JFSAJournal of Function Spaces and Applications0972-68022090-8997Hindawi Publishing Corporation47468110.1155/2012/474681474681Research ArticlePotential Operators on Cones of Nonincreasing FunctionsMeskhiAlexander1, 2MurtazaGhulam3KokilashviliV. M.1A. Razmadze Mathematical InstituteIvane Javakhishvili Tbilisi State University2 University Street, 0143 TbilisiGeorgiatsu.ge2Faculty of Informatics and Control SystemsGeorgian Technical University77 Kostava Street, TbilisiGeorgiagtu.edu.ge3Abdus Salam School of Mathematical SciencesGC University68-B New Muslim TownLahorePakistangcu.edu.pk2012122011201216052011150620112012Copyright © 2012 Alexander Meskhi and Ghulam Murtaza.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Necessary and sufficient conditions on weight pairs guaranteeing the two-weight inequalities for the potential operators (Iαf)(x)=0(f(t)/|xt|1α)dt and (α1,α2f)(x,y)=00(f(t,τ)/|xt|1α1|yτ|1α2)dtdτ on the cone of nonincreasing functions are derived. In the case of α1,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 ∞.

1. Introduction

Our aim is to derive necessary and sufficient conditions on weight pairs governing the boundedness of the following potential operators: (Iαf)(x)=0f(t)|x-t|1-αdt,0<α<1,(Iα1,α2f)(x,y)=0f(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 function u, for which the boundedness of the one-dimensional Hardy transform (Hf)(x)=1x0xf(t)dt from Ldecp(u,+) to Lp(u,+) holds, was established in . Two-weight Hardy inequality criteria on cones of nonincreasing functions were derived in the paper . The multidimensional analogues of these results were studied in . Some characterizations of the two-weight inequality for the single integral operators involving Hardy-type transforms for monotone functions were given in . The same problem for the Riesz potentials (Tαf)(x)=Rnf(y)|x-y|α-ndy,0<α<n, for nonnegative nonincreasing radial functions was studied in .

In the paper  necessary and sufficient conditions governing the boundedness of the multiple Riemann-Liouville transform (Rα1,α2f)(x,y)=0x0yf(t,τ)(x-t)1-α1(y-τ)1-α2dtdτ,0<α1,α2<1, from Ldecp(w,+2) to Lp(v,+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=1,1 from Ldecp(w,+2) to Lp(v,+2) was studied in  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  and references cited therein.

The monograph  is dedicated to the two-weight problem for multiple integral operators in classical Lebesgue spaces (see also the papers  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 .

Finally we mention that constants (often different constants in the same series of inequalities) will generally be denoted by c or C; by the symbol TfKf, 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); + 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 function f:+n+ is nonincreasing if f is nonincreasing in each variable separately.

Let 𝒟 be the class of all nonnegative nonincreasing functions on +n. Suppose that u is measurable a.e. positive function (weight) on +n. We denote by Lp(u,+n), 0<p<, the class of all nonnegative functions on +n for which fLp(u,R+n)(R+nfp(x1,,xn)u(x1,,xn)dx1dxn)1/p=(R+nfp(x)u(x)dx)1/p<. By the symbol Ldecp(u,+n) we mean the class Lp(u,+n)𝒟.

The next statement regarding two-weight criteria for the Hardy operator H on the cone of nonincreasing functions was proved in .

Theorem A.

Let v and w be weight functions on +, and let W()=.

Suppose that 1<pq<. Then the inequality [0(Hf(x))qv(x)dx]1/qC[0(f(x))pw(x)dx]1/p,fLdecp(w,R+), holds if and only if the following two conditions are satisfied: supa>0(0av(x)dx)1/q(0aw(x)dx)-1/p<,supa>0(av(x)xqdx)1/q(0aW-p(x)xpw(x)dx)1/p<.

Let 1<q<p<. Then H is bounded from Ldecp(w,+) to Lq(v,+) if and only if the following two conditions are satisfied: [0[(0tv(x)dx)1/pW-1/p(t)]rv(t)dt]1/r<,[0[(tx-qv(x)dx)1/p(0txpW-p(x)w(x)dx)1/p]rtpW-p(t)w(t)dt]1/r<, where r=pq/(p-q).

The following statement was proved in  for n=1. For n1 we refer to .

Proposition A.

Let 1<p,  q<. Suppose that T is a positive integral operator defined on functions f:+n+, 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 +n. Then the operator T is bounded from Ldecp(w,+n) to Lp(v,+n) if and only if the inequality (R+n(0x10xnT*g)pW-p(x1,,xn)w(x1,,xn)dx1dxn)1/pc(R+ng(x)qv1-q(x)dx)1/q holds for all g0.

Let Rα be the Riemann-Liouville transform with single kernel (Rαf)(x)=0xf(t)(x-t)1-αdt,xR+,  α>0.

If α=1, then Rα is the Hardy transform. The Lp(w,+)Lq(v,+) boundedness for R1 was characterized by Muckenhoupt () for p=q, and by Kokilashvili  and Bradley  for p<q (see also the monograph by Maz'ya  for these and relevant results).

