Interpolation of Generalized Gamma Spaces in a Critical Case

We establish some interpolation formulae for generalized gamma spaces with double weights in a critical case. Our approach is based on identifying generalized gamma spaces as appropriate K-interpolation spaces with general weights and then applying the reiteration technique for K-interpolation spaces.


Introduction
The scale of generalized gamma spaces with double weights (see Sect. 2 for definitions) was introduced in [18] in order to characterize the following real interpolation spaces (L p),α , L (q,β ) θ,r between grand Lebesgue spaces L p),α (with α = 1) and small Lebesgue spaces L (q,β (with β = 1) in the critical case p = q. Later on, it turned out (see [2,4]) that the following real interpolation spaces (with appropriate conditions on α and β) (L p) , L q ) θ,r , (L p , L (q ) θ,r , (L p),α , L q),β ) θ,r , (L ( p,α , L (q,β ) θ,r also coincided with appropriate G -spaces in the critical case p = q. Thus, it becomes imperative to investigate the interpolation properties of G -spaces themselves in the critical case. The aim of the present paper is to pursue this goal. The main finding of our investigation is this: in our special critical case, the scale of G -spaces remains stable under real interpolation method. We emphasize that this is not the case in non-critical cases as it is clear from the results in [3,[15][16][17][18].
Let us illustrate our special case critical. Consider the following real interpolation spaces ( p (w 0 ), G (q, m; v, w 1 )) θ,r between classical Lorentz spaces p (w 0 ) and G -spaces G (q, m; v, w 1 ). We characterize these interpolation spaces in the critical case p = q with an extra restriction w 0 = w 1 (see Theorem 6.1 below).
The key feature of our approach is to identify G -spaces as K -interpolation spaces (with general weights) between the classical Lorentz and L ∞ spaces. This is done in Sect. 3. Then, in order to apply the reiteration technique, we formulate appropriate reiteration theorems for K -interpolation spaces involving general weights (see Sect. 5). The proofs of these reiteration theorems are essentially based on certain Holmstedttype estimates (from [1]) and weighted Hardy-type inequalities (presented in Sect. 4). The interpolation formulae for G -spaces (our main results) are contained in Sect. 6. Finally, in Sect. 7, we single out some special cases from Sect. 6 in order to illustrate how our obtained results generalize/complement the existing results in previous papers [2-4, 9, 25].

Notation
Throughout the paper we will stick to the following notations. We 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. If both the estimates A B and B A hold, we simply put A ≈ B. We let · q,(a,b) denote the standard L q -quasi-norm on an interval (a, b) ⊂ R. We write X → Y for two quasi-normed spaces X and Y to mean that X is continuously embedded in Y . By a weight w on (0, 1), we always mean a positive locally integrable function on (0, 1). We let denote a bounded Lebesgue measurable domain in R n with measure 1. Finally, the symbol f * will denote the non-increasing rearrangement of a real-valued Lebesgue measurable function f on (see, for instance, [7]).

Slowly Varying Functions
Following [22], we say a weight b is slowly varying on (0, 1) if for every ε > 0, there are positive functions g ε and g −ε on (0, 1) such that g ε is non-decreasing and g −ε is non-increasing, and we have We denote the class of all slowly varying functions by SV . The class SV contains, for example, positive constant functions, and the functions t → (1 − ln t) and t → 1 + ln(1 − ln t). We collect in the next Proposition some properties of slowly varying functions. The proofs of these assertions can be carried out as in [22,Lemma 2.1] or [11,Proposition 3.4.33]. Proposition 2.1 Given b, b 1 , b 2 ∈ SV , the following are true: Thenb ∈ SV .

K-Interpolation Spaces
Let A 0 and A 1 be two quasi-normed spaces. We say (A 0 , A 1 ) is a compatible couple if A 0 and A 1 are continuously embedded in the same Hausdorff topological vector space. For each f ∈ A 0 + A 1 and t > 0, the Peetre K -functional is defined by Note that K (t, f ) is, as a function of t, non-decreasing on (0, ∞). In the sequel, we will refer to this fact simply as monotonicity of K -functional.
In what follows, we always assume that the couple Let 0 < q ≤ ∞, and let w be a positive weight on (0, 1) satisfying the following condition is finite; see, for instance, [1]. If 0 < q < ∞ and w(t) = t −θ with 0 < θ < 1, then we recover the classical real interpolation spacesĀ θ,q (see [7,8,24,27]). Note that, thanks to the condition (2.1), the spacesĀ w,q are intermediate for the couple (A 0 , A 1 ), that is, Next let f ∈Ā w,q . By monotonicity of K -functional and Thus we can conclude that we always have to work under the following condition on w so that the trivial caseĀ w,q = A 0 is excluded. If w ∈ SV , then the condition (2.1) is met thanks to Proposition 2.1 (iv) (if 0 < q < ∞) or to the very definition of a slowly varying function (if q = ∞).

