# Potential Operators on Cones of Nonincreasing Functions

Authors: Alexander Meskhi; Ghulam Murtaza
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

Necessary and sufficient conditions on weight pairs guaranteeing the two-weight inequalities for the potential operators(Iαf)(x)=∫0∞(f(t)/|x−t|1−α)dt and (ℐα1,α2f)(x,y)=∫0∞∫0∞(f(t,τ)/|x−t|1−α1|y−τ|1−α2)dtdτ on the cone of nonincreasing functions are derived. In the case of ℐα1,α2, we assume that the right-hand side weight is
of product type. The same problem for other mixed-type double potential operators is also studied. Exponents of the Lebesgue spaces are assumed to be between 1 and ∞.


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## Body

## 1. Introduction

Our aim is to derive necessary and sufficient conditions on weight pairs governing the boundedness of the following potential operators:(1.1)(Iαf)(x)=∫0∞f(t)|x-t|1-αdt,0<α<1,(Iα1,α2f)(x,y)=∬0∞f(t,τ)|x-t|1-α1|y-τ|1-α2dtdτ,0<α1,α2<1,
from Ldecp to Lq, where 1<p,q<∞.Historically, necessary and sufficient condition on a weight functionu, for which the boundedness of the one-dimensional Hardy transform
(1.2)(Hf)(x)=1x∫0xf(t)dt
from Ldecp(u,ℝ+) to Lp(u,ℝ+) holds, was established in [1]. Two-weight Hardy inequality criteria on cones of nonincreasing functions were derived in the paper [2]. The multidimensional analogues of these results were studied in [3–5]. Some characterizations of the two-weight inequality for the single integral operators involving Hardy-type transforms for monotone functions were given in [6–8]. The same problem for the Riesz potentials
(1.3)(Tαf)(x)=∫Rnf(y)|x-y|α-ndy,0<α<n,
for nonnegative nonincreasing radial functions was studied in [9].In the paper [10] necessary and sufficient conditions governing the boundedness of the multiple Riemann-Liouville transform
(1.4)(Rα1,α2f)(x,y)=∫0x∫0yf(t,τ)(x-t)1-α1(y-τ)1-α2dtdτ,0<α1,α2<1,
from Ldecp(w,ℝ+2) to Lp(v,ℝ+2) were derived, provided that w is a product of one-dimensional weights. Earlier, the problem of the boundedness of the two-dimensional Hardy transform H2=ℛ1,1 from Ldecp(w,ℝ+2) to Lp(v,ℝ+2) was studied in [4] under the condition that w and v have the following form: w(x,y)=w1(x)w2(y),v(x,y)=v1(x)v2(y).It should be emphasized that the two-weight problem for the Hardy-type transforms and fractional integrals with single kernels has been already solved. For the weight theory and history of these operators in classical Lebesgue spaces, we refer to the monographs [11–15] and references cited therein.The monograph [13] is dedicated to the two-weight problem for multiple integral operators in classical Lebesgue spaces (see also the papers [16–18] for criteria guaranteeing trace inequalities for potential operators with product kernels).Unfortunately, in the case of double potential operator, we assume that the right-hand weight is of product type and the left-hand one satisfies the doubling condition with respect to one of the variables. Even under these restrictions the two-weight criteria are written in terms of several conditions on weights. We hope to remove these restrictions on weights in our future investigations.Some of the results of this paper were announced without proofs in [19].Finally we mention that constants (often different constants in the same series of inequalities) will generally be denoted byc or C; by the symbol Tf≈Kf, where T and K are linear positive operators defined on appropriate classes of functions, we mean that there are positive constants c1 and c2 independent of f and x such that (Tf)(x)≤c1(Kf)(x)≤c2(Tf)(x); ℝ+ denotes the interval (0,∞) and p′ means the number p/(p-1) for 1<p<∞; W(x):=∫0xw(t)dt; Wj(xj):=∫0xjwj(t)dt; W(t1,…,tn):=Πi=1nWi(ti).


## 2. Preliminaries

We say that a functionf:ℝ+n→ℝ+ is nonincreasing if f is nonincreasing in each variable separately.Let𝒟 be the class of all nonnegative nonincreasing functions on ℝ+n. Suppose that u is measurable a.e. positive function (weight) on ℝ+n. We denote by Lp(u,ℝ+n), 0<p<∞, the class of all nonnegative functions on ℝ+n for which
(2.1)‖f‖Lp(u,R+n)≔(∫R+nfp(x1,…,xn)u(x1,…,xn)dx1⋯dxn)1/p=(∫R+nfp(x)u(x)dx)1/p<∞.
By the symbol Ldecp(u,ℝ+n) we mean the class Lp(u,ℝ+n)∩𝒟.The next statement regarding two-weight criteria for the Hardy operatorH on the cone of nonincreasing functions was proved in [2].Theorem A.
Letv and  w be weight functions on  ℝ+, and let  W(∞)=∞. (i)
Suppose that1<p≤q<∞. Then the inequality
(2.2)[∫0∞(Hf(x))qv(x)dx]1/q≤C[∫0∞(f(x))pw(x)dx]1/p,f∈Ldecp(w,R+),
holds if and only if the following two conditions are satisfied:
(2.3)sup⁡a>0(∫0av(x)dx)1/q(∫0aw(x)dx)-1/p<∞,sup⁡a>0(∫a∞v(x)xqdx)1/q(∫0aW-p′(x)xp′w(x)dx)1/p′<∞.(ii)
Let1<q<p<∞. Then H is bounded from Ldecp(w,ℝ+) to Lq(v,ℝ+) if and only if the following two conditions are satisfied:
(2.4)[∫0∞[(∫0tv(x)dx)1/pW-1/p(t)]rv(t)dt]1/r<∞,[∫0∞[(∫t∞x-qv(x)dx)1/p(∫0txp′W-p′(x)w(x)dx)1/p′]rtp′W-p′(t)w(t)dt]1/r<∞,
where r=pq/(p-q).The following statement was proved in [2] for n=1. For n≥1 we refer to [4].Proposition A.
Let1<p,q<∞. Suppose that  T is a positive integral operator defined on functions  f:ℝ+n→ℝ+, which are nonincreasing in each variable separately. Suppose that  T* is its formal adjoint. Let  w(x1,…,xn)=w1(x1)⋯wn(xn) be a product weight such that  Wi(∞)=∞,  i=1,…,n. Let  v be a general weight on  ℝ+n. Then the operator  T is bounded from  Ldecp(w,ℝ+n) to  Lp(v,ℝ+n) if and only if the inequality
(2.5)(∫R+n(∫0x1⋯∫0xnT*g)p′W-p′(x1,…,xn)w(x1,…,xn)dx1⋯dxn)1/p′≤c(∫R+ng(x)q′v1-q′(x)dx)1/q′
holds for all g≥0.LetRα be the Riemann-Liouville transform with single kernel
(2.6)(Rαf)(x)=∫0xf(t)(x-t)1-αdt,x∈R+,α>0.Ifα=1, then Rα is the Hardy transform. The Lp(w,ℝ+)→Lq(v,ℝ+) boundedness for R1 was characterized by Muckenhoupt ([20]) for p=q, and by Kokilashvili [21] and Bradley [22] for p<q (see also the monograph by Maz'ya [23] for these and relevant results).In the case when0<α<1, the Riemann-Liouville transform has singularity. For the results regarding the two-weight problem, in this case we refer, for example, to the monograph [11] and the references cited therein.The next result deals with the caseα>1 (see [24]).Theorem B.