In the case when 0<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  and the references cited therein.

The next result deals with the case α>1 (see ).

Theorem B.

Let α>1. Then the operator Rα is bounded from Lp(w,+) to Lq(v,+) if and only if supt>0(t(x-t)(α-1)qv(x)dx)1/q(0tw1-p(y)dy)1/p<,supt>0(tv(x)dx)1/q(0t(t-x)(α-1)pw1-p(y)dy)1/p<, for 1<pq< and {0(t(x-t)(α-1)qv(x)dx)r/q(0tw1-p(y)dy)r/qw1-p(t)dt}1/r<,{0(tv(x)dx)r/p(0t(t-y)(α-1)pw1-p(y)dy)r/pv(t)dt}1/r<, for 1<q<p<, where r is defined as follows: 1/r=1/q-1/p.

Theorem C (see [<xref ref-type="bibr" rid="B22">10</xref>]).

Let 1<pq<, and let 0<αi<1, i=1,2. Assume that v and w are weights on +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:

α1,α2 is bounded from Ldecp(w,+2) to Lq(v,+2);

the following four conditions hold simultaneously:

supa1,a2>0(0a10a2w(t1,t2)dt1dt2)-1/p(0a10a2(t1α1t2α2)qv(t1,t2)dt1dt2)1/q<,

supa1,a2>0(0a10a2(t1t2)pW-p(t1,t2)w(t1,t2)dt1dt2)1/p×(a1a2(t1α1-1t2α2-1)qv(t1,t2)dt1dt2)1/q<,

supa1,a2>0(0a1w1(t1)dt1)-1/p(0a2t2pW2-p(t2)w2(t2)dt2)1/p×(0a1a2t1qα1t2q(α2-1)v(t1,t2)dt1dt2)1/q<,

supa1,a2>0(0a1t1pW1-p(t1)w1(t1)dt1)1/p(0a2w2(t2)dt2)-1/p×(a10a2t1q(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 [<xref ref-type="bibr" rid="B22">10</xref>]).

Let 1<pq<, and let 0<αi<1, i=1,2. Then the following conditions are equivalent:

the boundedness of α1,α2 from Ldecp(w,+2) to Lq(v,+2) holds for w1;

B1supa1,a2>0B1(a1,a2)supa1,a2>0(a1a2)-1/p(0a10a2x1qα1x2qα2v(x1,x2)dx1dx2)1/q<;

B2supa1,a2>0B2(a1,a2)supa1,a2>0(a1a2)1/p(a1a2x1q(α1-1)x2q(α2-1)v(x1,x2)dx1dx2)1/q<;

B3supa1,a2>0B3(a1,a2)supa1,a2>0a1-1/pa21/p(0a1a2x1qα1x2q(α2-1)v(x1,x2)dx1dx2)1/q<;

B4supa1,a2>0B4(a1,a2)supa1,a2>0a11/pa2-1/p(a10a2x1q(α1-1)x2qα2v(x1,x2)dx1dx2)1/q<.

3. Potentials on <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M132"><mml:mrow><mml:msub><mml:mrow><mml:mi>ℝ</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>

In this section we discuss the two-weight problem for the operator Iα. We begin with the following lemma.

Lemma 3.1.

The following relation holds for nonnegative and nonincreasing function f: (Rαf)(x)xαHf(x), where H is the Hardy operator defined above.

Proof.

We follow the proof of Proposition  3.1 of . We have (Rαf)(x)=0x/2f(t)(x-t)1-αdt+x/2xf(t)(x-t)1-αdtJ1(x)+J2(x).

Observe that if 0<t<x/2, then (x-t)α-121-αxα-1. Hence, J1(x)21-αxα-10xf(t)dt=21-αxα(Hf)(x). Further, since f is nonincreasing, we have that J2(x)α-1(x2)αf(x2)cαxα(Hf)(x).

Finally we have the upper estimate for Rα.

The lower estimate is obvious because (x-t)α-1xα-1 for tx.

In the next statement we assume that Wα is the operator given by (Wαf)(x)=xf(t)(t-x)1-αdt,α>0.

Lemma 3.2.

Let 1<pq<, and let α>0. Suppose that W()=. Then the operator Wα is bounded from Ldecp(w,+) to Lq(v,+) if and only if (0(0xg(t)(x-t)-αdt)pW-p(x)w(x)dx)1/pc(0g(t)qv1-q(t)dt)1/q,g0.

Proof.

Taking Proposition A into account (for n=1), an integral operator (Tf)(x)=0k(x,y)f(y)dy is bounded from Ldecp(w,+) to Lq(v,+) if and only if (0(0x(T*f)(τ)dτ)pW-p(x)w(x)dx)1/pc(0f(t)qv1-q(t)dt)1/q,f0, where T* is a formal adjoint to T.

