Potential Operators on Cones of Nonincreasing Functions

Abstract

Necessary and sufficient conditions on weight pairs guaranteeing the two-weight inequalities for the potential operators (𝐼_𝛼𝑓)(𝑥)=∫^∞0(𝑓(𝑡)/|𝑥−𝑡|^1−𝛼)𝑑𝑡 and (ℐ_𝛼1,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 ℐ_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: 𝐼_𝛼𝑓(𝑥)=^∞0𝑓(𝑡)|𝑥−𝑡|^1−𝛼ℐ𝑑𝑡,0<𝛼<1,_𝛼1,𝛼2𝑓(𝑥,𝑦)=^∞0𝑓(𝑡,𝜏)|𝑥−𝑡|^1−𝛼1||𝑦−𝜏||^1−𝛼2𝑑𝑡𝑑𝜏,0<𝛼₁,𝛼₂<1,(1.1) from 𝐿^𝑝_dec to 𝐿^𝑞, 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)=1𝑥^𝑥0𝑓(𝑡)𝑑𝑡(1.2) from 𝐿^𝑝_dec(𝑢,ℝ₊) to 𝐿^𝑝(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 𝑇_𝛼𝑓(𝑥)=_ℝ𝑛𝑓(𝑦)||𝑥−𝑦||^𝛼−𝑛𝑑𝑦,0<𝛼<𝑛,(1.3) 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 ℛ_𝛼1,𝛼2𝑓(𝑥,𝑦)=^𝑥0^𝑦0𝑓(𝑡,𝜏)(𝑥−𝑡)^1−𝛼1(𝑦−𝜏)^1−𝛼2𝑑𝑡𝑑𝜏,0<𝛼₁,𝛼₂<1,(1.4) from 𝐿^𝑝_dec(𝑤,ℝ²⁺) to 𝐿^𝑝(𝑣,ℝ²⁺) 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=ℛ_1,1 from 𝐿^𝑝_dec(𝑤,ℝ²⁺) to 𝐿^𝑝(𝑣,ℝ²⁺) 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 by 𝑐 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)∶=∫^𝑥0𝑤(𝑡)𝑑𝑡; 𝑊_𝑗(𝑥_𝑗)∶=∫^𝑥𝑗₀𝑤_𝑗(𝑡)𝑑𝑡; 𝑊(𝑡₁,…,𝑡_𝑛)∶=Π^𝑛𝑖=1𝑊_𝑖(𝑡_𝑖).

2. Preliminaries

We say that a function 𝑓∶ℝ^𝑛+→ℝ₊ is nonincreasing if 𝑓 is nonincreasing in each variable separately.

Let 𝒟 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 𝐿^𝑝(𝑢,ℝ^𝑛+), 0<𝑝<∞, the class of all nonnegative functions on ℝ^𝑛+ for which ‖𝑓‖_𝐿^𝑝_(𝑢,ℝ^𝑛+₎∶=_ℝ𝑛+𝑓^𝑝𝑥₁,…,𝑥_𝑛𝑢𝑥₁,…,𝑥_𝑛𝑑𝑥₁⋯𝑑𝑥_𝑛^1/𝑝=_ℝ𝑛+𝑓^𝑝(𝑥)𝑢(𝑥)𝑑𝑥^1/𝑝<∞.(2.1) By the symbol 𝐿^𝑝_dec(𝑢,ℝ^𝑛+) we mean the class 𝐿^𝑝(𝑢,ℝ^𝑛+)∩𝒟.

The next statement regarding two-weight criteria for the Hardy operator 𝐻 on the cone of nonincreasing functions was proved in [2].

