Lemma 1 If , (), , then we have
and then (6) is valid. □
Definition 1 If , , , is a measurable function in such that for any and , , then we call the homogeneous function of −λ-degree in .
Lemma 2 Suppose , (), , is a homogeneous function of −λ-degree. If
satisfying , then each and for any ,
Proof Setting () in the integral , we find
Setting in the above integral, we obtain . Setting in (7), since , we find
Setting () and in the above integral, we find . □
Lemma 3 With the assumptions given in Lemma 2, then
is finite in a neighborhood of if any only if is continuous at .
Proof The sufficiency property is obvious. We prove the necessary property of the condition by mathematical induction in the following. For , there exists such that for any , . Since for (),
and , then by the Lebesgue control convergence theorem (cf. ), it follows (). Assuming that for n (≥2), is continuous at , then for , by the result of , since is finite in a neighborhood of , we find
then by the assumption for n, it follows
By mathematical induction, we prove that for , is continuous at . □
Lemma 4 With the assumptions given in Lemma 2, if there exists such that for , , (), , then we have
Proof Setting in (8), we find
Setting () in the above integral, since , we find (replacing by )
by (9), it follows
Without loss of generality, we estimate the case of , e.t.
. In fact, setting , such that , since (), there exists such that (), and then by the Fubini theorem, it follows
Hence by (10), we have
Since by Lemma 3 we find
then combining with (11), we have (8). □
Lemma 5 Suppose that , (), , , , , (≥0) is a measurable function of −λ-degree in such that
If are measurable functions in (), , then (1) for (), we have
(2) for , (), we have the reverse of (12).
Proof (1) For (), by the Hölder inequality (cf. ) and (7), it follows
For , by the Hölder inequality again, it follows
Then by (7), we have (12). (Note: for , we do not use the Hölder inequality again in the above.) (2) For , (), by the reverse Hölder inequality and in the same way, we obtain the reverses of (12). □