# Reiteration Theorems for Two-Parameter Limiting Real Interpolation Methods

Authors: Irshaad Ahmed; Tuba Ejaz
Journal: Journal of Function Spaces
 (2016)

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## Abstract

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.


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## Body

## 1. Introduction

Let(A0,A1) be a compatible couple of quasi-normed spaces; that is, we assume that both A0 and A1 are continuously embedded in some common quasi-normed space. Peetre’s K-functional is defined, for each f∈A0+A1 and t>0, by(1)Kt,f=Kt,f;A0,A1=inf⁡f0A0+tf1A1:f=f0+f1,where the infimum extends over all representations f=f0+f1 of f with f0∈A0 and f1∈A1. Let Φ be a quasi-normed function space with a monotone quasi-norm on (0,∞) and Haar measure dt/t, such that (2)t⟼min⁡1,t∈Φ. The general real interpolation space AΦ=(A0,A1)Φ consists of those f∈A0+A1 for which the following quasi-norm (3)fAΦ=Kt,fΦ is finite. We refer to [1–4] for a full development of the real interpolation method.Let0<θ<1 and 0<q≤∞; then the classical scale of real interpolation spaces Aθ,q=(A0,A1)θ,q is obtained when Φ is the weighted Lebesgue space Lq(t-θ) defined by the quasi-norm(4)gLqt-θ=∫0∞t-θqgtqdtt1/q. (When q=∞, the integral should be replaced by appropriate supremum.) The reiteration theorem for this scale states that (see [1])(5)Aθ0,q0,Aθ1,q1θ,q=A0,A11-θθ0+θθ1,q,where 0<θ0<θ1<1,0<θ<1, and 0<q0,q1,q≤∞. Moreover, for extreme cases, we have(6)Aθ0,q0,A1θ,q=A0,A11-θθ0+θ,q,A0,Aθ1,q1θ,q=A0,A1θθ1,q.We note that the scale (A0,A1)θ,q makes no sense for θ=0,1 unless q=∞. However, several authors (see, e.g., [5–7]) have investigated the limiting reiteration (with θ=0 or θ=1) by taking the parameter Φ to be more general weighted Lebesgue space Lq(w), with weight w being of the form t-θg(t),0≤θ≤1, where g is a broken-logarithmic function or, more generally, a slowly varying function. See [8] for more general weights.Recently, Cobos and Segurado [9] have defined a new limiting interpolation method (A0,A1)q;K, corresponding to limiting values θ=0,1, in a different way. Namely, if Φ is given by the quasi-norm (7)gΦ=∫01gtqdtt1/q+∫1∞gttqdtt1/q,where 1≤q≤∞, then (A0,A1)Φ=(A0,A1)q;K=Aq;K. The reiteration spaces (Aq;K,A1)q,K and (A0,Aq;K)q;K have been characterized in [10].In this paper we extend the limiting interpolation methodAq;K by considering the parameter space Φ to be (8)gΦ=∫01gtpdtt1/p+∫1∞gttqdtt1/q, where 0<p,q≤∞. We denote the resulting real interpolation method AΦ by A{p,q} without using the subindex K for convenience. Clearly, A{q,q}=Aq;K. In addition, note that we do not use the notation Ap,q for our method in order to avoid notational confusion (see, e.g., [2]). Since, for p=∞ and/or q=∞, A{p,q} coincides with A0+A1 (see [9, Lemma 3.2]), we only pay attention to the case when 0<p,q<∞.The motivation for introducing the two-parameter limiting spaces(A0,A1){p,q} mainly stems from the fact the sum of the limiting spaces (A0,A1)0,p;K and (A0,A1)1,q;K, introduced by Cobos et al. [11] in connection with the interpolation over the unit square, is precisely (A0,A1)p,q. This fact is established in [9, Proposition 3.4] for p=q, and the same argument also works for arbitrary values of p and q. We further note that two different parameters have already been used in defining certain J-limiting spaces (see [12, Definition 3.1]).The main goal of this paper is to characterize the reiteration spaces(9)Ap0,q0,Ap1,q1p,q. Moreover, the assertions of [10, Theorem 4.3] have been extended by identifying the spaces (A{p0,q0},A1){p,q} and (A0,A{p1,q1}){p,q}.The classical identities (5)-(6) are based on the estimates which relate the K-functionals of the interpolated couples (Aθ0,q0,Aθ1,q1), (Aθ0,q0,A1), and (A0,Aθ1,q1) with that of the original couple (A0,A1) (see [13]). The main ingredient of our proofs will be the similar estimates for the limiting spaces (A0,A1){p,q}. 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.


