Problem. If , then we have
This is the Ladyzhenskaya’s inequality.
To prove this, we need the following lemma.
Lemma. For , we have
Proof. e proceed by induction on . The case is trivial since and this leads to
We assume that the inequality holds for . Write with , . Set
For temporary set , . Then
Since , we have
by Hölder inequality. So
By integrating this with respect to and multiple Hölder inequality, we get
Since , we get
Proof of the Ladyzhenskaya’s inequality. For , the result of Lemma is
Put instead of . Then we have
Note that for we have
By Cauchy-Schwarz inequality, we have
1. Elias M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1971.