The Extremal Case of Hölder’s Inequality
We will soon need to know that Hölder’s inequality is in a sense the best we can do, at least for finite . That is, not only do we know that for any
and
we have
, but for any
there is some
for which we actually have equality. We will actually prove that
That is, not only is the integral bounded above by — and thus by
— but there actually exists some
in the unit ball which achieves this maximum.
Hölder’s inequality tells us that
so must be at least as big as every element of the given set. If
, then it’s clear that the asserted equality holds, since
a.e., and so
is the only element of the set on the right. Thus from here we can assume
.
We now define a function . At every point
where
we set
as well. At all other
we define
In the case where we will verify that
. That is, the essential supremum of
is
. And, indeed, we find that
at points where
, and
at points where
.
If , then we check
In either case, it’s easy to see that
as asserted.