We have 0x(Rαf)(t)dt=0x(0tf(τ)(t-τ)1-αdτ)dt=0xf(τ)(0x-τduu1-α)dτ=1α0xf(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.

Let 1<pq<, and let 0<α<1. Suppose that W()=. Then Iα is bounded from Ldecp(w,+) to Lq(v,+) if and only if supa>0A1(a,v,w)supa>0(0aw(t)dt)-1/p(0atαqv(t)dt)1/q<,  supa>0A2(a,v,w)supa>0(0atpW-p(t)w(t)dt)1/p(at(α-1)qv(t)dt)1/q<,supa>0A3(a,v,w)supa>0(aW-p(x)w(x)(x-a)αp  dx)1/p(0av(x)dx)1/q<,supa>0A4(a,v,w)supa>0(0aw(x)dx)-1/p(0av(x)(a-x)αqdx)1/q<.

Theorem 3.4.

Let 1<q<p<, and let 0<α<1. W()=. Then Iα is bounded from Ldecp(w,+) to Lq(v,+) if and only if [R+[(0txαqv(x)dx)1/pW-1/p(t)]rtαqv(t)dt]1/r<,[R+[(tv(x)x(1-α)qdx)1/p(0tW-p(x)w(x)x-p)1/p]rtpW-p(t)w(t)dt]1/r<,[R+[(tW-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 Theorems <xref ref-type="statement" rid="thm2.1">3.3</xref> and <xref ref-type="statement" rid="thm2.2">3.4</xref>.

By using the representation (Iαf)(x)=(Rαf)(x)+(Wαf)(x),  x>0, the obvious equality tW-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.

Let 1<pq<, and let 0<α<1/p. Then the operator Iα is bounded from Ldecp(1,+) to Lq(v,+) if and only if Bsupa>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 (0v(x)(Iαf(x))qdx)1/qc(0(f(x))pdx)1/p and observing that Iαfa(x)0a(dt/|x-t|1-α)aα for x(0,a).

Sufficiency.

By Theorem 3.3, it is enough to show that 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 estimates AicB, i=1,4, are obvious. We show that AicB for i=2,3. We have A2q(a,v,1)=aq/pk=02ka2k+1at(α-1)qv(t)dtaq/pk=0(2ka)(α-1)q(2ka2k+1av(t)dt)cBqaq/pk=0(2ka)(α-1)q(2k+1a)(1/p-α)q=cBqaq/p(k=02-kq/p)a-q/pcBq. Further, by the condition 0<α<1/p, we have that A3q(a,v,1)(ax(α-1)pdx)1/p(0av(t)dt)1/q=cα,paα-1/p(0av(t)dt)1/qcB.

Definition 3.6.

Let ρ be a locally integrable a.e. positive function on +. We say that ρ satisfies the doubling condition (ρDC(+)) if there is a positive constant b>1 such that for all t>0 the following inequality holds: 02tρ(x)dxbmin{0tρ(x)dx,t2tρ(x)dx}.

Remark 3.7.

It is easy to check that if ρDC(+), then ρ satisfies the reverse doubling condition: there is a positive constant b1>1 such that 02tρ(x)dxb1max{0tρ(x)dx,t2tρ(x)dx}. Indeed by (3.22) we have 02tρ(x)dx1b02tρ(x)dx+t2tρ(x)dx. Then 02tρ(x)dxbb-1t2tρ(x)dx. Analogously, 02tρ(x)dxbb-10tρ(x)dx. Finally, we have (3.23).

Corollary 3.8.

Let 1<pq<, and let 0<α<1. Suppose that W()=. Suppose also that vDC(+). Then Iα is bounded from Ldecp(w,+) to Lq(v,+) if and only if condition (3.11) is satisfied.

Proof.

Observe that by Remark 3.7, for m0, the inequality 02m0v(x)dxb1m0-k02kv(x)dx holds for all k>m0, where b1 is defined in (3.23).

Let a>0. Then there is m0 such that a[2m0,2m0+1). By applying (3.27) and the doubling condition for v, we find that (0aw(t)dt)-p/p(0atαqv(t)dt)p/q=c(aW-p(t)w(t)dt)(0atαqv(t)dt)p/qc(2m0W-p(t)w(t)dt)(02m0+1tαqv(t)dt)p/qck=m0(2k2k+1W-p(t)w(t)dt)(02m0+1v(t)dt)p/q2m0αpck=m0(2k2k+1W-p(t)w(t)dt)b1m0-k-1(02k+2v(t)dt)p/q2m0αpck=m0b1m0-k-1(2k2k+1W-p(t)w(t)dt)(2k+12k+2v(t)dt)p/q2k(α-1)p2kpck=m0b1m0-k-1(2k2k+1tpW-p(t)w(t)dt)(2k+12k+2v(t)t(α-1)qdt)p/qc(supa>0A2(a,v,w))pk=m0b1m0-k-1c(supa>0A2(a,v,w))p. So, we have seen that (3.11)(3.10). Let us check now that (3.13)(3.12).

