The Sobolev conjugate of p for , where n is space dimensionality, is

This is an important parameter in the Sobolev inequalities.

Motivation

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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

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References

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  • Lawrence C. Evans. Partial differential equations. Graduate Studies in Mathematics, Vol 19. American Mathematical Society. 1998. ISBN 0-8218-0772-2