## 2. Holmstedt-Type Estimates of theK-Functional

Letw be a positive weight on (0,∞), that is, a positive locally integrable function on (0,∞), and let 0<p,q<∞. Then, by Aw,p,q we will mean the real interpolation space AΦw,p,q, where Φw,p,q has the quasi-norm(10)gΦw,p,q=∫01wptgtpdtt1/p+∫1∞wqtgtqdtt1/q.Note that Aw,p,q=(A0,A1){p,q} for(11)wt=1,0<t<1,1t,t>1.In this section we present Holmstedt-type estimates for the real interpolation spacesAw,p,q, and we omit the proofs as they can be done as in [8, Section 2], where these estimates have been obtained for the case p=q.First, we formulate the results for the case(Aw0,p0,q0,A1). For this purpose we introduce some notations:(12)
I
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=
χ
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Φ
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h
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=
∫
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t
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p
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w
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d
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p
0
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0
<
t
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1
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max
⁡
ρ
0
t
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h
0
t
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t
>
0
.Subsequently, we will use the notationa1≲a2 for nonnegative quantities to mean that a1≤ca2 for some positive constant c which is independent of appropriate parameters involved in a1 and a2. If a1≲a2 and a2≲a1, we will put a1≈a2.Theorem 1.
For everyf∈A0+A1 and t>0, one has(13)KW0t,f;Aw0,p0,q0,A1≈It,f+ρ0tKt,ft.If ρ0≲h0, then(14)Kh0t,f;Aw0,p0,q0,A1≈It,f.Next, we present the estimates for the case(A0,Aw1,p1,q1). To this end, we denote (15)
J
t
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f
=
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t
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∞
s
K
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Φ
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1
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1
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min
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σ
1
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1
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>
0
.Theorem 2.
For everyf∈A0+A1 and t>0, one has(16)KW1t,f;A0,Aw1,p1,q1≈W1tJt,f+Kt,f.If g1≲σ1, then(17)Kg1t,f;A0,Aw1,p1,q1≈g1tJt,f.Finally, we state the results for the case(Aw0,p0,q0,Aw1,p1,q1). Set(18)gt=1χt,∞umin⁡u/h0u,u/ρ0uΦw1,p1,q1,(19)ht=χ0,tumin⁡g1u,σ1uΦw0,p0,q0.We assume that the weights w0 and w1 are such that h and 1/g are finite.Theorem 3.
For any weightw, all f∈A0+A1, and all t>0, one has(20)Kwt,f;Aw0,p0,q0,Aw1,p1,q1≲It,f+wtJt,f+wtσ1tKt,f+ρ0ttKt,f,(21)It,f≲Kht,f;Aw0,p0,q0,Aw1,p1,q1,(22)gtJt,f≲Kgt,f;Aw0,p0,q0,Aw1,p1,q1.Next we apply the above general results to obtain estimates for our limiting spaces.Corollary 4.