Theorem A. Let 𝑣 and 𝑤 be weight functions on ℝ₊, and let 𝑊(∞)=∞. (i)Suppose that 1<𝑝≤𝑞<∞. Then the inequality ^∞0(𝐻𝑓(𝑥))^𝑞𝑣(𝑥)𝑑𝑥^1/𝑞≤𝐶^∞0(𝑓(𝑥))^𝑝𝑤(𝑥)𝑑𝑥^1/𝑝,𝑓∈𝐿^𝑝_dec𝑤,ℝ₊,(2.2) holds if and only if the following two conditions are satisfied: sup_𝑎>0^𝑎0𝑣(𝑥)𝑑𝑥^1/𝑞^𝑎0sup_𝑎>0𝑤(𝑥)𝑑𝑥^−1/𝑝<∞,^∞𝑎𝑣(𝑥)𝑥^𝑞𝑑𝑥^1/𝑞^𝑎0𝑊^−𝑝′(𝑥)𝑥^𝑝′𝑤(𝑥)𝑑𝑥^1/𝑝′<∞.(2.3)(ii)Let 1<𝑞<𝑝<∞. Then 𝐻 is bounded from 𝐿^𝑝_dec(𝑤,ℝ₊) to 𝐿^𝑞(𝑣,ℝ₊) if and only if the following two conditions are satisfied: ^∞0^𝑡0𝑣(𝑥)𝑑𝑥^1/𝑝𝑊^−1/𝑝(𝑡)^𝑟𝑣(𝑡)𝑑𝑡^1/𝑟<∞,_∞0^∞𝑡𝑥^−𝑞𝑣(𝑥)𝑑𝑥^1/𝑝^𝑡0𝑥^𝑝′𝑊^−𝑝′(𝑥)𝑤(𝑥)𝑑𝑥^1/𝑝′^𝑟𝑡^𝑝′𝑊^−𝑝′(𝑡)𝑤(𝑡)𝑑𝑡^1/𝑟<∞,(2.4) where 𝑟=𝑝𝑞/(𝑝−𝑞).

The following statement was proved in [2] for 𝑛=1. For 𝑛≥1 we refer to [4].

Proposition A. Let 1<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,…,𝑥_𝑛)=𝑤₁(𝑥₁)⋯𝑤_𝑛(𝑥_𝑛) be a product weight such that 𝑊_𝑖(∞)=∞, 𝑖=1,…,𝑛. Let 𝑣 be a general weight on ℝ^𝑛+. Then the operator 𝑇 is bounded from 𝐿^𝑝_dec(𝑤,ℝ^𝑛+) to 𝐿^𝑝(𝑣,ℝ^𝑛+) if and only if the inequality ⎛⎜⎜⎝_ℝ𝑛+^𝑥1₀⋯^𝑥𝑛₀𝑇^∗𝑔^𝑝′𝑊^−𝑝′𝑥₁,…,𝑥_𝑛𝑤𝑥₁,…,𝑥_𝑛𝑑𝑥₁⋯𝑑𝑥_𝑛⎞⎟⎟⎠^1/𝑝′≤𝑐_ℝ𝑛+𝑔(𝑥)^𝑞′𝑣^1−𝑞′(𝑥)𝑑𝑥^1/𝑞′(2.5) holds for all 𝑔≥0.

Let 𝑅_𝛼 be the Riemann-Liouville transform with single kernel 𝑅_𝛼𝑓(𝑥)=^𝑥0𝑓(𝑡)(𝑥−𝑡)^1−𝛼𝑑𝑡,𝑥∈ℝ₊,𝛼>0.(2.6)

If 𝛼=1, then 𝑅_𝛼 is the Hardy transform. The 𝐿^𝑝(𝑤,ℝ₊)→𝐿^𝑞(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 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 [11] and the references cited therein.

The next result deals with the case 𝛼>1 (see [24]).

Theorem B. Let 𝛼>1. Then the operator 𝑅_𝛼 is bounded from 𝐿^𝑝(𝑤,ℝ₊) to 𝐿^𝑞(𝑣,ℝ₊) if and only if sup_𝑡>0^∞𝑡(𝑥−𝑡)^{(𝛼−1)𝑞}𝑣(𝑥)𝑑𝑥^1/𝑞^𝑡0sup_𝑡>0𝑤^1−𝑝′(𝑦)𝑑𝑦^1/𝑝′<∞,^∞𝑡𝑣(𝑥)𝑑𝑥^1/𝑞^𝑡0(𝑡−𝑥)^{(𝛼−1)𝑝′}𝑤^1−𝑝′(𝑦)𝑑𝑦^1/𝑝′<∞,(2.7) for 1<𝑝≤𝑞<∞ and ^∞0^∞𝑡(𝑥−𝑡)^{(𝛼−1)𝑞}𝑣(𝑥)𝑑𝑥^𝑟/𝑞^𝑡0𝑤^1−𝑝′(𝑦)𝑑𝑦^𝑟/𝑞′𝑤^1−𝑝′(𝑡)𝑑𝑡^1/𝑟<∞,_∞0^∞𝑡𝑣(𝑥)𝑑𝑥^𝑟/𝑝^𝑡0(𝑡−𝑦)^{(𝛼−1)𝑝′}𝑤^1−𝑝′(𝑦)𝑑𝑦^𝑟/𝑝′𝑣(𝑡)𝑑𝑡^1/𝑟<∞,(2.8) for 1<𝑞<𝑝<∞, where 𝑟 is defined as follows: 1/𝑟=1/𝑞−1/𝑝.

