Sobolev conjugate
The Sobolev conjugate of p for, where n is space dimensionality, is
This is an important parameter in the Sobolev inequalities.
Motivation
A question arises whether u from the Sobolev space belongs to for some q > p. More specifically, when does control ? It is easy to check that the following inequalitycan 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.