# Real Interpolation of Small Lebesgue Spaces in a Critical Case

Authors: Irshaad Ahmed; Aneesa Hafeez; Ghulam Murtaza
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We establish an interpolation formula for small Lebesgue spaces in a critical case.


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## Body

## 1. Introduction

The small Lebesgue spaces were introduced by A. Fiorenza in [1], and they have found applications, for example, in boundary value problems [2, 3], Besov embeddings [4, 5], and dimension-free Sobolev embeddings [6]. The reader is referred to a recent survey paper [7] for more details.LetΩ be a bounded domain in Rn with measure 1, and let 1<p<∞ and α>0. The small Lebesgue space L(p,α=L(p,α(Ω) is formed by all those real-valued Lebesgue measurable functions f on Ω, for which the norm (1)fLp,α=∫011-ln⁡t-α/p+α-1∫0tf∗updu1/pdttis finite; see [8]. Here f∗ denotes the nonincreasing rearrangement of f (see, for instance, [9]). For α=1, the spaces L(p,α coincide with the classical small Lebesgue spaces L(p (see [10, Corollary 3.3]).Let0<θ<1, 1<q<∞, 1<p<r<∞, and α>0. The real interpolation spaces (L(p,α,L(r,α)θ,q have been recently characterized in [11, Theorem 3.4] as the well-known Lorentz-Zygmund spaces. Here (·,·)θ,q denotes the classical real interpolation method (see Section 2). The critical case p=r has not been considered in [11], and the goal of the present paper is to fill this gap. More precisely, we will characterize the interpolation spaces (L(p,α,L(p,β)θ,q under an appropriate condition on α and β(β>0). Our approach is essentially different from that of [11] and allows the parameter q to lie also in the interval (0,1]. The key ingredient of our method is a general Holmstedt-type estimate from a recent paper [12].The plan of the paper is simple. We collect the necessary background, along with certain weighted Hardy-type inequalities, in Section2. The main result is contained in Section 3.


## 2. Preliminaries

Throughout the paper, we will writeA≲B or B≳A for two nonnegative 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. Moreover, we will write X↪Y for two quasi-normed spaces X and Y to mean that X is continuously embedded in Y.Let(A0,A1) be a compatible couple of quasi-normed spaces; that is, we assume that both A0 and A1 are continuously embedded in the same Hausdorff topological vector space. Peetre’s K-functional K(t,f)=K(t,f;A0,A1) is defined, for each f∈A0+A1 and t>0, by (2)Kt,f=inf⁡f0A0+tf1A1:f0∈A0,f1∈A1,f=f0+f1.In what follows, we always assume that the couple(A0,A1) is ordered in the sense that A1↪A0.Let0<q≤∞, 0≤θ≤1, and γ∈R. The real interpolation space A¯θ,q;γ=(A0,A1)θ,q;γ consists of all f∈A0 for which the quasi-norm (3)fA¯θ,q;γ=∫01t-θq1-ln⁡tqγKqt,fdtt1/qis finite (with usual modification when q=∞). When γ=0 and 0<θ<1, we put A¯θ,q=A¯θ,q;γ. Since A1↪A0, we have K(t,f)≈fA0,t>1. Combining this with the fact that, as a function of t, K(t,f)/t is nonincreasing, we can easily check that (4)fA¯θ,q≈∫0∞t-θqKqt,fdtt1/q.Thus, the spaces A¯θ,q are in fact the classical Lions-Peetre spaces (see [9, 13, 14]).In the sequel, we will work with the limiting spacesA¯0,q;γ and the classical spaces A¯θ,q. It is not hard to verify that the limiting spaces A¯0,q;γ are intermediate, without any condition on γ and q, for the couple (A0,A1), that is, (5)A1↪A¯0,q;γ↪A0.However, in order to exclude the trivial case A¯0,q;γ=A0, we have to work under the assumption γ≥-1/q (if q<∞) or γ>0 (if q=∞). This observation immediately follows from the elementary fact that, as a function of t, K(t,f) is nondecreasing.We close this section with the following weighted Hardy-type inequalities which will be needed in order to prove our main result in the next section.Lemma 1 ([15, Lemma 3.2]).
Let1≤s<∞, and assume that w and ϕ are nonnegative functions on (0,∞). Put (6)vt=wt1-sϕt∫t∞wudus.Then (7)∫0∞∫0tϕuhuduswtdt≲∫0∞hstvtdtholds for all nonnegative functions h on (0,∞).By an obvious change of variable, we get the following variant of the previous result.Lemma 2.
Let1≤s<∞, and assume that w and ϕ are nonnegative functions on (0,∞). Put (8)v0t=wt1-sϕt∫0twudus.Then (9)∫0∞∫t∞ϕuhuduswtdt≲∫0∞hstv0tdtholds for all nonnegative functions h on (0,∞).Lemma 3 ([16, Theorem 3.3 (b)]).
Let0<s<1. Assume w and v are positive functions on (0,1), and ψ is a positive function on (0,1)×(0,1). Then(10)∫01∫01ψt,uguduswtdt≲∫01gstvtdtholds for all positive and nondecreasing functions g on (0,1) if and only if(11)∫01∫x1ψt,uduswtdt≲∫x1vtdtholds for all 0<x<1.