Theorem C (see [10]). Let 1<𝑝≤𝑞<∞, and let 0<𝛼_𝑖<1, 𝑖=1,2. Assume that 𝑣 and 𝑤 are weights on ℝ²⁺. Suppose also that 𝑤(𝑥₁,𝑥₂)=𝑤₁(𝑥₁)𝑤₂(𝑥₂) for some one-dimensional weights 𝑤₁ and 𝑤₂ and that 𝑊_𝑖(∞)=∞, 𝑖=1,2. Then the following conditions are equivalent: (a)ℛ_𝛼1,𝛼2 is bounded from 𝐿^𝑝_dec(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ²⁺);(b)the following four conditions hold simultaneously:(i)sup_{𝑎1,𝑎2>0}^𝑎1₀^𝑎2₀𝑤𝑡₁,𝑡₂𝑑𝑡₁𝑑𝑡₂^−1/𝑝^𝑎1₀^𝑎2₀𝑡^𝛼1₁𝑡^𝛼2₂^𝑞𝑣𝑡₁,𝑡₂𝑑𝑡₁𝑑𝑡₂^1/𝑞<∞,(2.9)(ii)sup_{𝑎1,𝑎2>0}^𝑎1₀^𝑎2₀𝑡₁𝑡₂^𝑝′𝑊^−𝑝′𝑡₁,𝑡₂𝑤𝑡₁,𝑡₂𝑑𝑡₁𝑑𝑡₂^1/𝑝′×^∞𝑎₁^∞𝑎₂𝑡^𝛼11−1𝑡^𝛼22−1^𝑞𝑣𝑡₁,𝑡₂𝑑𝑡₁𝑑𝑡₂^1/𝑞<∞,(2.10)(iii)sup_{𝑎1,𝑎2>0}^𝑎1₀𝑤₁𝑡₁𝑑𝑡₁^−1/𝑝^𝑎2₀𝑡^𝑝′₂𝑊^−𝑝′₂𝑡₂𝑤₂𝑡₂𝑑𝑡₂^1/𝑝′×^𝑎1₀^∞𝑎₂𝑡^𝑞𝛼1₁𝑡^{𝑞(𝛼22−1)}𝑣𝑡₁,𝑡₂𝑑𝑡₁𝑑𝑡₂^1/𝑞<∞,(2.11)(iv)sup_{𝑎1,𝑎2>0}^𝑎1₀𝑡^𝑝′₁𝑊^−𝑝′₁𝑡₁𝑤₁𝑡₁𝑑𝑡₁^1/𝑝′^𝑎2₀𝑤₂𝑡₂𝑑𝑡₂^−1/𝑝×^∞𝑎₁^𝑎2₀𝑡^{𝑞(𝛼11−1)}𝑡^𝑞𝛼2₂v𝑡₁,𝑡₂𝑑𝑡₁𝑑𝑡₂^1/𝑞<∞.(2.12)

In particular, Theorem C yields the trace inequality criteria on the cone of nonincreasing functions.

Corollary A (see [10]). Let 1<𝑝≤𝑞<∞, and let 0<𝛼_𝑖<1, 𝑖=1,2. Then the following conditions are equivalent:(a)the boundedness of ℛ_𝛼1,𝛼2 from 𝐿^𝑝_dec(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ²⁺) holds for 𝑤≡1;(b)𝐵₁∶=sup_{𝑎1,𝑎2>0}𝐵₁𝑎₁,𝑎₂∶=sup_{𝑎1,𝑎2>0}𝑎₁𝑎₂^−1/𝑝^𝑎1₀^𝑎2₀𝑥^𝑞𝛼1₁𝑥^𝑞𝛼2₂𝑣𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂^1/𝑞<∞;(2.13)(c)𝐵₂∶=sup_{𝑎1,𝑎2>0}𝐵₂𝑎₁,𝑎₂∶=sup_{𝑎1,𝑎2>0}𝑎₁𝑎₂^1/𝑝′^∞𝑎₁^∞𝑎₂𝑥^{𝑞(𝛼11−1)}𝑥^{𝑞(𝛼22−1)}𝑣𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂^1/𝑞<∞;(2.14)(d)𝐵₃∶=sup_{𝑎1,𝑎2>0}𝐵₃𝑎₁,𝑎₂∶=sup_{𝑎1,𝑎2>0}𝑎^1−1/𝑝𝑎^1/𝑝′₂^𝑎1₀^∞𝑎₂𝑥^𝑞𝛼1₁𝑥^{𝑞(𝛼22−1)}𝑣𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂^1/𝑞<∞;(2.15)(e)𝐵₄∶=sup_{𝑎1,𝑎2>0}𝐵₄𝑎₁,𝑎₂∶=sup_{𝑎1,𝑎2>0}𝑎^1/𝑝′₁𝑎^2−1/𝑝^∞𝑎₁^𝑎2₀𝑥^{𝑞(𝛼11−1)}𝑥^𝑞𝛼2₂𝑣𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂^1/𝑞<∞.(2.16)

