with the elliptic case assuming that .
Lemma. Let . Then on ,
Proof. Set . Then we have
where , are certain constants, and
Take an and a such that in and outside and set
where is obtained in
Note that and are well-deifned since have compact supports and . Recalling some facts on Newtonian potential, we have
Note also that are infinitely differentiable and
So for any , we have
We estimate the integral oscillation of . Since in , we see that if , then
and for , , we have .
where is the surface measure on .
We write in polar coordinate. Then by using previous identity
By using mean value theorem, we obtain
Note that this estmiate allows shifting the origin. For this reason for any and any ball such that , we have
By taking the supremum of the left-hand side over all balls containing , we obtain the desired result.
The following result can be obtained by Calderon-Zygmund theory of singular integral by using the -boundedness of Riesz transform. However, by using Fefferman-Stein theorem, we can obtain it elegantly without using kernel method.
Theorem. Let , and . Then
We first prove the case . For the case , it can be obtained by using Fourier transform. For , by the Fefferman-Stein theorem and previous lemma and the -boundedness of Hardy-Littlewood maximal function, we obtain
For , we use duality argument. Assume that . Then integration by parts and Holder’s inequality give
Since is dense in ,
by density, we conclude that for any ,
So by taking supremum to with , we obtain