Reiteration Theorems for Two-Parameter Limiting Real Interpolation Methods

We introduce a limiting real interpolation method involving two scalar parameters. We derive Holmstedt-type estimates for this method that are applied to establish the reiteration theorems.

The motivation for introducing the two-parameter limiting spaces ( 0 ,  1 ) {,} mainly stems from the fact the sum of the limiting spaces ( 0 ,  1 ) 0,; and ( 0 ,  1 ) 1,; , introduced by Cobos et al. [11] in connection with the interpolation over the unit square, is precisely ( 0 ,  1 ) , .This fact is established in [9,Proposition 3.4] for  = , and the same argument also works for arbitrary values of  and .We further note that two different parameters have already been used in defining certain -limiting spaces (see [12,Definition 3.1]).
The main goal of this paper is to characterize the reiteration spaces Moreover, the assertions of [10,Theorem 4.3] have been extended by identifying the spaces ( { 0 , 0 } ,  1 ) {,} and The classical identities ( 5)-( 6) are based on the estimates which relate the -functionals of the interpolated couples (  0 , 0 ,   1 , 1 ), (  0 , 0 ,  1 ), and ( 0 ,   1 , 1 ) with that of the original couple ( 0 ,  1 ) (see [13]).The main ingredient of our proofs will be the similar estimates for the limiting spaces ( 0 ,  1 ) {,} .These estimates are derived in the next section as corollaries of more general Holmstedt-type estimates.Some Hardy-type inequalities, along with two other useful results, are given in Section 3. Finally, the reiteration theorems are established in Section 4.

Holmstedt-Type Estimates of the 𝐾-Functional
Let  be a positive weight on (0, ∞), that is, a positive locally integrable function on (0, ∞), and let 0 < ,  < ∞.Then, by  ,, we will mean the real interpolation space  Φ ,, , where Φ ,, has the quasi-norm Note that  ,, = ( 0 ,  1 ) {,} for In this section we present Holmstedt-type estimates for the real interpolation spaces  ,, , and we omit the proofs as they can be done as in [8,Section 2], where these estimates have been obtained for the case  = .
First, we formulate the results for the case (  0 , 0 , 0 ,  1 ).For this purpose we introduce some notations: Subsequently, we will use the notation  1 ≲  2 for nonnegative quantities to mean that  1 ≤  2 for some positive constant  which is independent of appropriate parameters involved in  1 and  2 .If  1 ≲  2 and  2 ≲  1 , we will put  1 ≈  2 .
Then, for all 0 <  < 1, and, for all  ≥ 1, Proof.This time we apply Theorem 2 to the weight  1 given by We see that Therefore, (30) and ( 31) are consequences of ( 17) and ( 16), respectively.

Auxiliary Results
In order to prove our main results in the next section, we need certain Hardy-type inequalities.We will derive them by verifying the sufficient conditions, for particular weights, for the general weighted Hardy-type inequalities.For the next two results we refer the reader to [14, Section 1].For 1 <  < ∞, put   = /(1 − ).holds for all nonnegative functions  on (0, 1) if and only if holds for all 0 <  < 1.

Then the inequality
holds for all nonnegative nondecreasing functions  on (0, 1) if and only if holds for all 0 <  < 1.
Proof.By interchanging the order of integration, we get the result for  = 1.For 1 <  < ∞, the estimate "≲" results from Lemma 8, applied with () = ℎ()/, ( The other estimate "≳" follows from and the fact that 1 + ln  and ln  are asymptotically the same as  → ∞.The proof is finished. holds for all nonincreasing nonnegative functions  on (1, ∞).
In order to facilitate certain change of variables in the first two theorems of the next section, we will make use of the next two lemmas concerning slowly varying functions.Here we say that a positive Lebesgue-measurable function  is slowly varying on (1, ∞) if, for all  > 0, the function   →   () is equivalent to a nondecreasing function and   →  − () is equivalent to a nonincreasing function.By symmetry, we say that  is slowly varying on (0, 1) if the function   → (1/) is slowly varying on (1, ∞).Finally,  is slowly varying on (0, ∞) if it is slowly varying on both (0, 1) and (1, ∞).For example,   → (1+| ln |)  is slowly varying on (0, ∞) for every real number .We refer to [16] for details on slowly varying functions.
Proof.Let  * be a nondecreasing function such that  /2 () ≈  * ().Set where Similarly, we can establish the following lemma.

Reiteration Theorems
Finally, we derive the reiteration theorems for our twoparameter limiting spaces  {,} by using the results of the previous two sections.
Theorem 16.Let 0 <  0 ,  0 , ,  < ∞.Then one has with equivalent norms where Proof. Put By Corollary 4, where We note that is nonincreasing since it is an integral average (with respect to the measure   0 −1 ) of a nonincreasing function   0 (, )/  0 .Consequently, which gives  21 ≲  11 .Next we apply Corollary 10 to the nondecreasing function ℎ() =   0 (, ) to conclude that  11 ≲  12 and apply Corollary 13 to the nonincreasing function () = ((, )/)  0 to find that Altogether, it follows that On the other hand, in (66), we replace (1 − ln ) 1/ 0 by an equivalent function  on (0, 1) (as obtained by Lemma 14) and make change of variable  = () to find that And making change of variable  = (1 + ln ) 1/ 0 in (67), we obtain Combining the previous two estimates, we achieve which completes the proof in view of (73).
Theorem 18.Let 0 <  1 ,  1 , ,  < ∞.Then we have with equivalent norms where Proof.This time, putting we obtain, by Corollary 5, that where Next we make change of variables  = (1 − ln ) −1/ 1 in (79) to obtain that As for change of variables in (80), we replace (1 + ln ) −1/ 1 by an equivalent function  (obtained this time by Lemma 15) and put  = () to get that Hence, which, along with (85), completes the proof.