Letα>1. Then the operator  Rα is bounded from  Lp(w,ℝ+) to  Lq(v,ℝ+) if and only if
(2.7)sup⁡t>0(∫t∞(x-t)(α-1)qv(x)dx)1/q(∫0tw1-p′(y)dy)1/p′<∞,sup⁡t>0(∫t∞v(x)dx)1/q(∫0t(t-x)(α-1)p′w1-p′(y)dy)1/p′<∞,
for 1<p≤q<∞ and
(2.8){∫0∞(∫t∞(x-t)(α-1)qv(x)dx)r/q(∫0tw1-p′(y)dy)r/q′w1-p′(t)dt}1/r<∞,{∫0∞(∫t∞v(x)dx)r/p(∫0t(t-y)(α-1)p′w1-p′(y)dy)r/p′v(t)dt}1/r<∞,
for 1<q<p<∞, where r is defined as follows: 1/r=1/q-1/p.Theorem C (see [10]).
Let1<p≤q<∞, and let  0<αi<1,  i=1,2. Assume that  v and  w are weights on  ℝ+2. Suppose also that  w(x1,x2)=w1(x1)w2(x2) for some one-dimensional weights  w1 and  w2 and that  Wi(∞)=∞,  i=1,2. Then the following conditions are equivalent: (a)
ℛα1,α2 is bounded from Ldecp(w,ℝ+2) to Lq(v,ℝ+2);(b)
the following four conditions hold simultaneously:
(i)
(2.9)sup⁡a1,a2>0(∫0a1∫0a2w(t1,t2)dt1dt2)-1/p(∫0a1∫0a2(t1α1t2α2)qv(t1,t2)dt1dt2)1/q<∞,(ii)
(2.10)sup⁡a1,a2>0(∫0a1∫0a2(t1t2)p′W-p′(t1,t2)w(t1,t2)dt1dt2)1/p′×(∫a1∞∫a2∞(t1α1-1t2α2-1)qv(t1,t2)dt1dt2)1/q<∞,(iii)
(2.11)sup⁡a1,a2>0(∫0a1w1(t1)dt1)-1/p(∫0a2t2p′W2-p′(t2)w2(t2)dt2)1/p′×(∫0a1∫a2∞t1qα1t2q(α2-1)v(t1,t2)dt1dt2)1/q<∞,(iv)
(2.12)sup⁡a1,a2>0(∫0a1t1p′W1-p′(t1)w1(t1)dt1)1/p′(∫0a2w2(t2)dt2)-1/p×(∫a1∞∫0a2t1q(α1-1)t2qα2v(t1,t2)dt1dt2)1/q<∞.In particular, Theorem C yields the trace inequality criteria on the cone of nonincreasing functions.Corollary A (see [10]).
Let1<p≤q<∞, and let 0<αi<1,  i=1,2.  Then the following conditions are equivalent:(a)
the boundedness ofℛα1,α2 from Ldecp(w,ℝ+2) to Lq(v,ℝ+2) holds for w≡1;(b)
(2.13)B1≔sup⁡a1,a2>0B1(a1,a2)≔sup⁡a1,a2>0(a1a2)-1/p(∫0a1∫0a2x1qα1x2qα2v(x1,x2)dx1dx2)1/q<∞;(c)
(2.14)B2≔sup⁡a1,a2>0B2(a1,a2)≔sup⁡a1,a2>0(a1a2)1/p′(∫a1∞∫a2∞x1q(α1-1)x2q(α2-1)v(x1,x2)dx1dx2)1/q<∞;(d)
(2.15)B3≔sup⁡a1,a2>0B3(a1,a2)≔sup⁡a1,a2>0a1-1/pa21/p′(∫0a1∫a2∞x1qα1x2q(α2-1)v(x1,x2)dx1dx2)1/q<∞;(e)
(2.16)B4≔sup⁡a1,a2>0B4(a1,a2)≔sup⁡a1,a2>0a11/p′a2-1/p(∫a1∞∫0a2x1q(α1-1)x2qα2v(x1,x2)dx1dx2)1/q<∞.


## 3. Potentials onℝ+

In this section we discuss the two-weight problem for the operatorIα. We begin with the following lemma.Lemma 3.1.
The following relation holds for nonnegative and nonincreasing functionf:
(3.1)(Rαf)(x)≈xαHf(x),
where H is the Hardy operator defined above.Proof.
We follow the proof of Proposition  3.1 of [10]. We have
(3.2)(Rαf)(x)=∫0x/2f(t)(x-t)1-αdt+∫x/2xf(t)(x-t)1-αdt≔J1(x)+J2(x).
Observe that if0<t<x/2, then (x-t)α-1≤21-αxα-1. Hence,
(3.3)J1(x)≤21-αxα-1∫0xf(t)dt=21-αxα(Hf)(x).
Further, since f is nonincreasing, we have that
(3.4)J2(x)≤α-1(x2)αf(x2)≤cαxα(Hf)(x).
Finally we have the upper estimate forRα.
The lower estimate is obvious because(x-t)α-1≥xα-1 for t≤x.In the next statement we assume thatWα is the operator given by
(3.5)(Wαf)(x)=∫x∞f(t)(t-x)1-αdt,α>0.Lemma 3.2.
Let1<p≤q<∞, and let  α>0. Suppose that  W(∞)=∞. Then the operator  Wα is bounded from  Ldecp(w,ℝ+) to  Lq(v,ℝ+) if and only if
(3.6)(∫0∞(∫0xg(t)(x-t)-αdt)p′W-p′(x)w(x)dx)1/p′≤c(∫0∞g(t)q′v1-q′(t)dt)1/q′,g≥0.Proof.
Taking Proposition A into account (forn=1), an integral operator
(3.7)(Tf)(x)=∫0∞k(x,y)f(y)dy
is bounded from Ldecp(w,ℝ+) to Lq(v,ℝ+) if and only if
(3.8)(∫0∞(∫0x(T*f)(τ)dτ)p′W-p′(x)w(x)dx)1/p′≤c(∫0∞f(t)q′v1-q′(t)dt)1/q′,f≥0,
where T* is a formal adjoint to T.
We have(3.9)∫0x(Rαf)(t)dt=∫0x(∫0tf(τ)(t-τ)1-αdτ)dt=∫0xf(τ)(∫0x-τduu1-α)dτ=1α∫0xf(τ)(x-τ)αdτ.
Taking T=Wα and T*=Rα, we derive the desired result.Now we formulate the main results of this section.Theorem 3.3.
