Characterizations of Sobolev Spaces
I will present here a few useful characterizations of Sobolev spaces. The source is Functional Analysis, Sobolev Spaces and Partial Differential Equations by Haim Brezis. Consider with . We denote by the conjugate of , i.e. . Then the following properties are equivalent:
 (i) ;
 (ii) there exists a constant such that
;
 (iii) there exists a constant such that for all and all with we have
(note that makes sense for and . Furthermore, we can take in (ii) and (iii). If we have
.
Proof: (i) (ii). Obvious. Write the definition and apply Holder’s inequality.
(ii) (i). The functional
is a continuous linear functional on a dense subspace of , and therefore can be extended to a continuous linear functional on the whole . By Riesz’s representation theorem there exists a function such that
Since this happens for every we find that .
(i) (iii). Assume first that . Let and set
Then we find that and therefore
For we have
and
If is small enough then there exists an open set such that for all and therefore
which concludes the proof for and . But we know that is dense in in the sense that there exists such that in and in for every . This finishes the argument. For apply the above for and make .
(iii) (ii). Let and consider such that . Take with small enough such that . Because of (iii) we have
which translates to
If we choose in the last inequality and make then we obtain (i).
In the case the following implications still hold:
The functions which satisfy (ii) and (iii) in the case when are the functions of bounded variation.

October 27, 2012 at 6:36 pmRegularity of the weak solution Part 1 « Beni Bogoşel's blog