# General Holmstedt’s Formulae for theK-Functional

Authors: Irshaad Ahmed; Georgi E. Karadzhov; Ali Raza
Journal: Journal of Function Spaces
 (2017)

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## Abstract

Explicit formulae for theK-functional for the general couple ((A0,A1)Φ0,(A0,A1)Φ1), where (A0,A1) is a compatible couple of quasi-normed spaces, are proved. As a consequence, the corresponding reiteration theorems are derived.


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## Body

## 1. Introduction

In recent papers [1–11] the classical Holmstedt formula for the K-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 K-method, but for the J-method as well. Here we consider the K-method in the most general case for quasi-normed spaces. Namely, let (A0,A1) be a compatible couple of quasi-normed spaces, that is, both A0 and A1 are linearly and continuously embedded in some Hausdorff topological vector space. By definition, the K-interpolation space (A0,A1)Φ has a quasi-norm(1)aA0,A1Φ=Kt,aΦ,where K(t,a)=K(t,a;A0,A1) is the K-functional of Peetre [13, 14], defined for 0<t<∞,  a∈A0+A1 as follows:(2)Kt,a=infa=a0+a1⁡a0A0+ta1A1,and Φ is a quasi-normed space of Lebesgue measurable functions, defined on (0,∞), with monotone quasi norm as follows: |g|≤|h| implies gΦ≤hΦ such that min⁡(1,t)∈Φ. If Φ=L∗q(w), that is,(3)gΦ=∫0∞wtgtqdtt1/q,0<q≤∞,we write (A0,A1)w(t),q instead of (A0,A1)Φ; if q=∞, the above integral has to be replaced by sup⁡w(t)g(t). Here w is a nonnegative Lebesgue measurable function defined on (0,∞) and called weight.In [1, 2] the case when w(t)=t-θv(t), 0<θ<1, and v-repeated logarithms was considered and this was extended to v -slowly varying in [3]. Certain limiting cases (when θ=0) are treated in [5, 8–10]. The case Φ=L∗q(w) with arbitrary w was investigated in [6]. In [4, 7, 11] the case when Φ=E(t-θ) and E is rearrangement invariant Banach space on (0,∞) with the Haar measure dt/t is treated in detail. Note that in the case w(t)=t-θv(t), 0<θ<1, v -slowly varying, the K- and J-methods are equivalent, but this is not true in the limiting cases θ=0 or θ=1. 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 K- and J-methods in the limiting cases are obtained in [15, 16].We use the notationsa1≲a2 or a2≳a1 for nonnegative functions or functionals to mean that the quotient a1/a2 is bounded; also, a1≈a2 means that a1≲a2 and a1≳a2. We say that a1 is equivalent to a2 if a1≈a2. Recall that the weight w is called slowly varying if for every ε>0 the function tεw(t) is equivalent to a nondecreasing one and the function t-εw(t) is equivalent to a nonincreasing one.The main results are announced in [17].