Let1<p≤q<∞, and let  0<α<1. Suppose that  W(∞)=∞. Then  Iα is bounded from  Ldecp(w,ℝ+) to  Lq(v,ℝ+) if and only if
(3.10)sup⁡a>0A1(a,v,w)≔sup⁡a>0(∫0aw(t)dt)-1/p(∫0atαqv(t)dt)1/q<∞,(3.11)sup⁡a>0A2(a,v,w)≔sup⁡a>0(∫0atp′W-p′(t)w(t)dt)1/p′(∫a∞t(α-1)qv(t)dt)1/q<∞,(3.12)sup⁡a>0A3(a,v,w)≔sup⁡a>0(∫a∞W-p′(x)w(x)(x-a)αp′dx)1/p′(∫0av(x)dx)1/q<∞,(3.13)sup⁡a>0A4(a,v,w)≔sup⁡a>0(∫0aw(x)dx)-1/p(∫0av(x)(a-x)αqdx)1/q<∞.Theorem 3.4.
Let1<q<p<∞, and let  0<α<1.  W(∞)=∞. Then  Iα is bounded from  Ldecp(w,ℝ+) to  Lq(v,ℝ+) if and only if
(3.14)[∫R+[(∫0txαqv(x)dx)1/pW-1/p(t)]rtαqv(t)dt]1/r<∞,[∫R+[(∫t∞v(x)x(1-α)qdx)1/p(∫0tW-p′(x)w(x)x-p′)1/p′]rtp′W-p′(t)w(t)dt]1/r<∞,[∫R+[(∫t∞W-p′(x)w(x)(x-t)-αp′)1/p′(∫0tv(x)dx)1/p]rv(t)dt]1/r<∞,[∫R+[W-1(t)∫0tv(x)(t-x)-αqdx]r/qw(t)dt]1/r<∞,
where 1/r=1/q-1/p.Proof of Theorems3.3 and 3.4.
By using the representation(3.15)(Iαf)(x)=(Rαf)(x)+(Wαf)(x),x>0,
the obvious equality
(3.16)∫t∞W-p′(x)w(x)dx=cpW1-p′(t).
Theorems A and B and Lemmas 3.1 and 3.2, we have the desired results.Corollary 3.5.
Let1<p≤q<∞, and let  0<α<1/p. Then the operator  Iα is bounded from  Ldecp(1,ℝ+) to  Lq(v,ℝ+) if and only if
(3.17)B≔sup⁡a>0a(α-1/p)(∫0av(t)dt)1/q<∞.Proof.
Necessity follows immediately taking the test function fa(x)=χ(0,a)(x) in the two-weight inequality
(3.18)(∫0∞v(x)(Iαf(x))qdx)1/q≤c(∫0∞(f(x))pdx)1/p
and observing that Iαfa(x)≥∫0a(dt/|x-t|1-α)≥aα for x∈(0,a).
Sufficiency.
By Theorem3.3, it is enough to show that
(3.19)max⁡{A1,A2,A3,A4}≤cB,
where Ai:=supa>0Ai(a,v,1), i=1,2,3,4 (see Theorem 3.3 for the definition of Ai(a,v,w)).
The estimatesAi≤cB, i=1,4, are obvious. We show that Ai≤cB for i=2,3. We have
(3.20)A2q(a,v,1)=aq/p′∑k=0∞∫2ka2k+1at(α-1)qv(t)dt≤aq/p′∑k=0∞(2ka)(α-1)q(∫2ka2k+1av(t)dt)≤cBqaq/p′∑k=0∞(2ka)(α-1)q(2k+1a)(1/p-α)q=cBqaq/p′(∑k=0∞2-kq/p′)a-q/p′≤cBq.
Further, by the condition 0<α<1/p, we have that
(3.21)A3q(a,v,1)≤(∫a∞x(α-1)p′dx)1/p′(∫0av(t)dt)1/q=cα,paα-1/p(∫0av(t)dt)1/q≤cB.Definition 3.6.
Letρ be a locally integrable a.e. positive function on ℝ+. We say that ρ satisfies the doubling condition (ρ∈DC(ℝ+)) if there is a positive constant b>1 such that for all t>0 the following inequality holds:
(3.22)∫02tρ(x)dx≤bmin⁡⁡{∫0tρ(x)dx,∫t2tρ(x)dx}.Remark 3.7.
It is easy to check that ifρ∈DC(ℝ+), then ρ satisfies the reverse doubling condition: there is a positive constant b1>1 such that
(3.23)∫02tρ(x)dx≥b1max⁡⁡{∫0tρ(x)dx,∫t2tρ(x)dx}.
Indeed by (3.22) we have
(3.24)∫02tρ(x)dx≥1b∫02tρ(x)dx+∫t2tρ(x)dx.
Then
(3.25)∫02tρ(x)dx≥bb-1∫t2tρ(x)dx.
Analogously,
(3.26)∫02tρ(x)dx≥bb-1∫0tρ(x)dx.
Finally, we have (3.23).Corollary 3.8.
Let1<p≤q<∞, and let  0<α<1. Suppose that  W(∞)=∞. Suppose also that  v∈DC(ℝ+). Then  Iα is bounded from  Ldecp(w,ℝ+) to  Lq(v,ℝ+) if and only if condition (3.11) is satisfied.Proof.
Observe that by Remark3.7, for m0∈ℤ, the inequality
(3.27)∫02m0v(x)dx≤b1m0-k∫02kv(x)dx
holds for all k>m0, where b1 is defined in (3.23).
Leta>0. Then there is m0∈ℤ such that a∈[2m0,2m0+1). By applying (3.27) and the doubling condition for v, we find that
(3.28)(∫0aw(t)dt)-p′/p(∫0atαqv(t)dt)p′/q=c(∫a∞W-p′(t)w(t)dt)(∫0atαqv(t)dt)p′/q≤c(∫2m0∞W-p′(t)w(t)dt)(∫02m0+1tαqv(t)dt)p′/q≤c∑k=m0∞(∫2k2k+1W-p′(t)w(t)dt)(∫02m0+1v(t)dt)p′/q2m0αp′≤c∑k=m0∞(∫2k2k+1W-p′(t)w(t)dt)b1m0-k-1(∫02k+2v(t)dt)p′/q2m0αp′≤c∑k=m0∞b1m0-k-1(∫2k2k+1W-p′(t)w(t)dt)(∫2k+12k+2v(t)dt)p′/q2k(α-1)p′2kp′≤c∑k=m0∞b1m0-k-1(∫2k2k+1tp′W-p′(t)w(t)dt)(∫2k+12k+2v(t)t(α-1)qdt)p′/q≤c(sup⁡a>0A2(a,v,w))p′∑k=m0∞b1m0-k-1≤c(sup⁡a>0A2(a,v,w))p′.
So, we have seen that (3.11)⇒(3.10). Let us check now that (3.13)⇒(3.12).