Indeed, for a>0, we choose m0 so that a[2m0,2m0+1). Then, by using the condition vDC(+) and Remark 3.7, (aW-p(x)w(x)(x-a)αpdx)(0av(x)dx)p/q(2m0W-p(x)w(x)xαpdx)(02m0+1v(x)dx)p/qck=m02kαp(2k2k+1W-p(x)w(x)dx)(02m0+1v(x)dx)p/qck=m02kαp(2k2k+1W-p(x)w(x)dx)b1m0-k+2(02k-1v(x)dx)p/qck=m0b1m0-k+2(2kW-p(x)w(x)dx)(02kv(x)(2k-x)αqdx)p/qc(supa>0A4(a,v,w))pk=m0b1m0-k+2c(supa>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 kernels α1,α2.

To derive the main results, we introduce the following multiple potential operators: Wα1,α2f(x1,x2)=x1x2f(t1,t2)dt1dt2(t1-x1)1-α1(t2-x2)1-α2,(RW)α1,α2f(x1,x2)=0x1x2f(t1,t2)dt1dt2(x1-t1)1-α1(t2-x2)1-α2,(WR)α1,α2f(x1,x2)=x10x2f(t1,t2)dt1dt2(t1-x1)1-α1(x2-t2)1-α2, where x1,x2+, f0, and 0<αi<1, i=1,2.

Definition 4.1.

One says that a locally integrable a.e. positive function ρ on +2 satisfies the doubling condition with respect to the second variable (ρDC(y)) if there is a positive constant c such that for all t>0 and almost every x>0 the following inequality holds: 02tρ(x,y)dycmin{0tρ(x,y)dy,t2tρ(x,y)dy}.

Analogously is defined the class of weights DC(x).

Remark 4.2.

If ρDC(y), then ρ satisfies the reverse doubling condition with respect to the second variable; that is, there is a positive constant c1 such that 02tρ(x,y)dyc1max{0tρ(x,y)dy,t2tρ(x,y)dy}.

Analogously, ρDC(x)ρRDC(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.

If vDC(x), then for the boundedness of α1,α2 from Ldecp(w,+2) to Lq(v,+2), it is necessary and sufficient that conditions (2.10) and (2.12) are satisfied.

If vDC(y), then α1,α2 is bounded from Ldecp(w,+2) to Lq(v,+2) if and only If conditions (2.10) and (2.11) are satisfied.

If vDC(x)DC(y), then α1,α2 is bounded from Ldecp(w,+2) to Lq(v,+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 Corollary 3.8 (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 operator α1,α2 with α1,α2>1 (see , [13, Section  1.6]).

Theorem D.

Let 1<pq<, and let α1,α21.

Suppose that w1-pDC(y). Then the operator α1,α2 is bounded from Lp(w,+2) to Lq(v,+2) if and only if P1supa,b>0(0a0bw1-p(x1,x2)(a-x1)(1-α1)pdx1dx2)1/p(abv(x1,x2)x2(1-α2)qdx1dx2)1/q<,P2supa,b>0(0a0bw1-p(x1,x2)dx1dx2)1/p(abv(x1,x2)(x1-a)(1-α1)qx2(1-α2)qdx1dx2)1/q<. Moreover, α1,α2max{P1,P2}.

Let w1-pDC(x). Then the operator α1,α2 is bounded from Lp(w,+2) to Lq(v,+) if and only if P̃1supa,b>0(0a0bw1-p(x1,x2)(b-x2)(1-α2)pdx1dx2)1/p(abv(x1,x2)x1(1-α1)qdx1dx2)1/q<,P̃2supa,b>0(0a0bw1-p(x1,x2)dx1dx2)1/p(abv(x1,x2)(x2-b)(1-α2)qx1(1-α1)qdx1dx2)1/q<. Moreover, α1,α2max{P̃1,P̃2}.

Let us introduce the following multiple integral operators: (HR)α1,α2f(x1,x2)=x1α1-10x10x2f(t1,t2)dt1dt2(x2-t2)1-α2,(RH)α1,α2f(x1,x2)=x2α2-10x10x2f(t1,t2)dt1dt2(x1-t1)1-α1,(HW)α1,α2f(x1,x2)=x1α1-10x1x2f(t1,t2)dt1dt2(t2-x2)1-α2,(WH)α1,α2f(x1,x2)=x2α2-1x10x2f(t1,t2)dt1dt2(t1-x1)1-α1,(HR)α1,α2f(x1,x2)=x10x2f(t1,t2)dt1dt2t11-α1(x2-t2)1-α2,(RH)α1,α2f(x1,x2)=0x1x2f(t1,t2)dt1dt2(x1-t1)1-α1t21-α2,(HW)α1,α2f(x1,x2)=x1x2f(t1,t2)dt1dt2t11-α1(t2-x2)1-α2,(WH)α1,α2f(x1,x2)=x1x2f(t1,t2)dt1dt2(t1-x1)1-α1t21-α2.

Now we prove some auxiliary statements.

Proposition 4.3.