Let0<p0,q0<∞. Set (23)ψ0t=t1-ln⁡t1/p0,0<t<1,1+ln⁡t1/q0,t≥1.Then, for all 0<t<1,(24)Kψ0t,f;Ap0,q0,A1≈∫0tKp0s,fdss1/p0+ψ0tKt,ft,and, for all t≥1,(25)Kψ0t,f;Ap0,q0,A1≈∫01Kp0s,fdss1/p0+∫1tKs,fsq0dss1/q0.Proof.
We apply Theorem1 to the weight w0 given by(26)w0t=1,0<t<1,1t,t>1.We can easily compute that(27)ρ0t≈t1-ln⁡t1/p0,0<t<1,1,t≥1,(28)h0t≈t,0<t<1,1+ln⁡t1/q0,t≥1.Hence, (24) follows from (13), and (25) follows from (14).Corollary 5.
Let0<p1,q1<∞. Set (29)ψ1t=1-ln⁡t-1/p1,0<t<1,t1+ln⁡t-1/q1,t≥1.Then, for all 0<t<1,(30)Kψ1t,f;A0,Ap1,q1≈ψ1t∫t1Kp1s,fdss1/p1+ψ1t∫1∞Ks,fsq1dss1/q1,and, for all t≥1,(31)Kψ1t,f;A0,Ap1,q1≈ψ1t∫t∞Ks,fsq1dss1/q1+Kt,f.Proof.
This time we apply Theorem2 to the weight w1 given by(32)w1t=1,0<t<1,1t,t>1.We see that(33)σ1t≈1,0<t<1,t1+ln⁡t-1/q1,t≥1,(34)g1t≈1-ln⁡t-1/p1,0<t<1,t,t≥1.Therefore, (30) and (31) are consequences of (17) and (16), respectively.Corollary 6.
Let0<p1<p0<∞, and 0<q0<q1<∞. Set (35)ψ2t=1-ln⁡t1/p0-1/p1,0<t<1,1+ln⁡t1/q0-1/q1,t≥1. Then, for all 0<t<1,(36)Kψ2t,f;Ap0,q0,Ap1,q1≈∫0tKp0s,fdss1/p0+ψ2t∫t1Kp1s,fdss1/p1+ψ2t∫1∞Ks,fsq1dss1/q1, and, for all t≥1,(37)Kψ2t,f;Ap0,q0,Ap1,q1≈∫01Kp0s,fdss1/p0+∫1tKs,fsq0dss1/q0+ψ2t∫t∞Ks,fsq1dss1/q1.Proof.
We apply Theorem3 to the weights w0 and w1 defined in (26) and (32), respectively, and it will suffice to derive the following:(38)Kψ2t,f;AΦw0,p0,q0,AΦw1,p1,q1≈It,f+ψ2tJt,f,t>0.
Nowρ0, h0,  σ1, and g1 are given by (27), (28), (33), and (34), respectively. Firstly, we establish the estimate “≲” in (38). For this we note the following, since K(t,f) is nondecreasing in “t” and K(t,f)/t is nonincreasing in “t” (see, e.g., [3, Proposition 5.1.2]): (39)It,f≳h0tKt,ft,Jt,f≳Kt,fg1t.
In view ofρ0(t)≲h0(t),t>1, and g1(t)≲σ1(t),0<t<1, the previous two estimates yield(40)It,f≳ρ0tKt,ft,t>1,(41)Jt,f≳Kt,fσ1t,0<t<1.Furthermore, from (39), we can write(42)It,f≳ψ2tKt,fσ1t,t>1,(43)ψ2tJt,f≳ρ0tKt,ft,0<t<1.Therefore, if we take w=ψ2 in (20) then “≲” in (38) follows, for 0<t<1, from (41) and (43), and for t>1, from (40) and (42).
In order to obtain the reverse estimate, we exploit (21) and (22), and for this we need to compute h and g, defined by (18) and (19), respectively. Since (44)min⁡g1t,σ1t≈1-ln⁡t-1/p1,0<t<1,t1+ln⁡t-1/q1,t≥1, so, for all 0<t<1,(45)ht=∫0t1-ln⁡s-p0/p1dss1/p0≈1-ln⁡t1/p0-1/p1, as p1<p0. For all t>1,(46)ht=∫011-ln⁡s-p0/p1dss1/p0+∫1t1+ln⁡s-q0/q1dss1/q0, where the first integral is convergent thanks to the condition p1<p0. Therefore, (47)ht≈1+∫1t1+ln⁡s-q0/q1dss1/q0. Noting (48)∫1/et1+ln⁡s-q0/q1dss≲1+∫1t1+ln⁡s-q0/q1dss≤1+∫1tln⁡s-q0/q1dss, (where the convergence of the first and third integrals is being implied by the condition q0<q1), we get (49)1+ln⁡t1-q0/q1≲1+∫1t1+ln⁡s-q0/q1dss≲1+ln⁡t1-q0/q1.Therefore, h(t)≈(1+ln⁡t)1/q0-1/q1,t>1. Hence, h(t)≈ψ2(t),t>0. Similarly, we find that g(t)≈ψ2(t),t>0. Consequently, “≳” in (38) results from (21) and (22). The proof is finished.