Indeed, fora>0, we choose m0 so that a∈[2m0,2m0+1). Then, by using the condition v∈DC(ℝ+) and Remark 3.7,
(3.29)(∫a∞W-p′(x)w(x)(x-a)αp′dx)(∫0av(x)dx)p′/q≤(∫2m0∞W-p′(x)w(x)xαp′dx)(∫02m0+1v(x)dx)p′/q≤c∑k=m0∞2kαp′(∫2k2k+1W-p′(x)w(x)dx)(∫02m0+1v(x)dx)p′/q≤c∑k=m0∞2kαp′(∫2k2k+1W-p′(x)w(x)dx)b1m0-k+2(∫02k-1v(x)dx)p′/q≤c∑k=m0∞b1m0-k+2(∫2k∞W-p′(x)w(x)dx)(∫02kv(x)(2k-x)αqdx)p′/q≤c(sup⁡a>0A4(a,v,w))p′∑k=m0∞b1m0-k+2≤c(sup⁡a>0A4(a,v,w))p′.
Hence, (3.13)⇒(3.12) follows. Implication (3.11)⇒(3.13) follows in the same way as in the case of implication (3.11)⇒(3.10). The details are omitted.


## 4. Potentials with Multiple Kernels

In this section we discuss two-weight criteria for the potentials with product kernelsℐα1,α2.To derive the main results, we introduce the following multiple potential operators:(4.1)Wα1,α2f(x1,x2)=∫x1∞∫x2∞f(t1,t2)dt1dt2(t1-x1)1-α1(t2-x2)1-α2,(RW)α1,α2f(x1,x2)=∫0x1∫x2∞f(t1,t2)dt1dt2(x1-t1)1-α1(t2-x2)1-α2,(WR)α1,α2f(x1,x2)=∫x1∞∫0x2f(t1,t2)dt1dt2(t1-x1)1-α1(x2-t2)1-α2,
where x1,x2∈ℝ+, f≥0, and 0<αi<1, i=1,2.Definition 4.1.
One says that a locally integrable a.e. positive functionρ on ℝ+2 satisfies the doubling condition with respect to the second variable (ρ∈DC(y)) if there is a positive constant c such that for all t>0 and almost every x>0 the following inequality holds:
(4.2)∫02tρ(x,y)dy≤cmin⁡⁡{∫0tρ(x,y)dy,∫t2tρ(x,y)dy}.Analogously is defined the class of weightsDC(x).Remark 4.2.
Ifρ∈DC(y), then ρ satisfies the reverse doubling condition with respect to the second variable; that is, there is a positive constant c1 such that
(4.3)∫02tρ(x,y)dy≥c1max⁡⁡{∫0tρ(x,y)dy,∫t2tρ(x,y)dy}.Analogously,ρ∈DC(x)⇒ρ∈RDC(x). This follows in the same way as the single variable case (see Remark 3.7).Theorem C implies the next statement.Corollary B.
Let the conditions of Theorem C be satisfied.(i)
Ifv∈DC(x), then for the boundedness of ℛα1,α2 from Ldecp(w,ℝ+2) to Lq(v,ℝ+2), it is necessary and sufficient that conditions (2.10) and (2.12) are satisfied.(ii)
Ifv∈DC(y), then ℛα1,α2 is bounded from Ldecp(w,ℝ+2) to Lq(v,ℝ+2) if and only If conditions (2.10) and (2.11) are satisfied.(iii)
Ifv∈DC(x)∩DC(y), then ℛα1,α2 is bounded from Ldecp(w,ℝ+2) to Lq(v,ℝ+2) if and only if the condition (2.10) is satisfied.Proof of Corollary B.
The proof of this statement follows by using the arguments of the proof of Corollary3.8 (see Section 2) but with respect to each variable separately (also see Remark 4.2). The details are omitted.The following result concerns with the two-weight criteria for the two-dimensional operatorℛα1,α2 with α1,α2>1 (see [25], [13, Section  1.6]).Theorem D.
Let1<p≤q<∞, and let  α1,α2≥1. (i)
Suppose thatw1-p′∈DC(y). Then the operator ℛα1,α2 is bounded from Lp(w,ℝ+2) to Lq(v,ℝ+2) if and only if
(4.4)P1≔sup⁡a,b>0(∫0a∫0bw1-p′(x1,x2)(a-x1)(1-α1)p′dx1dx2)1/p′(∫a∞∫b∞v(x1,x2)x2(1-α2)qdx1dx2)1/q<∞,P2≔sup⁡a,b>0(∫0a∫0bw1-p′(x1,x2)dx1dx2)1/p′(∫a∞∫b∞v(x1,x2)(x1-a)(1-α1)qx2(1-α2)qdx1dx2)1/q<∞.
Moreover, ∥ℛα1,α2∥≈max{P1,P2}.(ii)
Letw1-p′∈DC(x). Then the operator ℛα1,α2 is bounded from Lp(w,ℝ+2) to Lq(v,ℝ+) if and only if
(4.5)P̃1≔sup⁡a,b>0(∫0a∫0bw1-p′(x1,x2)(b-x2)(1-α2)p′dx1dx2)1/p′(∫a∞∫b∞v(x1,x2)x1(1-α1)qdx1dx2)1/q<∞,P̃2≔sup⁡a,b>0(∫0a∫0bw1-p′(x1,x2)dx1dx2)1/p′(∫a∞∫b∞v(x1,x2)(x2-b)(1-α2)qx1(1-α1)qdx1dx2)1/q<∞.
Moreover, ∥ℛα1,α2∥≈max{P̃1,P̃2}.Let us introduce the following multiple integral operators:(4.6)(HR)α1,α2f(x1,x2)=x1α1-1∫0x1∫0x2f(t1,t2)dt1dt2(x2-t2)1-α2,(RH)α1,α2f(x1,x2)=x2α2-1∫0x1∫0x2f(t1,t2)dt1dt2(x1-t1)1-α1,(HW)α1,α2f(x1,x2)=x1α1-1∫0x1∫x2∞f(t1,t2)dt1dt2(t2-x2)1-α2,(WH)α1,α2f(x1,x2)=x2α2-1∫x1∞∫0x2f(t1,t2)dt1dt2(t1-x1)1-α1,(H′R)α1,α2f(x1,x2)=∫x1∞∫0x2f(t1,t2)dt1dt2t11-α1(x2-t2)1-α2,(RH′)α1,α2f(x1,x2)=∫0x1∫x2∞f(t1,t2)dt1dt2(x1-t1)1-α1t21-α2,(H′W)α1,α2f(x1,x2)=∫x1∞∫x2∞f(t1,t2)dt1dt2t11-α1(t2-x2)1-α2,(WH′)α1,α2f(x1,x2)=∫x1∞∫x2∞f(t1,t2)dt1dt2(t1-x1)1-α1t21-α2.Now we prove some auxiliary statements.Proposition 4.3.
