I just read this method in a book due to W.X. Chen and C.M. Li published by American Institute of Mathematical Sciences this year 2010.

Let be a topological vector space. Suppose there are two extended norms (i.e. the norm of an element in might be infinity) defined on

.

Let

and

.

Theorem(Regularity Lifting I). Let be a contraction map from into itself and from into itself. Assume that , and that there exits a function such that in . Then also belongs to .

*Proof*. Firstly, let

be a norm on . We first show that is a contraction. Since is a contraction on , there exists a constant , such that

.

Similarly, we can find a constant , such that

.

Let . Then, for any

.

Since is a contraction, given , we can find a solution such that . Notice that is also a contraction and , the solution of the equation must be unique in . Because both and are solutions of the same equation in , we deduce that .

**Remark**. In practice, we usually choose to be the space of distributions, and and to be function spaces, for instance, and . We start from a function in a lower regularity space , if we can show that is a contraction from to itself and from to itself, then we can lift the regularity of to be in

.

**Applications to PDEs**. Now, we explain how the Regularity Lifting Theorem proved in the previous subsection can be used to boost the regularity of week solutions involving critical exponent

.

Still assume that is a smooth bounded domain in with . Let

be a weak solution of the above PDE. Then by Sobolev embedding

.

We can split the right hand side of the PDE in two parts

.

Then obviously . Hence, more generally, we consider the regularity of the weak solution of the following equation

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Theorem. Assume that . Let be any weak solution of the foregoing PDE. Then for any .

The proof of this theorem can be found in the book mentioned above.

Let us go back to the first PDE. Assume that is a weak solution. From the above theorem, we first conclude that is in for any . Then by a standard regularity result known as the -regularity for a second order uniformly elliptic operator in divergence form, is in . This implies that via Sobolev embedding. Finally, by repeated applications of the Schauder estimates, we derive that .

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