Let 1<pq<, and let α1,α21. 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.

The operator ()α1,α2 is bounded from Lp(w,+2) to Lq(v,+) if and only if Ĩ1supa,b>0(0a0bw1-p(x1,x2)(a-x1)(1-α1)pdx1dx2)1/p(abv(x1,x2)x2(1-α2)qdx1dx2)1/q<,Ĩ2supa,b>0(0a0bw1-p(x1,x2)dx1dx2)1/p(abv(x1,x2)(x1-a)(1-α1)qx2(1-α2)qdx1dx2)1/q<. Moreover, ()α1,α2max{Ĩ1,Ĩ2}.

The operator (𝒲)α1,α2 is bounded from Lp(w,+2) to Lq(v+) if and only if J̃1supa,b>0(0abv(x1,x2)(a-x1)(1-α1)qx2(1-α2)qdx1dx2)1/q(a0bw1-p(x1,x2)dx1dx2)1/q<,J̃2supa,b>0(0abv(x1,x2)x2(1-α2)qdx1dx2)1/q(a0bw1-p(x1,x2)(x1-a)(1-α1)pdx1dx2)1/p<. Moreover, (𝒲)α1,α2max{J̃1,J̃2}.

The operator ()α1,α2 is bounded from Lp(w,+2) to Lq(v,+) if and only if J̃1supa,b>0(a0bv(x1,x2)dx1dx2)1/q(0abw1-p(x1,x2)x2(1-α2)p(a-x1)(1-α1)pdx1dx2)1/p<,J̃2supa,b>0(a0bv(x1,x2)(x1-a)(1-α1)qdx1dx2)1/q(0abw1-p(x1,x2)x2(1-α2)pdx1dx2)1/p<. Moreover, ()α1,α2max{J̃1,J̃2}.

The operator (𝒲)α1,α2 is bounded from Lp(w,+2) to Lq(v,+) if and only if Ĩ1supa,b>0(0a0bv(x1,x2)(a-x1)(1-α1)qdx1dx2)1/q(abw1-p(x1,x2)x2(1-α2)pdx1dx2)1/p<,Ĩ2supa,b>0(0a0bv(x1,x2)dx1dx2)1/q(abw1-p(x1,x2)x2(1-α2)p(x1-a)(1-α1)pdx1dx2)1/p<. Moreover, (𝒲)α1,α2max{Ĩ1,Ĩ2}.

Proof.

Let w(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  (see also the proof of Theorem  1.1.6 in ).

Sufficiency.

First suppose that S0w21-p(x2)dx2=. Let {ak}k=-+ be a sequence of positive numbers for which the equality 2k=0akw21-p(x2)dx2 holds for all k. It is clear that {ak} is increasing and +=k[ak,ak+1). Moreover, it is easy to verify that 2k=akak+1w21-p(x2)dx2. Let f0. We have that (RH)α1,α2fLq(v,R+2)q=R+2v(x1,x2)((RH)α1,α2f)q(x1,x2)dx1dx2kZ0akak+1v(x1,x2)x2(1-α2)q(0x10x2f(t1,t2)(x1-t1)1-α1dt1dt2)qdx1dx2kZ0(akak+1v(x1,x2)x2(1-α2)qdx2)(0x1(x1-t1)α1-1(0ak+1f(t1,t2)dt2)dt1)qdx1=kZ0Vk(x1)(0x1(x1-t1)(α1-1)Fk(t1)dt1)qdx1, where Vk(x1)akak+1v(x1,x2)x2(1-α2)qdx2,Fk(t1)0ak+1f(t1,t2)dt2.

It is obvious that Ĩ1qsupa>0jZ(aajaj+1v(x1,x2)(x1-a)(1-α1)qx2(1-α2)qdx1dx2)(0a0ajw1-p(x1,x2)(a-x1)(1-α1)pdx1dx2)q/p,Ĩ2qsupa>0jZ(aajaj+1v(x1,x2)x2(1-α2)qdx1dx2)(0a0ajw1-p(x1,x2)(a-x1)(1-α1)pdx1dx2)q/p. Hence, by using the two-weight criteria for the one-dimensional Riemann-Liouville operator without singularity (see ), we find that (RH)α1,α2fLq(v,R+2)qcĨqjZ[0w1(x1)(0ajw21-p(x2)dx2)1-p(Fj(x1))pdx1]q/pcĨq[0w1(x1)jZ(0ajw21-p(x2)dx2)1-p(k=-jakak+1f(x1,t2)dt2)pdx1]q/p, where Ĩ=max{I1̃,I2̃}.

On the other hand, (4.11) yields k=n+(0akw21-p(x2)dx2)1-p(k=-nakak+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. Hence by Hardy’s inequality in discrete case (see, for example, [25, 26]) and Hölder’s inequality we have that (RH)α1,α2fLq(v,R+2)qcĨq[0w1(x1)jZ(ajaj+1w21-p(x2)dx2)1-p(ajaj+1f(x1,t2)dt2)pdx1]q/pcĨq[0w1(x1)jZ(ajaj+1w2(t2)fp(x1,t2)dt2)dx1]q/p=cĨqfLp(w,R+2)q.

