New limiting variants of the classical reiteration theorem for the K-interpolation method

We establish some reiteration theorems for limiting K-interpolation methods, thereby obtaining new limiting variants of the classical reiteration theorem. An application to the Fourier transform is given.

Recently, Cobos et al. [13] have defined two new scales of limiting K -interpolation spaces A 0,q;K = (A 0 , A 1 ) 0,q;K andĀ 0,q;K = (A 0 , A 1 ) 1,q;K , corresponding to the limiting values θ = 0, 1, without using the extra function b. Namely,Ā 0,q;K andĀ 1,q;K consist of elements f ∈ A 0 +A 1 with the following finite quasi-norms: respectively. The main purpose in the paper [13] was to investigate the connection between these limiting K -interpolation methods and the interpolation over the unit square. Let us mention that these limiting K -interpolation methods were earlier considered in [14] and [15] in the case when the underlying couple (A 0 , A 1 ) is ordered, meaning that A 0 (or A 1 ) is continuously embedded in A 1 (or A 0 ). Henceforth, for the sake of simplicity, we denoteĀ 0,q:K byĀ {0},q andĀ 1,q:K byĀ {1},q . The main goal of the present paper is to obtain limiting variants of the classical reiteration formula (1.1) by characterizing the following limiting reiteration spaces: (Ā θ 0 ,q 0 ,Ā θ 1 ,q 1 ) {0},q and (Ā θ 0 ,q 0 ,Ā θ 1 ,q 1 ) {1},q .
The key ingredients of our proofs will be the two-sided Hardy-type inequalities involving power-type weights. These inequalities are derived in Section 2. The main results are contained in Section 3, whereas an application to the Fourier transform is given in Section 4.
Throughout the paper, we will write A B or B A for two non-negative quantities A and B to mean that A ≤ cB for some positive constant c which is independent of appropriate parameters involved in A and B. We put A ≈ B if A B and A B.

Weighted Hardy-type inequalities
To prove our main results, we shall need suitable two-sided Hardy-type inequalities involving power-type weights. We derive them from the following general weighted Hardytype inequalities.
hold for all non-negative functions h on (0, ∞).
The next assertion deals with the general weighted Hardy-type inequality restricted to non-increasing functions.
holds for all non-negative and non-increasing functions h on (a, ∞).

Remark 2.3
The previous assertion is proved in [6] for a = 0, but the same proof also works for all 0 < a < ∞.
Next, using the previous two theorems, we obtain the needed two-sided Hardy-type inequalities.
holds for all non-negative and non-increasing functions h on (1, ∞).
Proof The estimate " " is a simple consequence of the fact that h is non-increasing. For 1 ≤ s < ∞, the converse estimate " " follows from Theorem 2.1, applied with . When 0 < s < 1, the desired converse estimate results from Theorem 2.2, applied with a = 1, The proof is complete.
holds for all non-negative and non-decreasing functions g on (0, 1).
Proof The proof follows by applying Corollary 2.4 to the non-increasing function h(t) = g(1/t) on (1, ∞).

Limiting reiteration theorems
In this section, we establish our main results. First we need to introduce K -interpolation spaces of type L and R. Namely, let v and w be non-negative and locally integrable functions on (0, ∞), and let 0 < p, q ≤ ∞. Then the K -interpolation spacesĀ L w,p;v,q = (A 0 , A 1 ) L w,p;v,q andĀ R w,p;v,q = (A 0 , A 1 ) R w,p;v,q consist of elements f ∈ A 0 + A 1 with the following finite quasi-norms: respectively. For details onĀ L w,p;v,q andĀ R w,p;v,q , the reader is referred to [6] where slightly different notations are used for these spaces. The spaces L and R, with weights involving logarithmic functions or, more generally, slowly varying functions, have also appeared in [5,8,16]. It is also worthy of mention that the spaces L and R have been considered in [10][11][12] in the more general framework of rearrangement invariant spaces.
Proof LetĀ = (Ā θ 0 ,q 0 ,Ā θ 1 ,q 1 ) {0},q , and take f ∈ A 0 + A 1 . According to [4, Theorem 2.1], we have the following formula for the classical K -interpolation method: Therefore, it turns out that where Clearly, Next, let us estimate each of I 2 and I 4 . We observe the simple fact that t → K(t, f ) is nondecreasing and apply Corollary 2.5, with s = q/q 1 , α = θ 1θ 0 , β = θ 1 q 1 , and g(t) = K q 1 (t, f ), to get Moreover, noting the fact that t → t -1 K(t, f ) is non-increasing, we have whence, in view of (3.3), we get I 1 I 2 . Therefore, Now we estimate I 4 . Observe that In addition, we have as t → K(t, f ) is non-decreasing. Combining the previous two estimates, we arrive at This, along with (3.2), leads us to Now inserting estimates (3.4) and (3.5) in (3.1) yields which finishes the proof.
Proof In view of the following elementary identity: the symmetry property (A 0 , A 1 ) {0},q = (A 1 , A 0 ) {1},q holds. Together with the well-known symmetry property for the scaleĀ θ,q , this gives Now applying Theorem 3.1 yields which completes the proof.
The next two results provide limiting variants of the reiteration formula (1.2) corresponding to the limiting values θ = 0, 1.
Using the same symmetry argument as in the proof of Theorem 3.2, we can derive the following limiting variants of the reiteration formula (1.3) from the previous two theorems.

An application
The Fourier transform F of a function f ∈ L 1 (R n ) is defined as The next result provides an application of Theorem 3.4 to the mapping properties of the Fourier transform. For related results, the reader is referred to the papers [17,18], and [5] and a recent PhD dissertation [19]. As usual, let f * denote the non-increasing rearrangement (see, for instance, [3]) of f . Put f * * (t) = t 0 f * (u) du, t > 0.
Theorem 4.1 Let 0 < q < ∞, and set Then F is bounded as a map from E to F.
Proof We simply put L 1 = L 1 (R n ) and L ∞ = L ∞ (R n ). It is well known that F is bounded as a map from L 1 to L ∞ and also as a map from L 2 to L 2 . Therefore, by the interpolation property of the K -interpolation methodĀ {1},q (see [13,Proposition 3.2]), F is bounded as a map from (L 2 , L 1 ) {1},q to (L 2 , L ∞ ) {1},q . Thus, the proof is complete if we show that (L 2 , L 1 ) {1},q = E and (L 2 , L ∞ ) {1},q = F. To this end, we observe that (L 1 , L ∞ ) 1/2,2 = L 2 and (L ∞ , L 1 ) 1/2,2 = L 2 , and apply Theorem 3.4 to obtain that