Problem.If , then we have

This is the Ladyzhenskaya’s inequality.

To prove this, we need the following lemma.

Lemma.For , we have

Here .

*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

and

For temporary set , . Then

Since , we have

because .

Thus,

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

for .

Hence,

So

**Reference**

1. Elias M. Stein, *Singular Integrals and Differentiability Properties of Functions*, Princeton University Press, 1971.

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