The Sobolev conjugate of p for
, where n is space dimensionality, is This is an important parameter in the Sobolev inequalities.
A question arises whether u from the Sobolev space
belongs to
It is easy to check that the following inequality can not be true for arbitrary q.
, infinitely differentiable function with compact support.
λ
( x ) := u ( λ x )
λ
results in the following inequality for
λ
going to zero or infinity we obtain a contradiction.
Thus the inequality (*) could only be true for which is the Sobolev conjugate.