3. Potentials on ℝ₊

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

Lemma 3.1. The following relation holds for nonnegative and nonincreasing function 𝑓: 𝑅_𝛼𝑓(𝑥)≈𝑥^𝛼𝐻𝑓(𝑥),(3.1) where 𝐻 is the Hardy operator defined above.

Proof. We follow the proof of Proposition  3.1 of [10]. We have 𝑅_𝛼𝑓(𝑥)=^0𝑥/2𝑓(𝑡)(𝑥−𝑡)^1−𝛼𝑑𝑡+^𝑥𝑥/2𝑓(𝑡)(𝑥−𝑡)^1−𝛼𝑑𝑡∶=𝐽₁(𝑥)+𝐽₂(𝑥).(3.2)
Observe that if 0<𝑡<𝑥/2, then (𝑥−𝑡)^𝛼−1≤2^1−𝛼𝑥^𝛼−1. Hence, 𝐽₁(𝑥)≤2^1−𝛼𝑥^𝛼−1^𝑥0𝑓(𝑡)𝑑𝑡=2^1−𝛼𝑥^𝛼(𝐻𝑓)(𝑥).(3.3) Further, since 𝑓 is nonincreasing, we have that 𝐽₂(𝑥)≤𝛼⁻¹𝑥2^𝛼𝑓𝑥2≤𝑐^𝛼𝑥^𝛼(𝐻𝑓)(𝑥).(3.4)
Finally we have the upper estimate for 𝑅_𝛼.
The lower estimate is obvious because (𝑥−𝑡)^𝛼−1≥𝑥^𝛼−1 for 𝑡≤𝑥.

In the next statement we assume that 𝑊_𝛼 is the operator given by 𝑊_𝛼𝑓(𝑥)=^∞𝑥𝑓(𝑡)(𝑡−𝑥)^1−𝛼𝑑𝑡,𝛼>0.(3.5)

Lemma 3.2. Let 1<𝑝≤𝑞<∞, and let 𝛼>0. Suppose that 𝑊(∞)=∞. Then the operator 𝑊_𝛼 is bounded from 𝐿^𝑝_dec(𝑤,ℝ₊) to 𝐿^𝑞(𝑣,ℝ₊) if and only if ⎛⎜⎜⎝^∞0^𝑥0𝑔(𝑡)(𝑥−𝑡)^−𝛼𝑑𝑡^𝑝′𝑊^−𝑝′(𝑥)𝑤(𝑥)𝑑𝑥⎞⎟⎟⎠^1/𝑝′≤𝑐^∞0𝑔(𝑡)^𝑞′𝑣^1−𝑞′(𝑡)𝑑𝑡^1/𝑞′,𝑔≥0.(3.6)

Proof. Taking Proposition A into account (for 𝑛=1), an integral operator (𝑇𝑓)(𝑥)=^∞0𝑘(𝑥,𝑦)𝑓(𝑦)𝑑𝑦(3.7) is bounded from 𝐿^𝑝_dec(𝑤,ℝ₊) to 𝐿^𝑞(𝑣,ℝ₊) if and only if ⎛⎜⎜⎝^∞0^𝑥0𝑇^∗𝑓(𝜏)𝑑𝜏^𝑝′𝑊^−𝑝′(𝑥)𝑤(𝑥)𝑑𝑥⎞⎟⎟⎠^1/𝑝′≤𝑐^∞0𝑓(𝑡)^𝑞′𝑣^1−𝑞′(𝑡)𝑑𝑡^1/𝑞′,𝑓≥0,(3.8) where 𝑇^∗ is a formal adjoint to 𝑇.
We have ^𝑥0𝑅_𝛼𝑓(𝑡)𝑑𝑡=^𝑥0^𝑡0𝑓(𝜏)(𝑡−𝜏)^1−𝛼𝑑𝜏𝑑𝑡=^𝑥0𝑓(𝜏)^0𝑥−𝜏𝑑𝑢𝑢^1−𝛼𝑑𝜏=𝛼1^𝑥0𝑓(𝜏)(𝑥−𝜏)^𝛼𝑑𝜏.(3.9) Taking 𝑇=𝑊_𝛼 and 𝑇^∗=𝑅_𝛼, we derive the desired result.

