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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