Abstract
Explicit formulae for the -functional for the general couple , where is a compatible couple of quasi-normed spaces, are proved. As a consequence, the corresponding reiteration theorems are derived.
1. Introduction
In recent papers [1–11] the classical Holmstedt formula for the -functional [12] was extended to more general cases. See also [13] for more results about generalized Holmstedt’s formula and reiteration theorems not only for the -method, but for the -method as well. Here we consider the -method in the most general case for quasi-normed spaces. Namely, let be a compatible couple of quasi-normed spaces, that is, both and are linearly and continuously embedded in some Hausdorff topological vector space. By definition, the -interpolation space has a quasi-normwhere is the -functional of Peetre [13, 14], defined for , as follows:and is a quasi-normed space of Lebesgue measurable functions, defined on , with monotone quasi norm as follows: implies such that If , that is,we write instead of ; if , the above integral has to be replaced by . Here is a nonnegative Lebesgue measurable function defined on and called weight.
In [1, 2] the case when , , and -repeated logarithms was considered and this was extended to -slowly varying in [3]. Certain limiting cases (when ) are treated in [5, 8–10]. The case with arbitrary was investigated in [6]. In [4, 7, 11] the case when and is rearrangement invariant Banach space on with the Haar measure is treated in detail. Note that in the case , , -slowly varying, the - and -methods are equivalent, but this is not true in the limiting cases or . The problem of the relation between both real interpolation methods for Banach couples is treated in [13], where general theorems are proven and certain applications are given. More results about the relation between - and -methods in the limiting cases are obtained in [15, 16].
We use the notations or for nonnegative functions or functionals to mean that the quotient is bounded; also, means that and We say that is equivalent to if Recall that the weight is called slowly varying if for every the function is equivalent to a nondecreasing one and the function is equivalent to a nonincreasing one.
The main results are announced in [17].
2. Formulae for the -Functional
Using the Holmstedt argument, we can prove general formulae for the -functional. Let and let
Theorem 1 (case ). If , thenwhere .
Proof. Let , , and . Using monotonicity of , we getLet , , and ThenwhenceAlso,HenceNow we estimate from above. To this end, following Holmstedt, we choose decomposition such that ThenFurther,Denote by the first term on the right-hand side and by the second one. We have whenceAlso,Thus
Remark 2. If the spaces and are too close, then formula (5) might be useless. For example, if , then In applications we require to be increasing (strictly).
Let
Theorem 3 (case ). If , thenwhere .
Proof. Let , , and Using monotonicity of , we getLet , , and ThenwhenceAlso,HenceNow we estimate from above using decomposition such that (12) are satisfied. Then We havewhenceHence
Formulae for the couple are more complicated. We need the following condition that in some sense require these two spaces to be not too close. Let
Theorem 4 (case ). Let , and let
Case 1. IfthenCase 2. IfthenCase 3. IfthenCase 4. If at least one of the conditions , , , , , , , , is satisfied, then
Proof. From (6) and (20) it follows thatLet , , and Then, using estimates (7) and (21), we have whenceAlso,Hence,To estimate from above we use the same decomposition as before with properties (12). ThenwhenceAlso,whenceTherefore for all weights we have
We give examples that are not entirely covered by the previous papers.
Example 5 (distant spaces). Letwhere , , , and are slowly varying weights and . We call these spaces distant if or , in opposition to the case and We haveThese integrals are convergent due to the property , Moreover,Also,In particular,On the other hand,for andfor Hence,and thereforewhere , , .
Also,where where
Moreover, if and , then and ; hence, in (56), (57), and (58), we can drop the last term (this is the case treated in [3]).
Example 6 (nondistant spaces). Let be as in Example 5 but with , , and , and , .
IfthenAlso,In particular,On the other hand, since , we haveand thereforewhere
3. Reiteration
The formulae for the -functional imply immediately theorems of reiteration or stability of the -method. In particular, we recover many classical results. For another type of general reiteration theorems see [13]. For example, Theorems 1 and 3 imply the following results.
Theorem 7. Let be a compatible couple of quasi-normed spaces. Thenwhereand is the same as in Theorem 1 and , , and ; is increasing.
Proof. We only need to check that Sinceand , therefore ; we see that the above quantity is finite. AlsoFurther, for , we have and ; hence . Therefore
Theorem 8. Let be a compatible couple of quasi-normed spaces. Then whereand is the same as in Theorem 3 and , , and ; is increasing.
Proof. We only need to check that We haveand, using also ,As above, we check that and for Hence for Then
In some cases the quasi norm of can be simplified.
Example 9 (distant spaces). Letwhere and , are slowly varying weights and letwhere and , are slowly varying weights. ThenwhereIndeed, we havefor andfor , whereLet Then where We can choose equivalent so that (see [6]); henceIt is sufficient to prove thatTo estimate for , we use monotonicity of the interpolation scale, while for we apply Hölder’s inequality. Namely, if ,and if , thenNext we check only the inequality for and the inequality for being similar. For , it follows from Fubini’s theorem. can be estimated from above if by the Muckenhoupt result [18].
Let , where , , and Then We apply this for ,Then and henceFurther, if we estimate from above using integration by parts. We haveIndeed, let Using that is equivalent to a decreasing function, we haveand since ,The integral above has a limit zero when Thus (93) is true. Further,orSince , we get (86).
Now we consider examples of nondistant spaces: , -slowly varying, , .
In order to handle this case, we need the following result.
Lemma 10. One has , and , if the following conditions are satisfied:where
Proof. The estimate (98) from above is a consequence of Lemmas , in [6]. Using (99), we get an estimate from below with Now condition (100) ensures the equivalence in (98).
Theorem 11. Let be a slowly varying weight and let , , satisfying , Ifthenwhere
Proof. We havehenceWe apply Lemma 10 with , , , and Condition (99) follows from monotonicity of . We have to check (100). To this end first we simplifyUsing also (102), we get Then (100) means that But this follows from the definition of .
For example, condition (102) is true if , , -slowly varying.
Competing Interests
The authors declare that they have no competing interests.