If S<, 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-. 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 xkw21-p(x)dx=2k (notice that xk is decreasing) and argue as in the proof of (i).

Proposition 4.4.

Let 1<pq<, and let α1,α21. 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.

The operator ()α1,α2 is bounded from Lp(w,+2) to Lq(v,+2) if and only if I1supa,b>0(0a0bw1-p(x1,x2)(b-x2)(1-α2)pdx1dx2)1/p(abv(x1,x2)x1(1-α1)qdx1dx2)1/q<,I2supa,b>0(0a0bw1-p(x1,x2)dx1dx2)1/p(abv(x1,x2)x1(1-α1)q(x2-b)(1-α2)qdx1dx2)1/q<. Moreover, ()α1,α2max{I1,I2}.

The operator (𝒲)α1,α2 is bounded from Lp(w,+2) to Lq(v,+) if and only if J1supa,b>0(a0bv(x1,x2)x1(1-α1)q(b-x2)(1-α2)qdx1dx2)1/q(0abw1-p(x1,x2)dx1dx2)1/p<,J2supa,b>0(a0bv(x1,x2)x1(α1-1)qdx1dx2)1/q(0abw1-p(x1,x2)(x2-b)(1-α2)pdx1dx2)1/p<. Moreover, (𝒲)α1,α2max{J1,J2}.

The operator ()α1,α2 is bounded from Lp(w,+2) to Lq(v,+) if and only if J1supa,b>0(0abv(x1,x2)dx1dx2)1/q(a0bw1-p(x1,x2)x1(1-α1)p(b-x2)(1-α2)pdx1dx2)1/p<,J2supa,b>0(0abv(x1,x2)(x2-b)(1-α2)qdx1dx2)1/q(a0bw1-p(x1,x2)x1(1-α1)pdx1dx2)1/p<. Moreover, ()α1,α2max{J1,J2}.

The operator (𝒲)α1,α2 is bounded from Lp(w,+2) to Lq(v,+) if and only if I1supa,b>0(0a0bv(x1,x2)(b-x2)(1-α2)qdx1dx2)1/q(abw1-p(x1,x2)x1(1-α1)pdx1dx2)1/p<,I2supa,b>0(0a0bv(x1,x2)dx1dx2)1/q(abw1-p(x1,x2)x1(1-α1)p(x2-b)(1-α2)pdx1dx2)1/p<. Moreover, (𝒲)α1,α2max{I1,I2}.

Proof of this proposition is similar to Proposition 4.3 by changing the order of variables.

Theorem 4.5.

Let 1<pq<, and let 0<α1,α21. Suppose that the weight function w on +2 is of product type, that is, w(x1,x2)=w1(x1)w2(x2). Suppose also that W1()=W2()=.

If vDC(y), then 𝒲α1,α2 is bounded from Ldecp(w,+2) to Lq(v,+2) if and only if A1supa,b>0(0a0bv(x1,x2)(a-x1)α1qdx1dx2)1/q×(0aw1(x1)dx1)-1/p(bW2-p(x2)w2(x2)x2α2pdx2)1/p<,A2supa,b>0(0a0bv(x1,x2)dx1dx2)1/q×(abW-p(x1,x2)w(x1,x2)(x1-a)α1px2α2pdx1dx2)1/p<.

If vDC(x), then 𝒲α1,α2 is bounded from Ldecp(w,+2) to Lq(v,+2) if and only if B1supa,b>0(0a0bv(x1,x2)(b-x2)α2qdx1dx2)1/q×(aW1-p(x1)w1(x1)x1α1pdx1)1/p(0bw2(x2)dx2)-1/p<,B2supa,b>0(0a0bv(x1,x2)dx1dx2)1/q×(abW-p(x1,x2)w(x1,x2)(x2-b)α2px1α1pdx1dx2)1/p<.

If vDC(x)DC(y), then 𝒲α1,α2 is bounded from Ldecp(w,+2) to Lq(v,+2) if and only if C1supa,b>0(abW-p(x1,x2)w(x1,x2)x2α2px1α1pdx1dx2)1/p×(0a0bv(x1,x2)dx1dx2)1/q<.

Proof.

By using Proposition A we see that the operator 𝒲α1,α2 is bounded from Ldecp(w,+2) to Lq(v,+2) if and only if the inequality (R+2(0x10x2[0τ10τ2g(t1,t2)dt1dt2(τ1-t1)1-α1(τ2-t2)1-α2]dτ1dτ2)p×W-p(x1,x2)w(x1,x2)dx1dx2R+2(0x10x2[0τ10τ2g(t1,t2)dt1dt2(τ1-t1)1-α1(τ2-t2)1-α2]dτ1dτ2)p)1/pc(R+2gqv1-q)1/q holds for all g0. Further, it is easy to see that 0x10x2[0τ10τ2g(t1,t2)dt1dt2(τ1-t1)1-α1(τ2-t2)1-α2]dτ1dτ2=0x10x2g(t1,t2)[t1x1t2x2dτ1dτ2(τ1-t1)1-α1(τ2-t2)1-α2]dt1dt2=cα1,α20x10x2g(t1,t2)(x1-t1)α1(x2-t2)α2dt1dt2. Hence 𝒲α1,α2 is bounded from Ldecp(w,+2) to Lq(v,+2) if and only if α1+1,α2+1 is bounded from Lq(v1-q,+2) to Lp(W-pw,+2).