Now we formulate the main results of this section.

Theorem 3.3. Let 1<𝑝≤𝑞<∞, and let 0<𝛼<1. Suppose that 𝑊(∞)=∞. Then 𝐼_𝛼 is bounded from 𝐿^𝑝_dec(𝑤,ℝ₊) to 𝐿^𝑞(𝑣,ℝ₊) if and only if sup_𝑎>0𝐴₁(𝑎,𝑣,𝑤)∶=sup_𝑎>0^𝑎0𝑤(𝑡)𝑑𝑡^−1/𝑝^𝑎0sup_𝑎>0𝑡^𝛼𝑞𝑣(𝑡)𝑑𝑡^1/𝑞<∞,(3.10)𝐴₂(𝑎,𝑣,𝑤)∶=sup_𝑎>0^𝑎0𝑡^𝑝′𝑊^−𝑝′(𝑡)𝑤(t)𝑑𝑡^1/𝑝′^∞𝑎sup_𝑎>0𝑡^{(𝛼−1)𝑞}𝑣(𝑡)𝑑𝑡^1/𝑞<∞,(3.11)𝐴₃(𝑎,𝑣,𝑤)∶=sup_𝑎>0^∞𝑎𝑊^−𝑝′(𝑥)𝑤(𝑥)(𝑥−𝑎)^𝛼𝑝′𝑑𝑥^1/𝑝′^𝑎0sup_𝑎>0𝑣(𝑥)𝑑𝑥^1/𝑞<∞,(3.12)𝐴₄(𝑎,𝑣,𝑤)∶=sup_𝑎>0^𝑎0𝑤(𝑥)𝑑𝑥^−1/𝑝^𝑎0𝑣(𝑥)(𝑎−𝑥)^𝛼𝑞𝑑𝑥^1/𝑞<∞.(3.13)

Theorem 3.4. Let 1<𝑞<𝑝<∞, and let 0<𝛼<1. 𝑊(∞)=∞. Then 𝐼_𝛼 is bounded from 𝐿^𝑝_dec(𝑤,ℝ₊) to 𝐿^𝑞(𝑣,ℝ₊) if and only if _ℝ+^𝑡0⎡⎢⎢⎣𝑥^𝛼𝑞𝑣(𝑥)𝑑𝑥^1/𝑝𝑊^−1/𝑝(𝑡)^𝑟𝑡^𝛼𝑞𝑣(𝑡)𝑑𝑡^1/𝑟<∞,_ℝ+⎡⎢⎢⎣^∞𝑡𝑣(𝑥)𝑥^{(1−𝛼)𝑞}𝑑𝑥^1/𝑝^𝑡0𝑊^−𝑝′(𝑥)𝑤(𝑥)𝑥^−𝑝′^1/𝑝′⎤⎥⎥⎦^𝑟⎡⎢⎢⎣𝑡^𝑝′𝑊^−𝑝′(𝑡)𝑤(𝑡)𝑑𝑡⎤⎥⎥⎦^1/𝑟<∞,_ℝ+⎡⎢⎢⎣^∞𝑡𝑊^−𝑝′(𝑥)𝑤(𝑥)(𝑥−𝑡)^{−𝛼𝑝′}^1/𝑝′^𝑡0𝑣(𝑥)𝑑𝑥^1/𝑝⎤⎥⎥⎦^𝑟𝑣(𝑡)𝑑𝑡⎤⎥⎥⎦^1/𝑟<∞,_ℝ+𝑊⁻¹(𝑡)^𝑡0𝑣(𝑥)(𝑡−𝑥)^−𝛼𝑞𝑑𝑥^𝑟/𝑞𝑤(𝑡)𝑑𝑡^1/𝑟<∞,(3.14) where 1/𝑟=1/𝑞−1/𝑝.