## 3. 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<α<∞, put α′=α/1-α.Lemma 7.
Let1<α<∞. Then the inequality(50)∫01∫0tgsdsαwtdt≲∫01gαtvtdt holds for all nonnegative functions g on (0,1) if and only if (51)∫t1wsds1/α≲∫0tv1-α′sds-1/α′holds for all 0<t<1.Lemma 8.
Let1<α<∞. Then the inequality(52)∫01∫t1gsdsαwtdt≲∫01gαtvtdtholds for all nonnegative functions g on (0,1) if and only if (53)∫0twsds1/α≲∫t1v1-α′sds-1/α′holds for all 0<t<1.For the proof of the next result, we refer to [15, Theorem 3.3 (b)].Lemma 9.
Let0<α<1, and let k:(0,1)×(0,1)→(0,∞). Then the inequality(54)∫01∫01kt,sgsdsαwtdt≲∫01gαtvtdtholds for all nonnegative nondecreasing functions g on (0,1) if and only if (55)∫01∫r1kt,sdsαwtdt≲∫r1vtdtholds for all 0<r<1.Corollary 10.
Let0<α<∞; then (56)∫01∫0thsdssαdtt≲∫011-ln⁡tαhαtdtt holds for all nondecreasing nonnegative functions h on (0,1).Proof.
Forα=1, the inequality follows by interchanging the order of integration on the left side of the inequality. For 1<α<∞, the result follows from Lemma 7, applied with g(t)=h(t)/t,w(t)=1/t and v(t)=tα-11-ln⁡tα. Finally, putting g=h,k(t,s)=1/sχ(0,t)(s),w(t)=1/t and v(t)=t-11-ln⁡tα, we obtain the result, for 0<α<1, from Lemma 9.Corollary 11.
Let0<α<∞; then (57)∫1∞∫t∞lsdssαdtt≲∫1∞1+ln⁡tαlαtdttholds for all nonincreasing nonnegative functions l on (1,∞).Proof.
The proof follows by applying the previous lemma to the nondecreasing functionh(s)=l(1/s) on (0,1).Corollary 12.
Let0<α<∞; then (58)∫011-ln⁡t-α-1∫t1hsdssαdtt≈∫011-ln⁡t-1hαtdttholds for all nondecreasing nonnegative functions h on (0,1).Proof.
By interchanging the order of integration, we get the result forα=1. For 1<α<∞, the estimate “≲” results from Lemma 8, applied with gt=ht/t,w(t)=t-1(1-ln⁡t)-α-1 and v(t)=tα-11-ln⁡t-1. For 0<α<1, it follows from Lemma 9, by taking g=h,k(t,s)=1/sχ(t,1)(s),w(t)=t-1(1-ln⁡t)-α-1 and v(t)=t-11-ln⁡t-1. The other estimate “≳” follows from(59)∫1thsdss≥htln⁡tand the fact that 1+ln⁡t and ln⁡t are asymptotically the same as t→∞. The proof is finished.Corollary 13.
Let0<α<∞; then (60)∫1∞1+ln⁡t-α-1∫1tlsdssαdtt≈∫1∞1+ln⁡t-1hαtdtt holds for all nonincreasing nonnegative functions l on (1,∞).Proof.
Apply the previous lemma to the nonincreasing functionh(s)=l(1/s) 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 functionb is slowly varying on (1,∞) if, for all ε>0, the function t↦tεb(t) is equivalent to a nondecreasing function and t↦t-εb(t) is equivalent to a nonincreasing function. By symmetry, we say that b is slowly varying on (0,1) if the function t↦b(1/t) is slowly varying on (1,∞). Finally, b is slowly varying on (0,∞) if it is slowly varying on both (0,1) and (1,∞). For example, t↦(1+|ln⁡t|)α is slowly varying on (0,∞) for every real number α. We refer to [16] for details on slowly varying functions.Lemma 14.
Letψ be a positive function on (0,1) of the following form: (61)ψt=tαbt, where α>0 and b is a slowly varying function. Then there exists a positive function ϕ on (0,1) having the following properties:(i)
ϕ≈ψ;(ii)
ϕ is strictlyincreasing, locally absolutely continuous;(iii)
ϕ′(t)/ϕ(t)≈1/t;(iv)
limt→0+⁡ϕ(t)=0, and limt→1-⁡ϕ(t)=1.Proof.
Letb∗ be a nondecreasing function such that tα/2b(t)≈b∗(t). Set (62)ϕt=ct∫0tuα/2b∗udu,where c-1=∫01uα/2b∗(u)du; ϕ satisfies (i)–(iv) (see [8, Lemma 3.1]).Similarly, we can establish the following lemma.Lemma 15.
Letψ be a positive function on (1,∞) of the following form: (63)ψt=tαvt, where α>0 and v is a slowly varying function. Then there exists a positive function ϕ on (1,∞) having the following properties: (i)
ϕ≈ψ;(ii)
ϕ is strictlyincreasing, locally absolutely continuous;(iii)
ϕ′(t)/ϕ(t)≈1/t;(iv)
limt→1+⁡ϕ(t)=1, and limt→∞⁡ϕ(t)=∞.