## 2. Formulae for theK-Functional

Using the Holmstedt argument, we can prove general formulae for theK-functional. Let K(t,a)≔K(t,a;A0,A1) and let(4)It,a≔χ0,tuKu,aΦ0,g0t≔tχt,∞uΦ0,h0t≔uχ0,tuΦ0.Theorem 1 (case((A0,A1)Φ0,A1)).
Ifa∈(A0,A1)Φ0+A1, then(5)Kρt,a;A0,A1Φ0,A1≈It,a+ρtKt,at,where ρ(t)≈g0(t)+h0(t).Proof.
Letw≔g0+h0, B0≔(A0,A1)Φ0, and B1≔A1. Using monotonicity of K, we get(6)It,a≥h0tKt,at,(7)aB0≳g0t+h0tKt,at.Let a=a0+a1, aj∈Aj, and j=0,1. Then(8)Kt,a≲Kt,a0+Kt,a1≲twta0B0+ta1A1,whence(9)wtKt,at≲Kwt,a;B0,B1.Also,(10)It,a≲χ0,tuKu,a0Φ0+χ0,tuKu,a1Φ0≲a0B0+h0ta1B1.Hence(11)It,a≲Kwt,a;B0,B1.Now we estimate K(w(t),a;B0,B1) from above. To this end, following Holmstedt, we choose decomposition a=a0+a1 such that K(t,a)≈a0A0+ta1A1. Then(12)Ku,a0≤a0A0≲Kt,a,Ku,a1≤ua1A1≲uKt,at.Further,(13)a0B0=Ku,a0Φ0≲χ0,tuKu,a0Φ0+χt,∞uKu,a0Φ0.Denote by C the first term on the right-hand side and by D the second one. We have (14)C≲It,a+χ0,tuKu,a1Φ0≲It,a+h0tKt,at,D≲g0tKt,at,whence(15)a0B0≲It,a+wtKt,at.Also,(16)wta1B1=wta1A1≲wtKt,at.Thus(17)Kwt,a;B0,B1≲It,a+wtKt,at.Remark 2.
If the spaces(A0,A1)Φ0 and A1 are too close, then formula (5) might be useless. For example, if Φ0=L∞(1/t), then g0=1=h0. In applications we require ρ to be increasing (strictly).Let(18)Jt,a≔χt,∞uKu,aΦ1,g1t≔tχt,∞uΦ1,h1t≔uχ0,tuΦ1.Theorem 3 (case(A0,(A0,A1)Φ1)).
Ifa∈A0+(A0,A1)Φ1, then(19)Kρt,a;A0,A0,A1Φ1≈ρtJt,a+Kt,a,where ρ(t)≈t/g1(t)+h1(t).Proof.
Letw(t)≔t/g1(t)+h1(t), B0≔A0, and B1≔(A0,A1)Φ1. Using monotonicity of K, we get(20)Jt,a≥g1tKt,at,(21)aB1≳g1t+h1tKt,at.Let a=a0+a1, aj∈Aj, and j=0,1. Then(22)Kt,a≲Kt,a0+Kt,a1≲a0B0+wta1B1,whence(23)Kt,a≲Kwt,a;B0,B1.Also,(24)Jt,a≲χt,∞uKu,a0Φ1+χt,∞uKu,a1Φ1≲g1tta0B0+a1B1.Hence(25)wtJt,a≲Kwt,a;B0,B1.Now we estimate K(w(t),a;B0,B1) from above using decomposition a=a0+a1 such that (12) are satisfied. Then (26)a0B0=a0A0≲Kt,a.We have(27)a1B1≲χ0,tuKu,a1Φ1+χt,∞uKu,a1Φ1≔C+D,C≲h1tKt,at,D≲Jt,a+g1tKt,at,whence(28)wta1B1≲wtJt,a+Kt,a.Hence(29)Kwt,a;B0,B1≲wtJt,a+Kt,a.Formulae for the couple((A0,A1)Φ0,(A0,A1)Φ1) are more complicated. We need the following condition that in some sense require these two spaces to be not too close. Let(30)1gt≔χt,∞uug0u+h0uΦ1<∞,ht≔χ0,tuug1u+h1uΦ0<∞.Theorem 4 (case((A0,A1)Φ0,(A0,A1)Φ1)).
LetBj≔(A0,A1)Φj, j=0,1 and let a∈B0+B1.
Case 1. If(31)g0g1+h1≲ρ≲g0+h0h1,h≲ρ≲g,then(32)Kρt,a;B0,B1≈It,a+ρtJt,a+g0t+ρth1tKt,at.
Case 2. If(33)g0g1+h1≲ρ≲h0h1,h≲ρ≲g,then(34)Kρt,a;B0,B1≈It,a+ρtJt,a+g0tKt,at.
Case 3. If(35)g0g1≲ρ≲g0+h0h1,h≲ρ≲g,then(36)Kρt,a;B0,B1≈It,a+ρtJt,a+ρth1tKt,at.