Proof of Theorems 3.3 and 3.4. By using the representation 𝐼_𝛼𝑓(𝑥)=𝑅_𝛼𝑓(𝑥)+𝑊_𝛼𝑓(𝑥),𝑥>0,(3.15) the obvious equality ^∞𝑡𝑊^−𝑝′(𝑥)𝑤(𝑥)𝑑𝑥=𝑐_𝑝𝑊^1−𝑝′(𝑡).(3.16) Theorems A and B and Lemmas 3.1 and 3.2, we have the desired results.

Corollary 3.5. Let 1<𝑝≤𝑞<∞, and let 0<𝛼<1/𝑝. Then the operator 𝐼_𝛼 is bounded from 𝐿^𝑝_dec(1,ℝ₊) to 𝐿^𝑞(𝑣,ℝ₊) if and only if 𝐵∶=sup_𝑎>0𝑎^{(𝛼−1/𝑝)}^𝑎0𝑣(𝑡)𝑑𝑡^1/𝑞<∞.(3.17)

Proof. Necessity follows immediately taking the test function 𝑓_𝑎(𝑥)=𝜒_(0,𝑎)(𝑥) in the two-weight inequality ^∞0𝑣(𝑥)𝐼_𝛼𝑓(𝑥)^𝑞𝑑x^1/𝑞≤𝑐^∞0(𝑓(𝑥))^𝑝𝑑𝑥^1/𝑝(3.18) and observing that 𝐼_𝛼𝑓_𝑎(𝑥)≥∫^𝑎0(𝑑𝑡/|𝑥−𝑡|^1−𝛼)≥𝑎^𝛼 for 𝑥∈(0,𝑎).
Sufficiency. By Theorem 3.3, it is enough to show that max𝐴₁,𝐴₂,𝐴₃,𝐴₄≤𝑐𝐵,(3.19) where 𝐴_𝑖∶=sup_𝑎>0𝐴_𝑖(𝑎,𝑣,1), 𝑖=1,2,3,4 (see Theorem 3.3 for the definition of 𝐴_𝑖(𝑎,𝑣,𝑤)).
The estimates 𝐴_𝑖≤𝑐𝐵, 𝑖=1,4, are obvious. We show that 𝐴_𝑖≤𝑐𝐵 for 𝑖=2,3. We have 𝐴^𝑞2(𝑎,𝑣,1)=𝑎^{𝑞/𝑝′∞}_𝑘=0^2𝑘+1𝑎2_𝑘𝑎≤𝑎_{𝑞/𝑝′∞}𝑡^{(𝛼−1)𝑞}𝑣(𝑡)𝑑𝑡_𝑘=02^𝑘𝑎^{(𝛼−1)𝑞}^{2𝑘+1𝑎2𝑘}_𝑎≤𝑐𝐵_𝑞𝑣(𝑡)𝑑𝑡𝑎^{𝑞/𝑝′∞}_𝑘=02^𝑘𝑎^{(𝛼−1)𝑞}2^𝑘+1𝑎^{(1/𝑝−𝛼)𝑞}=𝑐𝐵^𝑞𝑎^𝑞/𝑝′^∞_𝑘=02^{−𝑘𝑞/𝑝′}𝑎^{−𝑞/𝑝′}≤𝑐𝐵^𝑞.(3.20) Further, by the condition 0<𝛼<1/𝑝, we have that 𝐴^𝑞3(𝑎,𝑣,1)≤^∞𝑎𝑥^{(𝛼−1)𝑝′}𝑑𝑥^1/𝑝′^𝑎0𝑣(𝑡)𝑑𝑡^1/𝑞=𝑐_𝛼,𝑝𝑎^{𝛼−1/𝑝}^𝑎0𝑣(𝑡)𝑑𝑡^1/𝑞≤𝑐𝐵.(3.21)

Definition 3.6. Let 𝜌 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: ^02𝑡𝜌(𝑥)𝑑𝑥≤𝑏min^𝑡0𝜌(𝑥)𝑑𝑥,^𝑡2𝑡𝜌(𝑥)𝑑𝑥.(3.22)

Remark 3.7. It is easy to check that if 𝜌∈DC(ℝ₊), then 𝜌 satisfies the reverse doubling condition: there is a positive constant 𝑏₁>1 such that ^02𝑡𝜌(𝑥)𝑑𝑥≥𝑏₁max^𝑡0𝜌(𝑥)𝑑𝑥,^𝑡2𝑡𝜌(𝑥)𝑑𝑥.(3.23) Indeed by (3.22) we have ^02𝑡𝜌(𝑥)𝑑𝑥≥1𝑏^02𝑡𝜌(𝑥)𝑑𝑥+^𝑡2𝑡𝜌(𝑥)𝑑𝑥.(3.24) Then ^02𝑡𝜌(𝑥)𝑑𝑥≥𝑏𝑏−1^𝑡2𝑡𝜌(𝑥)𝑑𝑥.(3.25) Analogously, ^02𝑡𝜌(𝑥)𝑑𝑥≥𝑏𝑏−1^𝑡0𝜌(𝑥)𝑑𝑥.(3.26) Finally, we have (3.23).