By using Theorem D, (i) and (ii) follow immediately.

To prove (iii) we show that if vDC(x)DC(y), then (4.26) implies (4.23) and (4.24). Let a,b>0. Then a[2m0,2m0+1) for some m0. By using the doubling condition with respect to the first variable uniformly to the second one and Remark 4.2, we see that (0a0bv(x1,x2)(a-x1)α1qdx1dx2)p/q(0aw1(x1)dx1)-p/p=c(0a0bv(x1,x2)(a-x1)α1qdx1dx2)p/q(aW1-p(x1)w1(x1)dx1)ck=m0(2k2k+1W1-p(x1)w1(x1)dx1)2(m0+1)α1p(02m00bv(x1,x2)dx1dx2)p/qck=m0(2k2k+1x1α1pW1-p(x1)w1(x1)dx1)c1(m0-k)(p/q)(02k0bv(x1,x2)dx1dx2)p/qcC1p(bW2-p(x2)w2(x2)x2α2pdx2)-1. Hence, A1C1. In a similar manner we can show that A2C1.

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 (0a0bv(x1,x2)dx1dx2)p/q(aW1-p(x1)w(x1)x1α1pdx1)ck=m0(2k2k+1W1-p(x1)w(x1)dx1)2kα1p(02m0+10bv(x1,x2)dx1dx2)p/qck=m0(2k2k+1W1-p(x1)w(x1)dx1)c1(m0-k+2)(p/q)(02k-10b(2k-x1)α1qv(x1,x2)dx1dx2)p/qcA1p(bW2-p(x2)w2(x2)x2α2pdx2)-1. Hence, taking the supremum with respect to a and b, we find that C1cA1.

The following statements give analogous statement for the mixed-type operator (𝒲)α1,α2 and (𝒲)α1,α2.

Theorem 4.6.

Let 1<pq<, and let 0<α1,α21. Suppose that the weight function w on +2 is of product type, that is, w(x1,x2)=w1(x1)w2(x2). Suppose also that W1()=W2()=.

The operator (𝒲)α1,α2 is bounded from Ldecp(w,+2) to Lq(v,+2) if and only if supa,b>0(0a0bx1α1qv(x1,x2)(b-x2)-α2qdx1dx2)1/q(0a0bw1(x1)w2(x2)dx1dx2)-1/p<,supa,b>0(0a0bx1α1qv(x1,x2)dx1dx2)1/q(0aw1(x1)dx1)-1/p×(bW2-p(x2)w2(x2)(x2-b)α2pdx2)1/p<,supa,b>0(a0bv(x1,x2)x1(1-α1)q(b-x2)-α2qdx1dx2)1/q(0ax1pW1-p(x1)w1(x1)dx1)1/p×(0bw2(x2)dx2)-1/p<,supa,b>0(a0bx1(α1-1)qv(x1,x2)dx1dx2)1/q(0abW-p(x1,x2)w(x1,x2)x1p(x2-b)-α2pdx1dx2)1/p<.

The operator (𝒲)α1,α2 is bounded from Ldecp(w,+2) to Lq(v,+2) if and only if supa,b>0(0a0bx2α2qv(x1,x2)(a-x1)-α1qdx1dx2)1/q(0a0bw1(x1)w2(x2)dx1dx2)-1/p<,  supa,b>0(0a0bx2α2qv(x1,x2)dx1dx2)1/q(0bw2(x2)dx2)-1/p×(aW1-p(x1)w1(x1)(x1-a)α1pdx1)1/p<,supa,b>0(0abv(x1,x2)x2(1-α2)q(a-x1)-α1qdx1dx2)1/q(0aw1(x1)dx1)-1/p×(0bx2pW2-p(x2)w2(x2)dx2)1/p<,supa,b>0(0abx2(α2-1)qv(x1,x2)dx1dx2)1/q(a0bW-p(x1,x2)w(x1,x2)x2p(x1-a)-α1pdx1dx2)1/p<.

Proof.

We prove part (i). The proof of part (ii) is similar by changing the order of variables.

First we show that the two-sided pointwise relation (𝒲)α1,α2f(𝒲)α1,α2f, f, holds. Indeed, by using the fact that f is nonincreasing in the first variable, we find that (RW)α1,α2f(x1,x2)=0x1/2x2()+x1/2x1x2()cα1x1α1-10x1/2x2f(t1,t2)(t2-x2)1-α2dt1dt2+cα1′′x1α1-10x1/2x2f(t1,t2)(t2-x2)1-α2dt1dt2cα1,α2(HW)α1,α2f(x1,x2). The inequality (HW)α1,α2f(x1,x2)(RW)α1,α2f(x1,x2) is obvious because x1-t1x1 for 0<t1x1.