## 3. Interpolation Formula

First we compute theK-functional for the couple (L(p,α,L(p,β), 0<α<β. To this end, we need the following Holmstedt-type estimate.Theorem 4.
Let-1<γ0<γ1, and let f∈A0. Then (12)K1-ln⁡tγ0-γ1,f;A¯0,1;γ0,A¯0,1;γ1≈It,f+1-ln⁡tγ0-γ1Jt,f,0<t<1,where (13)It,f=∫0t1-ln⁡uγ0Ku,fduu,and (14)Jt,f=∫t11-ln⁡uγ1Ku,fduu.Proof.
Letj=0,1. Put (15)gjt=t1+∫t11-ln⁡uγjduu,0<t<1,and (16)hjt=∫0tu1-ln⁡uγjduu,0<t<1.According to case  4 in [12, Theorem 4], we have (17)Kg0tg1t,f;A¯0,1;γ0,A¯0,1;γ1≈It,f+1-ln⁡tγ0-γ1Jt,f,provided that(18)∫0tu1-ln⁡uγ0g1u+h1uduu≲g0tg1t,0<t<1,and(19)1+∫t1u1-ln⁡uγ1g0u+h0uduu≲g1tg0t,0<t<1.Since γj>-1, then gj(t)≈t(1-ln⁡t)γj+1. Moreover, note that hj≲gj. Finally, the condition γ0<γ1 implies that both (18) and (19) hold. The proof is complete.Remark 5.
Ifγ0<γ1, then we have A¯0,1;γ1↪A¯0,1;γ0. Thus, in this case, we trivially have (20)Kt,f;A¯0,1;γ0,A¯0,1;γ1≈fA¯0,1;γ0,t>1.Corollary 6.
Let1<p<∞, 0<α<β, and let f∈Lp(Ω). Then, for all 0<t<1, we have (21)K1-ln⁡t1-1/pα-β,f;Lp,α,Lp,β≈I1t,f+1-ln⁡t1-1/pα-βJ1t,f,where (22)I1t,f=∫0t1-ln⁡u-α/p+α-1∫0uf∗τpdτ1/pduu,and (23)J1t,f=∫t11-ln⁡u-β/p+β-1∫0uf∗τpdτ1/pduu.Proof.
First observe thatL∞(Ω)↪Lp(Ω) as Ω has finite measure. We put Lp=Lp(Ω) and L∞=L∞(Ω). Since (see [13, Theorem 5.2.1]) (24)Kt1/p,f;Lp,L∞≈∫0tf∗updu1/p, it follows immediately that L(p,η=(Lp,L∞)0,1;-η/p+η-1, where η>0. Thus, the proof simply follows by applying Theorem 4 to A0=Lp, A1=L∞, γ0=-α/p+α-1, and γ1=-β/p+β-1.Following [4], we define small Lebesgue spaces in a slightly general way as follows. Let 1≤p<∞, 0<q≤∞, and b>-1/q. The small Lebesgue space L(p,b,q=L(p,b,q(Ω) is formed by all those real-valued Lebesgue measurable functions f on Ω, for which the quasi-norm (25)fLp,b,q=∫011-ln⁡tqb∫0tf∗upduq/pdtt1/qis finite. The spaces L(p,b,q are a particular case of more general GΓ(p,m,w) spaces introduced and studied in [17–19]. Note that if 1<p<∞, q=1, and b=-α/p+α-1 with α>0, then we recover the spaces L(p,α.The following interpolation formula is the main contribution of this paper.Theorem 7.
Let1<p<∞, 0<α<β, 0<θ<1, and 0<q≤∞. Then (26)Lp,α,Lp,βθ,q=Lp,b,q,where b=(1-1/p)((1-θ)α+θβ)-1/q.Proof.
