Sobolev conjugate
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The Sobolev conjugate of 
is
This is an important parameter in the Sobolev inequalities.
[edit] Motivation
A question arises whether u from the Sobolev space W1,p(Rn) belongs to Lq(Rn) 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 uλ(x): = u(λx). We have that
The inequality (*) for uλ results in the following inequality for u
If 
, then by letting λ going to zero or infinity we obtain a contradiction. Thus the inequality (*) could only be true for
,
which is the Sobolev conjugate.