Let1<p≤q<∞, and let  α1,α2≥1. Suppose that either  w(x1,x2)=w1(x1)w2(x2) or  v(x1,x2)=v1(x1)v2(x2) for some one-dimensional weights  w1,  w2,  v1, and  v2. (i)
The operator(ℛℋ)α1,α2 is bounded from Lp(w,ℝ+2) to Lq(v,ℝ+) if and only if
(4.7)Ĩ1≔sup⁡a,b>0(∫0a∫0bw1-p′(x1,x2)(a-x1)(1-α1)p′dx1dx2)1/p′(∫a∞∫b∞v(x1,x2)x2(1-α2)qdx1dx2)1/q<∞,Ĩ2≔sup⁡a,b>0(∫0a∫0bw1-p′(x1,x2)dx1dx2)1/p′(∫a∞∫b∞v(x1,x2)(x1-a)(1-α1)qx2(1-α2)qdx1dx2)1/q<∞.
Moreover, ∥(ℛℋ)α1,α2∥≈max{Ĩ1,Ĩ2}.(ii)
The operator(𝒲ℋ)α1,α2 is bounded from Lp(w,ℝ+2) to Lq(vℝ+) if and only if
(4.8)J̃1≔sup⁡a,b>0(∫0a∫b∞v(x1,x2)(a-x1)(1-α1)qx2(1-α2)qdx1dx2)1/q(∫a∞∫0bw1-p′(x1,x2)dx1dx2)1/q<∞,J̃2≔sup⁡a,b>0(∫0a∫b∞v(x1,x2)x2(1-α2)qdx1dx2)1/q(∫a∞∫0bw1-p′(x1,x2)(x1-a)(1-α1)p′dx1dx2)1/p′<∞.
Moreover, ∥(𝒲ℋ)α1,α2∥≈max{J̃1,J̃2}.(iii)
The operator(ℛℋ′)α1,α2 is bounded from Lp(w,ℝ+2) to Lq(v,ℝ+) if and only if
(4.9)J̃1′≔sup⁡a,b>0(∫a∞∫0bv(x1,x2)dx1dx2)1/q(∫0a∫b∞w1-p′(x1,x2)x2(1-α2)p′(a-x1)(1-α1)p′dx1dx2)1/p′<∞,J̃2′≔sup⁡a,b>0(∫a∞∫0bv(x1,x2)(x1-a)(1-α1)qdx1dx2)1/q(∫0a∫b∞w1-p′(x1,x2)x2(1-α2)p′dx1dx2)1/p′<∞.
Moreover, ∥(ℛℋ′)α1,α2∥≈max{J̃1′,J̃2′}.(iv)
The operator(𝒲ℋ′)α1,α2 is bounded from Lp(w,ℝ+2) to Lq(v,ℝ+) if and only if
(4.10)Ĩ1′≔sup⁡a,b>0(∫0a∫0bv(x1,x2)(a-x1)(1-α1)qdx1dx2)1/q(∫a∞∫b∞w1-p′(x1,x2)x2(1-α2)p′dx1dx2)1/p′<∞,Ĩ2′≔sup⁡a,b>0(∫0a∫0bv(x1,x2)dx1dx2)1/q(∫a∞∫b∞w1-p′(x1,x2)x2(1-α2)p′(x1-a)(1-α1)p′dx1dx2)1/p′<∞.
Moreover, ∥(𝒲ℋ′)α1,α2∥≈max{Ĩ1′,Ĩ2′}.Proof.
Letw(x1,x2)=w1(x1)w2(x2). The proof of the case v(x1,x2)=v1(x1)v2(x2) is followed by duality arguments. We prove, for example, part (i). Proofs of other parts are similar and, therefore, are omitted. We follow the proof of Theorem  3.4 of [25] (see also the proof of Theorem  1.1.6 in [13]).
Sufficiency.
First suppose thatS≔∫0∞w21-p′(x2)dx2=∞. Let {ak}k=-∞+∞ be a sequence of positive numbers for which the equality
(4.11)2k=∫0akw21-p′(x2)dx2
holds for all k∈ℤ. It is clear that {ak} is increasing and ℝ+=∪k∈ℤ[ak,ak+1). Moreover, it is easy to verify that
(4.12)2k=∫akak+1w21-p′(x2)dx2.
Let f≥0. We have that
(4.13)‖(RH)α1,α2f‖Lq(v,R+2)q=∫R+2v(x1,x2)((RH)α1,α2f)q(x1,x2)dx1dx2≤∑k∈Z∫0∞∫akak+1v(x1,x2)x2(1-α2)q(∫0x1∫0x2f(t1,t2)(x1-t1)1-α1dt1dt2)qdx1dx2≤∑k∈Z∫0∞(∫akak+1v(x1,x2)x2(1-α2)qdx2)(∫0x1(x1-t1)α1-1(∫0ak+1f(t1,t2)dt2)dt1)qdx1=∑k∈Z∫0∞Vk(x1)(∫0x1(x1-t1)(α1-1)Fk(t1)dt1)qdx1,
where
(4.14)Vk(x1)≔∫akak+1v(x1,x2)x2(1-α2)qdx2,Fk(t1)≔∫0ak+1f(t1,t2)dt2.
It is obvious that(4.15)Ĩ1q≥sup⁡a>0j∈Z(∫a∞∫ajaj+1v(x1,x2)(x1-a)(1-α1)qx2(1-α2)qdx1dx2)(∫0a∫0ajw1-p′(x1,x2)(a-x1)(1-α1)p′dx1dx2)q/p′,Ĩ2q≥sup⁡a>0j∈Z(∫a∞∫ajaj+1v(x1,x2)x2(1-α2)qdx1dx2)(∫0a∫0ajw1-p′(x1,x2)(a-x1)(1-α1)p′dx1dx2)q/p′.
Hence, by using the two-weight criteria for the one-dimensional Riemann-Liouville operator without singularity (see [24]), we find that
(4.16)‖(RH)α1,α2f‖Lq(v,R+2)q≤cĨq∑j∈Z[∫0∞w1(x1)(∫0ajw21-p′(x2)dx2)1-p(Fj(x1))pdx1]q/p≤cĨq[∫0∞w1(x1)∑j∈Z(∫0ajw21-p′(x2)dx2)1-p(∑k=-∞j∫akak+1f(x1,t2)dt2)pdx1]q/p,
where Ĩ=max{I1̃,I2̃}.
On the other hand, (4.11) yields
(4.17)∑k=n+∞(∫0akw21-p′(x2)dx2)1-p(∑k=-∞n∫akak+1w21-p′(x2)dx2)p-1=∑k=n+∞(∫0akw21-p′(x2)dx2)1-p(∫0an+1w21-p′(x2)dx2)p-1=(∑k=n+∞2k(1-p))2(n+1)(p-1)≤c
for all n∈ℤ. Hence by Hardy’s inequality in discrete case (see, for example, [25, 26]) and Hölder’s inequality we have that
(4.18)‖(RH)α1,α2f‖Lq(v,R+2)q≤cĨq[∫0∞w1(x1)∑j∈Z(∫ajaj+1w21-p′(x2)dx2)1-p(∫ajaj+1f(x1,t2)dt2)pdx1]q/p≤cĨq[∫0∞w1(x1)∑j∈Z(∫ajaj+1w2(t2)fp(x1,t2)dt2)dx1]q/p=cĨq‖f‖Lp(w,R+2)q.