First note thatL(p,β↪L(p,α as α<β. Put X=(L(p,α,L(p,β)θ,q and Y=L(p,b,q. Let f∈Lp. Assume that 0<q<∞. By making an appropriate change of variable and using Corollary 6, we arrive at (27)fXq≈C1+C2,where (28)C1=∫011-ln⁡tθq1-1/pβ-α-1∫0t1-ln⁡u-α/p+α-1Fuduuqdtt,and (29)C2=∫011-ln⁡t1-θq1-1/pα-β-1∫t11-ln⁡u-β/p+β-1Fuduuqdtt,with (30)Ft=∫0tf∗updu1/p.Thus, the proof will be complete, in the case 0<q<∞, if we show that C1≲fYq and C2≈fYq. Since F is nondecreasing, we get (31)C2≥∫011-ln⁡t1-θq1-1/pα-β-1Fqt∫t11-ln⁡u-β/p+β-1duuqdtt,or (32)C2≳∫011-ln⁡t1-θq1-1/pα-β-1Fqt1-ln⁡t-β/p+β-1qdtt,whence we obtain C2≳fYq as (1-ln⁡t)-β/p+β-1 and (1-ln⁡t)-β/p+β are asymptotically the same as t→0+.
Next we establish the estimatesC1≲fYq and C2≲fYq. To this end, we distinguish two cases: q≥1 and q<1. Assume first that q≥1. Then, C1≲fYq follows from Lemma 1, applied with s=q, h=F, ϕ(t)=t-1(1-ln⁡u)-α/p+α-1, and w(t)=t-1(1-ln⁡t)θq(1-1/p)(β-α)-1χ(0,1)(t). Similarly, C2≲fYq follows from Lemma 2. Next assume that q<1. Since F is nondecreasing, we can apply Lemma 3 with s=q, g=F, w(t)=t-1(1-ln⁡t)θq(1-1/p)(β-α)-1, ψ(t,u)=u-1(1-ln⁡u)-α/p+α-1χ(0,t)(u), and v(t)=t-1(1-ln⁡t)q(1-1/p)((1-θ)α+θβ)-1. Note that (10) turns into C1≲fYq. Thus, we have to verify (11) for all 0<x<1. Observe that (11) holds trivially for all 1/2≤x<1, and thus we may assume that 0<x<1/2. Let us put η=-α/p+α and δ=θq(1-1/p)(β-α). Now we have (33)∫01∫x1ψt,uduswtdt=∫x1∫xt1-ln⁡uη-1duuq1-ln⁡tδ-1dtt≤∫x11-ln⁡uη-1duuq∫x11-ln⁡tδ-1dtt≈1-ln⁡xqη+δ≈∫x1vtdtwhich proves the validity of (11). Hence, the estimate C1≲fYq follows from (10). Similarly, we can obtain C2≲fYq from Lemma 3. This completes the proof in the case 0<q<∞. Next we assume that q=∞. This time we have(34)fX≈D1+D2,where (35)D1=sup0<t≤1⁡1-ln⁡tθ1-1/pβ-α∫0t1-ln⁡u-α/p+α-1Fuduu,and (36)D2=sup0<t≤1⁡1-ln⁡t1-θ1-1/pα-β∫t11-ln⁡u-β/p+β-1Fuduu.In order to estimate D1, we note that (37)D1≤fYsup0<t≤1⁡1-ln⁡tθ1-1/pβ-α∫0t1-ln⁡uθ1-1/pα-β-1duu,whence we get D1≲fY. Similarly, we can show that D2≲fY. On the other hand, again using the fact that F is nondecreasing, we get D2≳fY. Altogether, we have D1+D2≈fY which gives fX≈fY in view of (34). The proof of the theorem is complete.



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