Sobolev conjugate

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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 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.

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