IfS<∞, then without loss of generality we can assume that S=1. In this case we choose the sequence {ak}k=-∞0 for which (4.11) holds for all k∈ℤ-. Arguing as in the case of S=∞, we finally obtain the desired result.
Necessity follows by choosing the appropriate test functions. The details are omitted.
To prove, for example, (iii), we choose the sequence{xk} so that ∫xk∞w21-p′(x)dx=2k (notice that xk is decreasing) and argue as in the proof of (i).Proposition 4.4.
Let1<p≤q<∞, and let  α1,α2≥1. Suppose that either  w(x1,x2)=w1(x1)w2(x2) or  v(x1,x2)=v1(x1)v2(x2) for some one-dimensional weights:  w1,  w2,  v1, and  v2.
(i)
The operator(ℋℛ)α1,α2 is bounded from Lp(w,ℝ+2) to Lq(v,ℝ+2) if and only if
(4.19)I1≔sup⁡a,b>0(∫0a∫0bw1-p′(x1,x2)(b-x2)(1-α2)p′dx1dx2)1/p′(∫a∞∫b∞v(x1,x2)x1(1-α1)qdx1dx2)1/q<∞,I2≔sup⁡a,b>0(∫0a∫0bw1-p′(x1,x2)dx1dx2)1/p′(∫a∞∫b∞v(x1,x2)x1(1-α1)q(x2-b)(1-α2)qdx1dx2)1/q<∞.
Moreover, ∥(ℋℛ)α1,α2∥≈max{I1,I2}.(ii)
The operator(ℋ𝒲)α1,α2 is bounded from Lp(w,ℝ+2) to Lq(v,ℝ+) if and only if
(4.20)J1≔sup⁡a,b>0(∫a∞∫0bv(x1,x2)x1(1-α1)q(b-x2)(1-α2)qdx1dx2)1/q(∫0a∫b∞w1-p′(x1,x2)dx1dx2)1/p′<∞,J2≔sup⁡a,b>0(∫a∞∫0bv(x1,x2)x1(α1-1)qdx1dx2)1/q(∫0a∫b∞w1-p′(x1,x2)(x2-b)(1-α2)p′dx1dx2)1/p′<∞.
Moreover, ∥(ℋ𝒲)α1,α2∥≈max{J1,J2}.(iii)
The operator(ℋ′ℛ)α1,α2 is bounded from Lp(w,ℝ+2) to Lq(v,ℝ+) if and only if
(4.21)J1′≔sup⁡a,b>0(∫0a∫b∞v(x1,x2)dx1dx2)1/q(∫a∞∫0bw1-p′(x1,x2)x1(1-α1)p′(b-x2)(1-α2)p′dx1dx2)1/p′<∞,J2′≔sup⁡a,b>0(∫0a∫b∞v(x1,x2)(x2-b)(1-α2)qdx1dx2)1/q(∫a∞∫0bw1-p′(x1,x2)x1(1-α1)p′dx1dx2)1/p′<∞.
Moreover, ∥(ℋ′ℛ)α1,α2∥≈max{J1′,J2′}.(iv)
The operator(ℋ′𝒲)α1,α2 is bounded from Lp(w,ℝ+2) to Lq(v,ℝ+) if and only if
(4.22)I1′≔sup⁡a,b>0(∫0a∫0bv(x1,x2)(b-x2)(1-α2)qdx1dx2)1/q(∫a∞∫b∞w1-p′(x1,x2)x1(1-α1)p′dx1dx2)1/p′<∞,I2′≔sup⁡a,b>0(∫0a∫0bv(x1,x2)dx1dx2)1/q(∫a∞∫b∞w1-p′(x1,x2)x1(1-α1)p′(x2-b)(1-α2)p′dx1dx2)1/p′<∞.
Moreover, ∥(ℋ′𝒲)α1,α2∥≈max{I1′,I2′}.Proofof this proposition is similar to Proposition 4.3 by changing the order of variables.Theorem 4.5.
Let1<p≤q<∞, and let  0<α1,α2≤1. Suppose that the weight function  w on  ℝ+2 is of product type, that is,  w(x1,x2)=w1(x1)w2(x2). Suppose also that  W1(∞)=W2(∞)=∞.
(i)
Ifv∈DC(y), then 𝒲α1,α2 is bounded from Ldecp(w,ℝ+2) to Lq(v,ℝ+2) if and only if
(4.23)A1≔sup⁡a,b>0(∫0a∫0bv(x1,x2)(a-x1)α1qdx1dx2)1/q×(∫0aw1(x1)dx1)-1/p(∫b∞W2-p′(x2)w2(x2)x2α2p′dx2)1/p′<∞,(4.24)A2≔sup⁡a,b>0(∫0a∫0bv(x1,x2)dx1dx2)1/q×(∫a∞∫b∞W-p′(x1,x2)w(x1,x2)(x1-a)α1p′x2α2p′dx1dx2)1/p′<∞.(ii)
Ifv∈DC(x), then 𝒲α1,α2 is bounded from Ldecp(w,ℝ+2) to Lq(v,ℝ+2) if and only if
(4.25)B1≔sup⁡a,b>0(∫0a∫0bv(x1,x2)(b-x2)α2qdx1dx2)1/q×(∫a∞W1-p′(x1)w1(x1)x1α1p′dx1)1/p′(∫0bw2(x2)dx2)-1/p<∞,B2≔sup⁡a,b>0(∫0a∫0bv(x1,x2)dx1dx2)1/q×(∫a∞∫b∞W-p′(x1,x2)w(x1,x2)(x2-b)α2p′x1α1p′dx1dx2)1/p′<∞.(iii)
Ifv∈DC(x)∩DC(y), then 𝒲α1,α2 is bounded from Ldecp(w,ℝ+2) to Lq(v,ℝ+2) if and only if
(4.26)C1≔sup⁡a,b>0(∫a∞∫b∞W-p′(x1,x2)w(x1,x2)x2α2p′x1α1p′dx1dx2)1/p′×(∫0a∫0bv(x1,x2)dx1dx2)1/q<∞.Proof.
By using Proposition A we see that the operator𝒲α1,α2 is bounded from Ldecp(w,ℝ+2) to Lq(v,ℝ+2) if and only if the inequality
(4.27)(∫R+2(∫0x1∫0x2[∫0τ1∫0τ2g(t1,t2)dt1dt2(τ1-t1)1-α1(τ2-t2)1-α2]dτ1dτ2)p′×W-p′(x1,x2)w(x1,x2)dx1dx2∫R+2(∫0x1∫0x2[∫0τ1∫0τ2g(t1,t2)dt1dt2(τ1-t1)1-α1(τ2-t2)1-α2]dτ1dτ2)p′)1/p′≤c(∫R+2gq′v1-q′)1/q′
holds for all g≥0. Further, it is easy to see that
(4.28)∫0x1∫0x2[∫0τ1∫0τ2g(t1,t2)dt1dt2(τ1-t1)1-α1(τ2-t2)1-α2]dτ1dτ2=∫0x1∫0x2g(t1,t2)[∫t1x1∫t2x2dτ1dτ2(τ1-t1)1-α1(τ2-t2)1-α2]dt1dt2=cα1,α2∫0x1∫0x2g(t1,t2)(x1-t1)α1(x2-t2)α2dt1dt2.