Classical Lorentz Spaces
Let 0 < q ≤ ∞ and let w be weight on (0, 1). Assume that The classical Lorentz spaces q (w) = q (w)( ) consists of those real-valued Lebesgue measurable functions f on , for which the quasi-norm is finite; see [26]. Thanks to the condition (c2), we always have q (w) = {0}; more precisely, we have the embedding L ∞ → q (w). The classical Lorentz spaces cover many well-known spaces: for instance, when w(t) = t 1/ p b(t) (with 0 < p ≤ ∞ and b ∈ SV ) the spaces q (w) become the Lorentz-Karamata spaces L p,q;b (see, for instance, [20]). In particular, when b(t) = (1 − ln t) α , α ∈ R, we put L p,q (log L) α = L p,q;b . The space L p,q (log L) α is called the Lorent-Zygmund space and it was introduced by Bennett and Rudnick [6]. If α = 0, the Lorentz-Zygmund space L p,q (log L) α coincides with the Lorentz space L p,q which becomes the Lebesgue space L p if p = q.
Since f * is non-increasing, we can verify easily that Thus, in the case q = ∞ we can assume that w is non-decreasing.

Generalized Gamma Spaces
We first introduce a notation. For 0 < m, q ≤ ∞, we say a pair (w, v) of weights is admissible if the following conditions are met: is finite.
is non-increasing, we can check that the following function is equivalent to a non-increasing function. Consequently (thanks to the Condition (d4)), it follows that Moreover, the Condition (d3) guarantees that the converse embedding does not hold. Thus, the trivial case G (q, m; v, w) = q (w) is excluded. However, note that for q = m the spaces G (q, m; v, w) again coincide with m (w) for an appropriate weightw.

Remark 2.5
The scale of G (q, m; v, w) spaces is very general and covers many wellknown scales of spaces. If we take q = 1 and w(t) = t, then we recover the classical gamma spaces m (ṽ) (see [26]) for an appropriate weightṽ. Let 0 < m, p, q < ∞, w(t) = t 1/ p and v ∈ SV , then the spaces G (q, m; v, w) coincide with the small Lorentz spaces L ( p,q,m v from [3]. As a still more special case, if become the small Lebesgue spaces L (q,α ; see [18,19]. Finally, since we also allow the case m = ∞ in our definition in contrast to [18], we observe that the spaces S p,α considered in [12] are also a special case of the spaces G (q, m; v, w).

Generalized Gamma Spaces as K -Interpolation Spaces
In this section we characterize the generalized gamma spaces as K -interpolation spaces with general weights. To this end, we first need the following computation of Kfunctional for the couple ( q (w), L ∞ ). While this computation is a special case of a far more general formula in [13, p. 84], we present a simple proof for reader's convenience.
Using the elementary inequality whence we get the estimate " " in (3.1), by taking the infimum over all decompositions of f . To prove the converse estimate " ", we fix 0 < t < 1 and take the following particular decomposition of f : 1) . Therefore, we can check easily that Thus, we arrive at from which follows the estimate " ". The proof is complete.
The next two results describe the characterization of G (q, m; v, w) spaces as K -interpolation spaces.
be a pair of admissible weights. Let φ be the inverse of the following function Moreover, define Proof We give the argument only in the case m < ∞ since the other case m = ∞ is analogous. Set temporarily X = ( q (w), L ∞ ) ρ,m , and let f ∈ q (w). In view of the simple fact that an application of Lemma 3.1 yields finally, the following simple computation completes the proof.
We omit the proof of the next result since it can be carried out by using the same argument as in the proof of the previous theorem.
If we assume additionally that w is differentiable on (0, 1), then

Weighted Hardy-Type Inequalities
The weighted Hardy-type inequalities presented in this section will be the key ingredients in the proofs of our reiteration theorems in the next section.
holds for all non-negative functions h on (0, ∞).
We also have the following variant of the previous result; see [3,Theorem 3.3].
Theorem 4.2 Let 1 < α < ∞, and assume that g and φ are non-negative functions holds for all non-negative functions h on (0, ∞).
The next result is a simple consequence of [1, Lemma 3.3].

Theorem 4.3
Let 0 < α < 1, and assume that g and φ are non-negative functions on holds for all non-negative and non-decreasing functions h on (0, ∞).

Reiteration
First of all, we recall (from Sect. 2.3) that a weight w appearing in the K -interpolation spaceĀ w,q has to satisfy the conditions (2.1) and (2.2) so that both the trivial cases A w,q = {0} andĀ w,q = A 0 are excluded. For convenience we introduce a further notation: for 0 < m < ∞, we say a weight w satisfies the condition (H m ) if the following estimate holds: Moreover, we say a weight w satisfies the condition (H ∞ ) if the following estimate holds:

Remark 5.1
Let w ∈ SV . Then, by Proposition 2.1 (iv)-(vi), w satisfies (H m ). Clearly, by the very definition of a slowly varying function, w also satisfies (H ∞ ).
Next define Let f ∈ A 0 . Since w satisfies (H m ), we can apply the estimate (2.19) in [1] to obtain whence, by an appropriate change of variable, we get In view of monotonicity of K -functional, it follows immediately from (5.1) that Next we establish the converse estimate f X f Y . To this end, we note that, from (5.1), we have f r X ≈ I 1 + I 2 , where In view of Thus, it remains to establish that I 1 f r Y . The case r = m immediately follows from Fubini's theorem. For the case r = m,

and apply Theorem 4.2 (if r > m) or Theorem 4.3 (if r < m). It is not hard to verify that
and consequently, the estimate I 1 f r Y holds. The proof is complete. Remark 5. 3 If we take w(t) = t −θ 1 , 0 < θ 1 < 1, then we get back the classical result from [24]. If we take w ≡ 1 and m = 1, then we recover the first assertion in [21,Theorem 3.21]. The case when w ∈ SV also follows from [5,Theorem 11]. The particular case when w is a logarithmic function has earlier been considered in [10, Theorem 4 (a)].
Next we treat the case m = ∞. In this regard, an elementary but important observation is made in the next remark.

Remark 5.4
Let (A 0 , A 1 ) be a compatible couple of quasi-normed spaces. Using monotonicity of K -functional, we observe that the following identity holds for every f ∈ A 0 . Therefore, while working withĀ w,∞ , we can always assume, without loss of generality, that w is non-increasing. Theorem 5.5 Let 0 < r < ∞ , 0 < θ < 1, and suppose w is strictly decreasing and differentiable on (0, 1) and satisfies (H ∞ ). Put ρ = 1/w, and assume that lim t→1 − ρ(t) =

Then we have
Proof Put X = (A 0 ,Ā w,∞ ) θ,r and Y =Āw ,r . Next, in view of (2.2), we observe that lim t→0 + ρ(t) = 0. Let f ∈ A 0 . Since w satisfies (H ∞ ), we can apply the estimate (2.19) in [1] to obtain whence we arrive at Now the estimate f X f Y follows immediately from (5.2). Next we establish the converse estimate f X f Y . Put and noting we can write and Now by monotonicity of K -functional, we obtain and whence we get and

Now an application of Fubini's theorem gives
Altogether, we arrive at f r X ≈ I f r Y which completes the proof.

Remark 5.6
To the best of our knowledge, the assertion of Theorem 5.5 is new. We note that the particular case when w is a general slowly varying function is entirely missing from [1,5,20], and also not covered by [14,Theorem 5.5].

Remark 5.7
Let 0 < m < ∞. Suppose that w 0 and w 1 are two weights such that w 0 /w 1 is non-decreasing. Then it is not hard to check thatĀ w 1 ,m →Ā w 0 ,m . If we assume, additionally, that  and c 2 ∈ (0, 1) such that

3)
and Then we have Let f ∈ A 0 , and put In view of Remark 5.7 and (5.3), we have ρ ≈ σ on (0, 1). Moreover, since ρ is strictly increasing, we have in fact ρ < σ on (0, 1). As a consequence, we obtain σ > 0 on (0, 1), that is, σ is also strictly increasing on (0, 1). Now, according to the estimates (2.30) and (2.35) in [1], for all 0 < t < 1 we have and where By monotonicity of K -functional, we get Since w 1 satisfies (H m ), we also have Altogether, (5.5) reduces to Thus, from (5.6) and (5.7), we have the following two-sided Holmstedt-type estimate whence it turns out that where and Next, using (5.4), we can compute that Now following the same line of argument which we used while estimating the quantity on right hand side of (5.1), we can show that f r Y ≈ I 2 . Thus it remains to establish the estimate I 1 f r Y . In the case when r ≥ m, this desired estimate follows from Fubini's theorem (if r = m) or from Theorem 4.1 (if r > m). For the remaining case r < m, we apply Theorem 4.
Observe that (4.2) holds trivially for 1/2 < x < 1, and for 0 < x < 1/2 we have Thus, (4.2) is valid. Hence, the estimate I 1 f r Y follows from Theorem 4.4 in the case r < m. This completes the proof.

Interpolation Formulae
Finally, we are in a position to describe the interpolation properties of generalized gamma spaces. In view of well-known reiteration technique, our interpolation formulae are rather straightforward consequences of reiteration theorems (from previous section) and characterization of generalized gamma spaces as K -interpolation spaces (Theorems 3.2 and 3.3). Thus, we illustrate how the reiteration technique works only in a single case, and omit the proofs of the remaining assertions.
Throughout this section, ψ and φ are same as defined in Theorem 3.2.

Assume that η 1 satisfies (H m ). Then
(b) Let 0 < m < ∞ and q = ∞. Assume that w is strictly increasing and differentiable on (0, 1) with lim and assume that η 2 satisfies (H m ). Then (c) Let m = ∞ and 0 < q < ∞. Assume that ρ = 1/v is strictly increasing and differentiable on (0, 1) with lim and that ρ = 1/v is strictly increasing and differentiable on (0, 1) with lim Proof We give the argument only in the first case. Using Theorem 3.2, we can write now an application of Theorem 5.2 yields Temporarily set X = ( q (w), L ∞ )η 1 ,r and take f ∈ q (w). Then now a change of variable t = ψ(s) gives next using Lemma 3.1, we arrive at or, whence we get X = G (q, r ; V 1 , w) as desired.