Corollary 3.8. Let 1<𝑝≤𝑞<∞, and let 0<𝛼<1. Suppose that 𝑊(∞)=∞. Suppose also that 𝑣∈DC(ℝ₊). Then 𝐼_𝛼 is bounded from 𝐿^𝑝_dec(𝑤,ℝ₊) to 𝐿^𝑞(𝑣,ℝ₊) if and only if condition (3.11) is satisfied.

Proof. Observe that by Remark 3.7, for 𝑚₀∈ℤ, the inequality ^2𝑚0₀𝑣(𝑥)𝑑𝑥≤𝑏^{𝑚01−𝑘}^2𝑘₀𝑣(𝑥)𝑑𝑥(3.27) holds for all 𝑘>𝑚₀, where 𝑏₁ is defined in (3.23).
Let 𝑎>0. Then there is 𝑚₀∈ℤ such that 𝑎∈[2^𝑚0,2^𝑚0+1). By applying (3.27) and the doubling condition for 𝑣, we find that ^𝑎0𝑤(𝑡)𝑑𝑡^{−𝑝′/𝑝}^𝑎0𝑡^𝛼𝑞𝑣(𝑡)𝑑𝑡^𝑝′/𝑞=𝑐^∞𝑎𝑊^−𝑝′(𝑡)𝑤(𝑡)𝑑𝑡^𝑎0𝑡^𝛼𝑞𝑣(𝑡)𝑑𝑡^𝑝′/𝑞≤𝑐^∞2_𝑚0𝑊^−𝑝′(𝑡)𝑤(𝑡)𝑑𝑡^2𝑚0+1₀𝑡^𝛼𝑞𝑣(𝑡)𝑑𝑡^𝑝′/𝑞≤𝑐^∞_𝑘_=𝑚0^2𝑘+1₂^𝑘𝑊^−𝑝′(𝑡)𝑤(𝑡)𝑑𝑡^2𝑚0+1₀𝑣(𝑡)𝑑𝑡^𝑝′/𝑞2^{𝑚0𝛼𝑝′}≤𝑐^∞_𝑘_=𝑚0^2𝑘+1₂^𝑘𝑊^−𝑝′(𝑡)𝑤(𝑡)𝑑𝑡𝑏^{𝑚01−𝑘−1}^2𝑘+2₀𝑣(𝑡)𝑑𝑡^𝑝′/𝑞2^{𝑚0𝛼𝑝′}≤𝑐^∞_𝑘_=𝑚0𝑏^{𝑚01−𝑘−1}^2𝑘+1₂^𝑘𝑊^−𝑝′(𝑡)𝑤(𝑡)𝑑𝑡^2𝑘+2_2𝑘+1𝑣(𝑡)𝑑𝑡^𝑝′/𝑞2^{𝑘(𝛼−1)𝑝′}2^𝑘𝑝′≤𝑐^∞_𝑘_=𝑚0𝑏^{𝑚01−𝑘−1}^2𝑘+1₂^𝑘𝑡^𝑝′𝑊^−𝑝′(𝑡)𝑤(𝑡)𝑑𝑡^2𝑘+2_2𝑘+1𝑣(𝑡)𝑡^{(𝛼−1)𝑞}𝑑𝑡^𝑝′/𝑞≤𝑐sup_𝑎>0𝐴₂(𝑎,𝑣,𝑤)^𝑝′∞_𝑘_=𝑚0𝑏^{𝑚01−𝑘−1}≤𝑐sup_𝑎>0𝐴₂(𝑎,𝑣,𝑤)^𝑝′.(3.28) 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∈[2^𝑚0,2^𝑚0+1). Then, by using the condition 𝑣∈DC(ℝ₊) and Remark 3.7, ^∞𝑎𝑊^−𝑝′(𝑥)𝑤(𝑥)(𝑥−𝑎)^𝛼𝑝′𝑑𝑥^𝑎0𝑣(𝑥)𝑑𝑥^𝑝′/𝑞≤^∞2_𝑚0𝑊^−𝑝′(𝑥)𝑤(𝑥)𝑥^𝛼𝑝′𝑑𝑥^2𝑚0+1₀𝑣(𝑥)𝑑𝑥^𝑝′/𝑞≤𝑐^∞_𝑘_=𝑚02^{𝑘𝛼𝑝′}^2𝑘+1₂^𝑘𝑊^−𝑝′(𝑥)𝑤(𝑥)𝑑𝑥^2𝑚0+1₀𝑣(𝑥)𝑑𝑥^𝑝′/𝑞≤𝑐^∞_𝑘_=𝑚02^{𝑘𝛼𝑝′}^2𝑘+1₂^𝑘𝑊^−𝑝′(𝑥)𝑤(𝑥)𝑑𝑥𝑏^{𝑚01−𝑘+2}^2𝑘−1₀𝑣(𝑥)𝑑𝑥^𝑝′/𝑞≤𝑐^∞_𝑘_=𝑚0𝑏^{𝑚01−𝑘+2}^∞2_𝑘𝑊^−𝑝′(𝑥)𝑤(𝑥)𝑑𝑥^2𝑘₀𝑣(𝑥)2^𝑘−𝑥^𝛼𝑞𝑑𝑥^𝑝′/𝑞≤𝑐sup_𝑎>0𝐴₄(𝑎,𝑣,𝑤)^𝑝′∞_𝑘_=𝑚0𝑏^{𝑚01−𝑘+2}≤𝑐sup_𝑎>0𝐴₄(𝑎,𝑣,𝑤)^𝑝′.(3.29) 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 ℐ_𝛼1,𝛼2.