## 4. Reiteration Theorems

Finally, we derive the reiteration theorems for our two-parameter limiting spacesA{p,q} by using the results of the previous two sections.Theorem 16.
Let0<p0,q0,p,q<∞. Then one has with equivalent norms(64)Ap0,q0,A1p,q=Aw,p,q, where (65)wt=1-ln⁡t1/p0,0<t<1,t-11+ln⁡t-1/q,t>1.Proof.
Put(66)I1=∫01Kpt1-ln⁡t1/p0,f;Ap0,q0,A1dtt1/p,(67)I2=∫1∞1+ln⁡t-q/q0-1Kq1+ln⁡t1/q0,f;Ap0,q0,A1dtt1/q.By Corollary 4, (68)I1≈I11+I12,I2≈I21+I22, where(69)I11=∫01∫0tKp0s,fdssp/p0dtt1/p,I12=∫011-ln⁡tp/p0Kpt,fdtt1/p,I21=∫01Kp0s,fdss1/p0,I22=∫1∞1+ln⁡t-q/q0-1∫1tKs,fsq0dssq/q0dtt1/q.
We note that(70)t⟼1tp0∫0tKp0s,fdssis nonincreasing since it is an integral average (with respect to the measure tp0-1) of a nonincreasing function Kp0(t,f)/tp0. Consequently, (71)I11≥∫01Kp0s,fdss1/p0∫01tp0dtt1/p,which gives I21≲I11. Next we apply Corollary 10 to the nondecreasing function h(t)=Kp0(t,f) to conclude that I11≲I12 and apply Corollary 13 to the nonincreasing function l(t)=K(t,f)/tq0 to find that(72)I22≈∫1∞1+ln⁡t-1Kt,ftqdtt1/q.Altogether, it follows that(73)I1+I2≈fAw,p,q.
On the other hand, in (66), we replace t(1-ln⁡t)1/p0 by an equivalent function ϕ on (0,1) (as obtained by Lemma 14) and make change of variable s=ϕ(t) to find that (74)I1≈∫01Kps,f;Ap0,q0,A1dss1/p.And making change of variable s=(1+ln⁡t)1/q0 in (67), we obtain (75)I2≈∫1∞Ks,f;Ap0,q0,A1sqdtt1/q.Combining the previous two estimates, we achieve(76)I1+I2≈fAp0,q0,A1p,q,which completes the proof in view of (73).Remark 17.
By takingp0=q0=p=q, we get back the first assertion of [10, Theorem 4.3].Theorem 18.
Let0<p1,q1,p,q<∞. Then we have with equivalent norms (77)A0,Ap1,q1p,q=Aw,p,q,where(78)wt=1+ln⁡1t-1/q,0<t<1,t-11+ln⁡t1/q1,t>1.Proof.
This time, putting(79)I1=∫011-ln⁡t-1Kp1-ln⁡t-1/p1,f;A0,Ap1,q1dtt1/p,(80)I2=∫1∞t-q1+ln⁡tq/q1Kqt1+ln⁡t-1/q1,f;A0,Ap1,q1dtt1/q,we obtain, by Corollary 5, that (81)I1≈I11+I12,I2≈I21+I22,where (82)I11=∫011-ln⁡t-p/p1-1∫t1Kp1s,fdssp/p1dss1/p,I12=∫1∞Ks,fsq1dss1/q1,I21=∫1∞∫t∞Ks,fsq1dssq/q1dss1/q,I22=∫1∞t-q1+ln⁡tq/q1Kqt,fdtt1/q.