Hence 𝒲α1,α2 is bounded from Ldecp(w,ℝ+2) to Lq(v,ℝ+2) if and only if ℛα1+1,α2+1 is bounded from Lq′(v1-q′,ℝ+2) to Lp′(W-p′w,ℝ+2).
By using Theorem D, (i) and (ii) follow immediately.
To prove (iii) we show that ifv∈DC(x)∩DC(y), then (4.26) implies (4.23) and (4.24). Let a,b>0. Then a∈[2m0,2m0+1) for some m0∈ℤ. By using the doubling condition with respect to the first variable uniformly to the second one and Remark 4.2, we see that
(4.29)(∫0a∫0bv(x1,x2)(a-x1)α1qdx1dx2)p′/q(∫0aw1(x1)dx1)-p′/p=c(∫0a∫0bv(x1,x2)(a-x1)α1qdx1dx2)p′/q(∫a∞W1-p′(x1)w1(x1)dx1)≤c∑k=m0∞(∫2k2k+1W1-p′(x1)w1(x1)dx1)2(m0+1)α1p′(∫02m0∫0bv(x1,x2)dx1dx2)p′/q≤c∑k=m0∞(∫2k2k+1x1α1p′W1-p′(x1)w1(x1)dx1)c1(m0-k)(p′/q)(∫02k∫0bv(x1,x2)dx1dx2)p′/q≤cC1p′(∫b∞W2-p′(x2)w2(x2)x2α2p′dx2)-1.
Hence, A1≤C1. In a similar manner we can show that A2≤C1.
For necessity, let us see, for example, that (4.23) implies (4.26). For a∈[2m0,2m0+1), by using the doubling condition for v with respect to the first variable and Remark 4.2, we have
(4.30)(∫0a∫0bv(x1,x2)dx1dx2)p′/q(∫a∞W1-p′(x1)w(x1)x1α1p′dx1)≤c∑k=m0∞(∫2k2k+1W1-p′(x1)w(x1)dx1)2kα1p′(∫02m0+1∫0bv(x1,x2)dx1dx2)p′/q≤c∑k=m0∞(∫2k2k+1W1-p′(x1)w(x1)dx1)c1(m0-k+2)(p′/q)(∫02k-1∫0b(2k-x1)α1qv(x1,x2)dx1dx2)p′/q≤cA1p′(∫b∞W2-p′(x2)w2(x2)x2α2p′dx2)-1.
Hence, taking the supremum with respect to a and b, we find that C1≤cA1.The following statements give analogous statement for the mixed-type operator(ℛ𝒲)α1,α2 and (𝒲ℛ)α1,α2.Theorem 4.6.
Let1<p≤q<∞, and let  0<α1,α2≤1. Suppose that the weight function  w on  ℝ+2 is of product type, that is,  w(x1,x2)=w1(x1)w2(x2). Suppose also that  W1(∞)=W2(∞)=∞.
(i)
The operator(ℛ𝒲)α1,α2 is bounded from Ldecp(w,ℝ+2) to Lq(v,ℝ+2) if and only if
(4.31)sup⁡a,b>0(∫0a∫0bx1α1qv(x1,x2)(b-x2)-α2qdx1dx2)1/q(∫0a∫0bw1(x1)w2(x2)dx1dx2)-1/p<∞,(4.32)sup⁡a,b>0(∫0a∫0bx1α1qv(x1,x2)dx1dx2)1/q(∫0aw1(x1)dx1)-1/p×(∫b∞W2-p′(x2)w2(x2)(x2-b)α2p′dx2)1/p′<∞,(4.33)sup⁡a,b>0(∫a∞∫0bv(x1,x2)x1(1-α1)q(b-x2)-α2qdx1dx2)1/q(∫0ax1p′W1-p′(x1)w1(x1)dx1)1/p′×(∫0bw2(x2)dx2)-1/p<∞,(4.34)sup⁡a,b>0(∫a∞∫0bx1(α1-1)qv(x1,x2)dx1dx2)1/q(∫0a∫b∞W-p′(x1,x2)w(x1,x2)x1p′(x2-b)-α2p′dx1dx2)1/p′<∞.(ii)
The operator(𝒲ℛ)α1,α2 is bounded from Ldecp(w,ℝ+2) to Lq(v,ℝ+2) if and only if
(4.35)sup⁡a,b>0(∫0a∫0bx2α2qv(x1,x2)(a-x1)-α1qdx1dx2)1/q(∫0a∫0bw1(x1)w2(x2)dx1dx2)-1/p<∞,(4.36)sup⁡a,b>0(∫0a∫0bx2α2qv(x1,x2)dx1dx2)1/q(∫0bw2(x2)dx2)-1/p×(∫a∞W1-p′(x1)w1(x1)(x1-a)α1p′dx1)1/p′<∞,(4.37)sup⁡a,b>0(∫0a∫b∞v(x1,x2)x2(1-α2)q(a-x1)-α1qdx1dx2)1/q(∫0aw1(x1)dx1)-1/p×(∫0bx2p′W2-p′(x2)w2(x2)dx2)1/p′<∞,(4.38)sup⁡a,b>0(∫0a∫b∞x2(α2-1)qv(x1,x2)dx1dx2)1/q(∫a∞∫0bW-p′(x1,x2)w(x1,x2)x2p′(x1-a)-α1p′dx1dx2)1/p′<∞.Proof.
We provepart (i). The proof of part (ii) is similar by changing the order of variables.
First we show that the two-sided pointwise relation(ℛ𝒲)α1,α2f≈(ℋ𝒲)α1,α2f, f↓, holds. Indeed, by using the fact that f is nonincreasing in the first variable, we find that
(4.39)(RW)α1,α2f(x1,x2)=∫0x1/2∫x2∞(⋯)+∫x1/2x1∫x2∞(⋯)≤c′α1x1α1-1∫0x1/2∫x2∞f(t1,t2)(t2-x2)1-α2dt1dt2+cα1′′x1α1-1∫0x1/2∫x2∞f(t1,t2)(t2-x2)1-α2dt1dt2≤cα1,α2(HW)α1,α2f(x1,x2).
The inequality
(4.40)(HW)α1,α2f(x1,x2)≤(RW)α1,α2f(x1,x2)
is obvious because x1-t1≤x1 for 0<t1≤x1.
