Note that if an estimate of the
form
holds with a constant
independent of
then
.
Such conclusion follows from the substitution of
in place of the
.
Proof
Since
has compact support we
state
Hence,
Consequently,
The
th
term of the product is independent of
,
hence, we
continue
Note that according to the formula (
Holder
inequality
2
),


(HI1)

We apply the last relationship with
and
obtain
Next, we integrate the above relationship with respect to the
:
We apply the formula (
HI1
) to the
integral
with
and
and
obtain
Next, we integrate with respect to the
:
and apply the formula (
HI1
) to the
integral
with
,
and
:
We continue the integration with respect to the variables
,...,
and arrive to the
estimate
The above is the desired estimate for
.
To obtain the estimate for
it is enough to apply the last result to the function
with
.