Case 4. If at least one of the conditionsg0/g1≲ρ, h1≲g1, h≲ρ≲g,g0≲h0, ρ≲h0/h1, h≲ρ≲g,g0≲h0, h1≲g1, h≲ρ≲gis satisfied, then(37)Kρt,a;B0,B1≈It,a+ρtJt,a.Proof.
From (6) and (20) it follows that(38)g0tKt,at≲It,aifg0≲h0,g0tKt,at≲ρtJt,aifg0g1≲ρ,ρth1tKt,at≲It,aifρ≲h0h1,ρth1tKt,at≲ρtJt,aifh1≲g1.Let a=a0+a1, aj∈Aj, and j=0,1. Then, using estimates (7) and (21), we have (39)Kt,a≲Kt,a0+Kt,a1≲tg0t+h0ta0B0+tg1t+h1ta1B1,whence(40)g0tKt,at≲Kρt,a;B0,B1ifg0g1+h1≲ρ,ρth1tKt,at≲Kρt,a;B0,B1ifρ≲g0+h0h1.Also,(41)It,a≲χ0,tuKu,a0Φ0+χ0,tuKu,a1Φ0≲a0B0+hta1B1,Jt,a≲χt,∞uKu,a0Φ1+χt,∞uKu,a1Φ1≲1gta0B0+a1B1.Hence,(42)It,a≲Kρt,a;B0,B1ifh≲ρ,ρtJt,a≲Kρt,a;B0,B1ifρ≲g.To estimate K(ρ(t),a;B0,B1) from above we use the same decomposition as before with properties (12). Then(43)a0B0≲χ0,tuKu,a0Φ0+χ0,tuKu,a1Φ0+χt,∞uKu,a0Φ0,whence(44)a0B0≲It,a+g0t+h0tKt,at≲It,a+g0tKt,at.Also,(45)a1B1≲χ0,tuKu,a1Φ1+χt,∞uKu,a1Φ1≔C+D,C≲h1tKt,at,D≲Jt,a+g1tKt,at≲Jt,a,whence(46)ρta1B1≲ρtJt,a+ρth1tKt,at.Therefore for all weights ρ we have(47)Kρt,a;B0,B1≲It,a+ρtJt,a+g0t+ρth1tKt,at.We give examples that are not entirely covered by the previous papers.Example 5 (distant spaces).
Let(48)gΦj≔∫01t-θjvjtgtpjdtt1/pj+∫1∞t-ηjwjtgtqjdtt1/qj,where 0≤θ0<θ1≤1, 0≤η0<η1≤1, 0<pj,qj≤∞, and vj,wj are slowly varying weights and j=0,1. We call these spaces distant if θ0<θ1 or η0<η1, in opposition to the case θ0=θ1 and η0=η1. We have(49)
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.In particular,(52)g0≳h0,g1≲h1,g≳h.On the other hand,(53)g0th1t≈t1+∫t1s-θ0v0sp0dss1/p0∫0ts1-θ1v1sp1dss-1/p1for 0<t<1 and(54)g0th1t≈t∫t∞s-η0w0sq0dss1/q01+∫1ts1-η1w1sq1dss1/q1-1for t≥1. Hence,(55)h≲g0h1≲gand therefore(56)Kρt,a;B0,B1≈It,a+ρtJt,a+g0tKt,at,where B0=(A0,A1)Φ0, B1=(A0,A1)Φ1, ρ(t)≈g0(t)/h1(t).
Also,(57)Kρt,a;A0,A1Φ0,A1≈It,a+g0tKt,at,where ρ(t)≈g0(t).(58)Kρt,a;A0,A0,A1Φ1≈ρtJt,a+Kt,a,where ρ(t)≈t/h1(t).
Moreover, if0<θ0<θ1<1 and 0<η0<η1<1, then gj≈hj and j=0,1; hence, in (56), (57), and (58), we can drop the last term (this is the case treated in [3]).Example 6 (nondistant spaces).
LetΦj be as in Example 5 but with θ0=θ1=0, η0=η1=1, and vj(t)=(1-log⁡t)aj, and wj(t)=(1+log⁡t)bj, j=0,1.
If(59)a0-a1<1p1-1p0,ajpj+1>0,b1-b0<1q0-1q1,bjqj+1>0,p1≤p0,then(60)gjt≈t1-log⁡taj+1/pj,0<t<1,1+log⁡tbj,t≥1,hjt≈t1-log⁡taj,0<t<1,1+log⁡tbj+1/qj,t≥1.
Also,(61)gt≈ht≈1-log⁡ta0-a1+1/p0-1/p1,0<t<1,1+log⁡tb0-b1+1/q0-1/q1,t≥1.In particular,(62)gjt≳hjt,0<t<1,gjt≲hjt,t≥1.On the other hand, since 1/p0≤1/p1, we have(63)g0g1+h1≲h≲h0h1and therefore(64)Kρt,a;A0,A1Φ0,A0,A1Φ1≈It,a+ρtJt,a+g0tKt,at,where(65)ρt=1-log⁡ta0-a1+1/p0-1/p1,0<t<1,1+log⁡tb0-b1+1/q0-1/q1,t≥1.


