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 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.
See also Sergei Lvovich Sobolev conjugate index
References Lawrence C. Evans. Partial differential equations. Graduate Studies in Mathematics, Vol 19. American Mathematical Society. 1998.