# Sobolev conjugate

Updated on
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The Sobolev conjugate of p for 1 p < n , where n is space dimensionality, is

p = p n n p > p

This is an important parameter in the Sobolev inequalities.

## Motivation

A question arises whether u from the Sobolev space W 1 , p ( R n ) belongs to L q ( R n ) for some q>p. More specifically, when does D u L p ( R n ) control u L q ( R n ) ? It is easy to check that the following inequality

u L q ( R n ) C ( p , q ) D u L p ( R n ) (*)

can not be true for arbitrary q. Consider u ( x ) C c ( R n ) , infinitely differentiable function with compact support. Introduce u λ ( x ) := u ( λ x ) . We have that

u λ L q ( R n ) q = R n | u ( λ x ) | q d x = 1 λ n R n | u ( y ) | q d y = λ n u L q ( R n ) q D u λ L p ( R n ) p = R n | λ D u ( λ x ) | p d x = λ p λ n R n | D u ( y ) | p d y = λ p n D u L p ( R n ) p

The inequality (*) for u λ results in the following inequality for u

u L q ( R n ) λ 1 n / p + n / q C ( p , q ) D u L p ( R n )

If 1 n / p + n / q 0 , then by letting λ going to zero or infinity we obtain a contradiction. Thus the inequality (*) could only be true for

q = p n n p ,

which is the Sobolev conjugate.

## References

Sobolev conjugate Wikipedia

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