## 3. Reiteration

The formulae for theK-functional imply immediately theorems of reiteration or stability of the K-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(A0,A1) be a compatible couple of quasi-normed spaces. Then(66)A0,A1Φ0,A1Φ=A0,A1Ψ,where(67)gΨ≔χ0,ρ-1tuguΦ0Φ+tgρ-1tρ-1tΦand ρ is the same as in Theorem 1 and ρ(0)=0, ρ(∞)=∞, and ρ(1)=1; ρ is increasing.Proof.
We only need to check thatmin⁡(1,t)∈Ψ. Since(68)χ0,1tχ0,ρ-1tuuΦ0Φ=χ0,1th0ρ-1tΦand h0≲ρ, therefore h0ρ-1t≲t; we see that the above quantity is finite. Also(69)χ1,∞tχ0,1uuΦ0Φ=χ1,∞Φh01<∞,χ1,∞tχ1,ρ-1tΦ0Φ≤χ1,∞Φχ1,∞Φ0<∞.Further, for t>1, we have g0(t)≤t and h0t≲1+uχ1,tuΦ0≲t; hence t≲ρ-1(t). Therefore(70)χ1,∞ttρ-1tΦ≲χ1,∞Φ<∞.Theorem 8.
Let(A0,A1) be a compatible couple of quasi-normed spaces. Then (71)A0,A0,A1Φ1Φ=A0,A1Ψ,where(72)gΨ≔tχρ-1t,∞gΦ1Φ+gρ-1tΦand ρ is the same as in Theorem 3 and ρ(0)=0, ρ(∞)=∞, and ρ(1)=1; ρ is increasing.Proof.
We only need to check thatmin⁡(1,t)∈Ψ. We have(73)χ0,1ttχρ-1t,∞min⁡1,uΦ1Φ≲χ0,1tth11Φ<∞and, using also ρ(u)≲u/g1(u),(74)χ1,∞ttχρ-1t,∞uΦ1Φ≲χ1,∞g1ρ-1tρ-1tΦ<∞.As above, we check that g1(t)≲1 and h1(t)≲1 for 0<t<1. Hence ρ(t)≳t for 0<t<1. Then(75)min⁡1,ρ-1Φ≲1+χ0,1ttΦ<∞.In some cases the quasi norm ofΨ can be simplified.Example 9 (distant spaces).
Let(76)gΦ0≔∫01v0tgtp0dtt1/p0+∫1∞w0tgtq0dtt1/q0,where 0<p0,q0<∞ and v0, w0 are slowly varying weights and let(77)gΦ≔∫01t-1vtgtpdtt1/p+∫1∞t-1wtgtqdtt1/q,where 0<p,q<∞ and v, w are slowly varying weights. Then(78)A0,A1Φ0,A1Φ=A0,A1Ψ,gΨ=∫01t-1vρtgtpdtt1/p+∫1∞t-1wρtgtqdtt1/q,where(79)
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.Indeed, we have(80)Kρt,a;A0,A1Φ0,A1≈It,a+ρttKt,a;A0,A1for 0<t<1 and(81)Kρt,a;A0,A1Φ0,A1≈I1t,a+∫01v0sKs,a;A0,A1p0sdss1/p0+ρttKt,a;A0,A1for t≥1, where(82)It,a=∫0tv0uKu,ap0duu1/p0,I1t,a=∫1tw0uKu,aq0duu1/q0.Let ρ(1)=1. Then(83)aA0,A1Φ0,A1Φ≈B01/p+B11/p+C01/q+C11/q+C21/q, where (84)B0=∫01vρtρtpIt,apdρρ,B1=∫01vρtρtpρtKt,atpdρρ,C0=∫01v0sKs,ap0sdss1/p0∫1∞wρtρtqdρρ,C1=∫1∞wρtρtqI1t,aqdρρ,C2=∫1∞wρtρtqρtKt,atqdρρ.We can choose equivalent ρ so that dρ/ρ≈dt/t (see [6]); hence(85)B1≈∫01u-1vρuKu,apduu,C2≈∫1∞u-1wρuKu,aqduu.It is sufficient to prove that(86)B0+C0≲∫01u-1vρuKu,apduu,(87)C1≲∫1∞u-1wρuKu,aqduu.To estimate C0 for p≤p0, we use monotonicity of the interpolation scale, while for p>p0 we apply Hölder’s inequality. Namely, if p≤p0,(88)∫01v0tKt,ap0dtt≲∫01v0tKt,apdttp0/p≲∫01t-1vρtKt,ap0dttp0/pand if p>p0, then(89)∫01v0tKt,ap0dtt=∫01tv0tvρtp0t-1vρtKt,ap0dtt≲∫01t-1vρtKt,apdttp0/p.Next we check only the inequality for B0 and the inequality for C1 being similar. For p=p0, it follows from Fubini’s theorem. B0 can be estimated from above if p>p0 by the Muckenhoupt result [18].
