Theorem 3.1 Let p and q be conjugate parameters with , and let , , , and , and , satisfy and . Then
(8)
and
(9)
where the constant factors and are the best possible and is defined in Lemma 2.2. Inequalities (8) and (9) are equivalent.
Proof If (7) takes the form of the equality for a , then there exist constants A and B such that they are not all zero, and
Hence, there exists a constant K such that
We suppose (otherwise ). Then a.e. in , which contradicts the fact that . Hence, (7) takes the form of a strict inequality, so does (6), and we have (9).
By Hölder’s inequality [8], we have
(10)
By (9), we have (8). On the other hand, suppose that (8) is valid. Set
then it follows . By (6), we have . If , then (9) is obviously valid. If , then by (8), we obtain
and
Hence, we have (9), which is equivalent to (8).
For , define functions , as follows:
Then
and
where
and
Taking , by the Fubini theorem, we obtain
and
In view of the above results, if the constant factor in (8) is not the best possible, then there exists a positive number with such that
(11)
By the Fatou lemma and (11), we have
which contradicts the fact that . Hence, the constant factor in (8) is the best possible.
If the constant factor in (9) is not the best possible, then by (10), we may get a contradiction that the constant factor in (8) is not the best possible. Thus the theorem is proved. □
Remark 1 Setting in Theorem 3.1, we have (3) and (4).
Remark 2 Setting in Theorem 3.1, we have the following particular results:
and