In view of the fact that(83)t⟼tq1∫t∞Ks,fsq1dss is nondecreasing, it follows that I12≲I21. Moreover, by Corollaries 11 and 12, we get I21≲I22 and (84)I11≈∫011-ln⁡t-1Kpt,fdss1/p.Collecting all the previous estimates yields(85)I1+I2≈fAw,p,q.Next we make change of variables s=(1-ln⁡t)-1/p1 in (79) to obtain that (86)I1≈∫01Kps,f;A0,Ap1,q1dss1/p.As for change of variables in (80), we replace t(1+ln⁡t)-1/q1 by an equivalent function ϕ (obtained this time by Lemma 15) and put s=ϕ(t) to get that(87)I2≈∫1∞Ks,f;A0,Ap1,q1sqdtt1/q.Hence,(88)I1+I2≈fA0,Ap1,q1p,qwhich, along with (85), completes the proof.Remark 19.
The above theorem provides an extension of the second assertion of [10, Theorem 4.3].In order to describe our final result, we need two more scales of real interpolation spacesAw,p;v,q- and Aw,p;v,q+. Namely, Aw,p;v,q-=AΦ- and Aw,p;v,q+=AΦ+, where(89)gΦ-=∫0∞vqt∫0twpugupduuq/pdtt1/q,gΦ+=∫0∞bqt∫t∞wpugupduuq/pdtt1/q. The reader is referred to [8] for details on Aw,p;v,q- and Aw,p;v,q+. Recall that the quasi-norm on the intersection X∩Y of two quasi-normed spaces X and Y is given by(90)fX∩Y=max⁡fX,fY.Theorem 20.
Let0<p1<p0<∞,0<q0<q1<∞, and 0<p,q<∞. Then one has with equivalent norms(91)Ap0,q0,Ap1,q1p,q=Aw1,p,q∩Aw2,p0;v2,p-∩Aw3,q1;v3,q+,where (92)w1t=1-ln⁡t1/p0-1/p,0<t<1,1+ln⁡t1/q0-1/q,t>1,w2(t)=1, w3(t)=1/t, v2(t)=(1-ln⁡t)-1/pχ(0,1)(t), and v3(t)=(1+ln⁡t)-1/qχ(1,∞)(t).Proof.
Putting(93)I1=∫011-ln⁡t-1Kp1-ln⁡t1/p0-1/p1,f;Ap0,q0,Ap1,q1dtt1/p,I2=∫1∞1+ln⁡tq/q1-q/q0-1Kq1+ln⁡t1/q0-1/q1,Ap0,q0,Ap1,q1dtt1/q,and making appropriate change of variables, we observe that(94)I1+I2≈fAp0,q0,Ap1,q1p,q.On the other hand, by Corollary 6, we obtain (95)I1≈I11+I12+I13,I2≈I21+I22+I23, where(96)I11=∫011-ln⁡t-1∫0tKp0s,fdssp/p0dtt1/p,I12=∫011-ln⁡tp/p0-p/p1-1∫t1Kp1s,fdssp/p1dtt1/p,I13=∫1∞Ks,fsq1dss1/q1,I21=∫01Kp0s,fdss1/p0,I22=∫1∞1+ln⁡tq/q1-q/q0-1∫1tKs,fsq0dssq/q0dtt1/q,I23=∫1∞1+ln⁡t-1∫t∞Ks,fsq1dssq/q1dtt1/q.
Note thatI11=fAw2,p0;v2,p- and I23=Aw3,q1;v3,q+. Furthermore, we have(97)I12≈∫011-ln⁡tp/p0-1Kpt,fdtt1/p,I22=∫1∞1+ln⁡tq/q1-1Kt,ftqdtt1/q,thanks to Corollaries 12 and 13, respectively. Thus, I12+I22≈fAw1,p,q. Making use of the fact that (98)t⟼1tp0∫0tKp0s,fdss,t⟼tq1∫t∞Ks,fsq1dssare, respectively, nonincreasing and nondecreasing, we can further deduce that I21≲I11 and I13≲I23. Collecting all these estimates, we get at the following:(99)I1+I2≈fAw1,p,q+fAw2,p0;v2,p-+Aw3,q1;v3,q+. Combining this estimate with (94) finishes the proof.



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*Source: 8214643-2016-03-13.xml*