Letv1=w11-s(φ∫t1w1(u)du)s, where 1<s<∞, φ>0, and w1>0. Then (90)∫01∫0tφuhudusw1tdt≲∫01hstv1tdt.We apply this for φ(t)=v0(t)p0/t,h(t)=Kp0(t,a),(91)w1t=vρt/ρtpρ′tρt,s=pp0.Then v1(t)≈[v0(t)v(ρ(t))/ρ(t)]pt-1 and hence(92)B0≲∫01v0tKt,apvρtρtpdtt.Further, if p<p0 we estimate B0 from above using integration by parts. We have(93)B0=-∫01It,ap/p0d∫t1vρtρtpdρρ.Indeed, let a∈(A0,A1)Ψ. Using that v(ρ)/ρ is equivalent to a decreasing function, we have(94)S≔It,a1/p0vρtρt≲∫0tvρuKu,av0uρup0duu,and since tv0(t)≲ρ(t),p<p0,(95)S≲∫0tvρuKu,aup0duu≲∫0tvρuKu,aupduu.The integral above has a limit zero when t→0. Thus (93) is true. Further,(96)B0≲∫01It,ap/p0-1v0tKt,a∫t1vρtρtpdρρdtt,or(97)B0≲∫01v0tKt,apvρtρtpdtt.Since ρ(t)=g0(t)≳tv0(t), we get (86).Now we consider examples of nondistant spaces:B≔(A0,(A0,A1)w1(t),p)h(t),q, w1-slowly varying, 0<p<∞,0<q<∞.In order to handle this case, we need the following result.Lemma 10.
One has(98)∫0∞∫t∞φugudusw0tdt≈∫0∞gstv0tdt;φ≥0,g≥0,w0≥0,v0≥0, and 0<s<∞, if the following conditions are satisfied:(99)∫t∞φugudu≳gt∫t∞φudu,(100)φt∫t∞φudu≲w0t∫0tw0udu,where(101)v0t=φt∫t∞φudus-1∫0tw0udufor0<s≤1,v0t=w01-stφt∫0tw0udusfor1≤s<∞.Proof.
The estimate (98) from above is a consequence of Lemmas 3.2, 3.3 in [6]. Using (99), we get an estimate from below with ∫0∞gs(t)w0(t)∫t∞φ(u)dusdt. Now condition (100) ensures the equivalence in (98).Theorem 11.
Letw1 be a slowly varying weight and let ρp(t)≔1/∫t∞w1p(u)du/u, 0<p<∞, satisfying ρ(0)=0, ρ(∞)=∞. If(102)∫0thqssqdss≈hqttq,0<q<∞,then(103)B≔A0,A0,A1w1t,pht,q=A0,A1wt,q,where(104)wqt=hqρtρptw1pt.Proof.
We have(105)Kρt,a;A0,A0,A1w1t,p≈ρt∫t∞w1psKps,adss1/p;hence(106)aBq≈∫0∞hqρtρqt∫t∞w1psKps,adssq/pdρtρt.We apply Lemma 10 with φ(t)=w1p(t)/t, g(t)=Kp(t,a), w0(t)=hq(ρ(t))ρq-1(t)ρ′(t), and s=q/p. Condition (99) follows from monotonicity of K. We have to check (100). To this end first we simplify(107)∫0tw0udu=∫0thqρuρqudρuρu=∫0ρthquuqduu.Using also (102), we get ∫0tw0(u)du≲hq(ρ(t))ρq(t). Then (100) means that (108)w1ptρpt≲tρ′tρt.But this follows from the definition of ρ.For example, condition (102) is true if h(t)=t-θa(t), 0≤θ<1, a -slowly varying.



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*Source: 4958073-2017-02-14.xml*