## Compensated compactness

Suppose is a vector field and . and is a scalar function. We have following theorem(Coifman-Lions-Meyers-Semmes)

**Theorem: **If and , then , which is the hardy space.

Given , it has harmonic extension

**Definition:** the non-tangential maximal function

It is easy to prove that the Hardy-Littlewood maximal function. From this we can Hardy norm as

Hardy space consists of all having finite hardy norm. There is well know fact that the dual space of is BMO, which is defined as the following.

Define , if for any cube ,

then . and but not in .

Let us see how do we use the main theorem. Suppose on , is the solution of the following elliptic equation

where , and is uniform elliptic. YanYan Li and Sagun Chanillo proved that the green function of this elliptic operator belongs to BMO. The right hand side of this equation can be rewritten as , where

therefore the right hand side belong to . Since

therefore from the theorem we stated at the beginning, we get

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