Further, it is easy to check that(4.41)∫0x1∫0x2(∫τ1∞∫0τ2g(t1,t2)t11-α1(τ2-t2)1-α2dt1dt2)dτ1dτ2=∫0x1∫0x2(∫τ1x1∫0τ2g(t1,t2)t11-α1(τ2-t2)1-α2dt1dt2)dτ1dτ2+∫0x1∫0x2(∫x1∞∫0τ2g(t1,t2)t11-α1(τ2-t2)1-α2dt1dt2)dτ1dτ2=∫0x1∫0x2g(t1,t2)t1α1-1(∫0t1∫t2x2(τ2-t2)α2-1dτ1dτ2)dt1dt2+∫x1∞∫0x2g(t1,t2)t1α1-1(∫0x1∫t2x2(τ2-t2)α2-1dτ1dτ2)dt1dt2=c∫0x1∫0x2g(t1,t2)t1α1(x2-t2)α2dt1dt2+cx1∫x1∞∫0x2g(t1,t2)t1α1-1(x2-t2)α2dt1dt2.
Hence, since the boundedness of(ℋ𝒲)α1,α2 from Ldecp(w,ℝ+2) to Lq(v,ℝ+2) is equivalent to the inequality (see also [4])
(4.42)(∫R+2(∫0x1∫0x2[∫τ1∞∫0τ2g(t1,t2)dt1dt2t11-α1(τ2-t2)1-α2]dτ1dτ2)p′W-p′(x1,x2)w(x1,x2)dx1dx2)1/p′≤c(∫R+2gq′v1-q′)1/q′,
we can conclude that Proposition 4.4 yields the desired result.Proposition 4.7.
Let the conditions of Theorem4.6 be satisfied. Then(i)
ifv∈DC(x), then (ℛ𝒲)α1,α2 is bounded from Ldecp(w,ℝ+2) to Lq(v,ℝ+2) if and only if (4.33) and (4.34) hold;(ii)
ifv∈DC(y), then (ℛ𝒲)α1,α2 is bounded from Ldecp(w,ℝ+2) to Lq(v,ℝ+2) if and only if (4.32) and (4.34) are satisfied;(iii)
ifv∈DC(x)∩DC(y), then (ℛ𝒲)α1,α2 is bounded from Ldecp(w,ℝ+2) to Lq(v,ℝ+2) if and only if (4.34) holds.Proof.
(i) Taking into account the arguments used in Theorem4.5, we can prove that (4.34) implies (4.32) and (4.33) implies (4.31).
(ii) It can be checked that (4.32) implies (4.31) and (4.34) implies (4.33). To show that, for example, (4.32) implies (4.31), we take a,b>0. Then b∈[2m0,2m0+1) for some integer m0. By using the doubling condition for v with respect to the second variable, we have
(4.43)(∫0a∫0bx1α1qv(x1,x2)(b-x2)α2qdx1dx2)p′/q(∫0bw2(x2)dx2)-p′/q≤c(∫0a∫02m0+1x1α1qv(x1,x2)dx1dx2)p′/q(∫2m0∞W2-p′(x2)w2(x2)dx2)2(m0+1)α2p′≤c∑k≥m0(∫2k2k+1W2-p′(x2)w2(x2)dx2)(∫0a∫02k-1x1α1qv(x1,x2)dx1dx2)p′/q×c1(m0-k)p′/q2(m0+1)α2p′≤c∑k≥m0(∫2k2k+1W2-p′(x2)w2(x2)(x2-2k-1)α2p′dx2)(∫0a∫02k-1x1α1qv(x1,x2)dx1dx2)p′/q×c1(m0-k)p′/q≤c(∫0aw1(x1)dx1)1/p.
By a similar manner it follows that (4.34) implies (4.33). The proof of (iii) is similar, and we omit it.The proof of the next statement is similar to that of Proposition4.7.Proposition 4.8.
Let the conditions of Theorem4.6 be satisfied. Then(i)
ifv∈DC(x), then (𝒲ℛ)α1,α2 is bounded from Ldecp(w,ℝ+2) to Lq(v,ℝ+2) if and only if (4.36) and (4.38) hold;(ii)
ifv∈DC(y), then (𝒲ℛ)α1,α2 is bounded from Ldecp(w,ℝ+2) to Lq(v,ℝ+2) if and only if (4.37) and (4.38) are satisfied;(iii)
ifv∈DC(x)∩DC(y), then (𝒲ℛ)α1,α2 is bounded from Ldecp(w,ℝ+2) to Lq(v,ℝ+2) if and only if (4.38) holds.Now we are ready to discuss the operatorsℐα1,α2 on the cone of nonincreasing functions.Theorem 4.9.
Let1<p≤q<∞, and let  0<α1,α2<1. Suppose that the weight  v belongs to the class  DC(y). Let  w(x1,x2)=w1(x1)w2(x2) for some one-dimensional weight functions  w1 and  w2 and  W1(∞)=W2(∞)=∞. Then the operator  ℐα1,α2 is bounded from  Ldecp(w,ℝ+2) to  Lq(v,ℝ+2) if and only if conditions (2.10), (2.11), (4.23), (4.24), (4.32), (4.34), (4.37), and (4.38) are satisfied.Theorem 4.10.
Let1<p≤q<∞, and let  0<α1,α2<1. Suppose that the weight  v belongs to the class  DC(x). Let  w(x1,x2)=w1(x1)w2(x2) for some one-dimensional weight functions  w1 and  w2 and  W1(∞)=W2(∞)=∞. Then the operator  ℐα1,α2 is bounded from  Ldecp(w,ℝ+2) to  Lq(v,ℝ+2) if and only if conditions (2.10), (2.12), (4.25), (4.33), (4.34), (4.36), and (4.38) are satisfied.Theorem 4.11.
Let1<p≤q<∞, and let  0<α1,α2<1. Suppose that the weight  v∈DC(x)∩DC(y). Let  w(x1,x2)=w1(x1)w2(x2) for some one-dimensional weight functions  w1 and  w2 and  W1(∞)=W2(∞)=∞. Then the operator  ℐα1,α2 is bounded from  Ldecp(w,ℝ+2) to  Lq(v,ℝ+2) if and only if conditions (2.10), (4.26), (4.34), and (4.38) are satisfied.Proofs of these statements follow immediately from the pointwise estimate
(4.44)Iα1,α2f=Rα1,α2f+Wα1,α2f+RWα1,α2f+WRα1,α2f.
Corollary B, Theorem 4.5, and Propositions 4.7 and 4.8.The next statement shows that the two-weight inequality forℐα1,α2 can be characterized by one condition when w≈1.Corollary 4.12.
Let1<p≤q<∞, and let  0<α1,α2<1/p. Suppose that  v∈DC(x)∪DC(y). Then the operator  ℐα1,α2 is bounded from  Ldecp(1,ℝ+2) to  Lq(v,ℝ+2) if and only if
(4.45)D≔sup⁡a,b>0a(α1-(1/p))b(α2-(1/p))(∫0a∫0bv(t,τ)dtdτ)1/q<∞.Proof.
Necessity can be derived by substituting the test function fa,b(x)=χ(0,a)×(0,b)(x) in the two-weight inequality for ℐα1,α2.
Sufficiency follows by using Theorems 4.9 and 4.10 and the arguments of the proof of Corollary 3.5 with respect to each variable. Details are omitted.



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*Source: 474681-2012-02-01.xml*