The Sobolev conjugate of p for , where n is space dimensionality, is
This is an important parameter in the Sobolev inequalities

Sobolev inequality

In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the Rellich–Kondrachov theorem showing that under...

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Motivation

A question arises whether u from the Sobolev space

Sobolev space

In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space...

belongs to for some q>p. More specifically, when does control ? It is easy to check that
the following inequality (*)
can not be true for arbitrary q. Consider , infinitely differentiable function with compact support. Introduce . We have that
The inequality (*) for results in the following inequality for
If , then by letting going to zero or infinity we obtain a contradiction. Thus the inequality (*) could only be true for,
which is the Sobolev conjugate.