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.
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.