Further, it is easy to check that 0x10x2(τ10τ2g(t1,t2)t11-α1(τ2-t2)1-α2dt1dt2)dτ1dτ2=0x10x2(τ1x10τ2g(t1,t2)t11-α1(τ2-t2)1-α2dt1dt2)dτ1dτ2+0x10x2(x10τ2g(t1,t2)t11-α1(τ2-t2)1-α2dt1dt2)dτ1dτ2=0x10x2g(t1,t2)t1α1-1(0t1t2x2(τ2-t2)α2-1dτ1dτ2)dt1dt2+x10x2g(t1,t2)t1α1-1(0x1t2x2(τ2-t2)α2-1dτ1dτ2)dt1dt2=c0x10x2g(t1,t2)t1α1(x2-t2)α2dt1dt2+cx1x10x2g(t1,t2)t1α1-1(x2-t2)α2dt1dt2.

Hence, since the boundedness of (𝒲)α1,α2 from Ldecp(w,+2) to Lq(v,+2) is equivalent to the inequality (see also ) (R+2(0x10x2[τ10τ2g(t1,t2)dt1dt2t11-α1(τ2-t2)1-α2]dτ1dτ2)pW-p(x1,x2)w(x1,x2)dx1dx2)1/pc(R+2gqv1-q)1/q, we can conclude that Proposition 4.4 yields the desired result.

Proposition 4.7.

Let the conditions of Theorem 4.6 be satisfied. Then

if vDC(x), then (𝒲)α1,α2 is bounded from Ldecp(w,+2) to Lq(v,+2) if and only if (4.33) and (4.34) hold;

if vDC(y), then (𝒲)α1,α2 is bounded from Ldecp(w,+2) to Lq(v,+2) if and only if (4.32) and (4.34) are satisfied;

if vDC(x)DC(y), then (𝒲)α1,α2 is bounded from Ldecp(w,+2) to Lq(v,+2) if and only if (4.34) holds.

Proof.

(i) Taking into account the arguments used in Theorem 4.5, 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 (0a0bx1α1qv(x1,x2)(b-x2)α2qdx1dx2)p/q(0bw2(x2)dx2)-p/qc(0a02m0+1x1α1qv(x1,x2)dx1dx2)p/q(2m0W2-p(x2)w2(x2)dx2)2(m0+1)α2pckm0(2k2k+1W2-p(x2)w2(x2)dx2)(0a02k-1x1α1qv(x1,x2)dx1dx2)p/q×c1(m0-k)p/q2(m0+1)α2pckm0(2k2k+1W2-p(x2)w2(x2)(x2-2k-1)α2pdx2)(0a02k-1x1α1qv(x1,x2)dx1dx2)p/q×c1(m0-k)p/qc(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 Proposition 4.7.

Proposition 4.8.

Let the conditions of Theorem 4.6 be satisfied. Then

if vDC(x), then (𝒲)α1,α2 is bounded from Ldecp(w,+2) to Lq(v,+2) if and only if (4.36) and (4.38) hold;

if vDC(y), then (𝒲)α1,α2 is bounded from Ldecp(w,+2) to Lq(v,+2) if and only if (4.37) and (4.38) are satisfied;

if vDC(x)DC(y), then (𝒲)α1,α2 is bounded from Ldecp(w,+2) to Lq(v,+2) if and only if (4.38) holds.

Now we are ready to discuss the operators α1,α2 on the cone of nonincreasing functions.

Theorem 4.9.

Let 1<pq<, 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 α1,α2 is bounded from Ldecp(w,+2) to Lq(v,+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.

Let 1<pq<, 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 α1,α2 is bounded from Ldecp(w,+2) to Lq(v,+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.

Let 1<pq<, and let 0<α1,  α2<1. Suppose that the weight vDC(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 α1,α2 is bounded from Ldecp(w,+2) to Lq(v,+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 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 for α1,α2 can be characterized by one condition when w1.

Corollary 4.12.

Let 1<pq<, and let 0<α1,α2<1/p. Suppose that vDC(x)DC(y). Then the operator α1,α2 is bounded from Ldecp(1,+2) to Lq(v,+2) if and only if Dsupa,b>0a(α1-(1/p))b(α2-(1/p))(0a0bv(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 α1,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.

Acknowledgments

The first author was partially supported by the Shota Rustaveli National Science Foundation Grant (Project no. GNSF/ST09_23_3-100). A part of this work was carried out at the Abdus Salam School of Mathematical Sciences, GC University, Lahore. The authors are thankful to the Higher Education Commission, Pakistan, for the financial support. The first author expresses his gratitude to Professor V. M. Kokilashvili for drawing his attention to the two-weight problem for potentials with product kernels.