To derive the main results, we introduce the following multiple potential operators: 𝒲_𝛼1,𝛼2𝑓𝑥₁,𝑥₂=^∞𝑥₁^∞𝑥₂𝑓𝑡₁,𝑡₂𝑑𝑡₁𝑑𝑡₂𝑡₁−𝑥₁^1−𝛼1𝑡₂−𝑥₂^1−𝛼2(ℛ𝒲)_𝛼1,𝛼2,𝑓𝑥₁,𝑥₂=^𝑥1₀^∞𝑥₂𝑓𝑡₁,𝑡₂𝑑𝑡₁𝑑𝑡₂𝑥₁−𝑡₁^1−𝛼1𝑡₂−𝑥₂^1−𝛼2(𝒲ℛ)_𝛼1,𝛼2,𝑓𝑥₁,𝑥₂=^∞𝑥₁^𝑥2₀𝑓𝑡₁,𝑡₂𝑑𝑡₁𝑑𝑡₂𝑡₁−𝑥₁^1−𝛼1𝑥₂−𝑡₂^1−𝛼2,(4.1) where 𝑥₁,𝑥₂∈ℝ₊, 𝑓≥0, and 0<𝛼_𝑖<1, 𝑖=1,2.

Definition 4.1. One says that a locally integrable a.e. positive function 𝜌 on ℝ²⁺ 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: ^02𝑡𝜌(𝑥,𝑦)𝑑𝑦≤𝑐min^𝑡0𝜌(𝑥,𝑦)𝑑𝑦,^𝑡2𝑡𝜌(𝑥,𝑦)𝑑𝑦.(4.2)

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

Remark 4.2. If 𝜌∈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 ^02𝑡𝜌(𝑥,𝑦)𝑑𝑦≥𝑐₁max^𝑡0𝜌(𝑥,𝑦)𝑑𝑦,^𝑡2𝑡𝜌(𝑥,𝑦)𝑑𝑦.(4.3)

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)If 𝑣∈DC(𝑥), then for the boundedness of ℛ_𝛼1,𝛼2 from 𝐿^𝑝_dec(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ²⁺), it is necessary and sufficient that conditions (2.10) and (2.12) are satisfied.(ii)If 𝑣∈DC(𝑦), then ℛ_𝛼1,𝛼2 is bounded from 𝐿^𝑝_dec(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ²⁺) if and only If conditions (2.10) and (2.11) are satisfied.(iii)If 𝑣∈DC(𝑥)∩DC(𝑦), then ℛ_𝛼1,𝛼2 is bounded from 𝐿^𝑝_dec(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ²⁺) 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 ℛ_𝛼1,𝛼2 with 𝛼₁,𝛼₂>1 (see [25], [13, Section  1.6]).

Theorem D. Let 1<𝑝≤𝑞<∞, and let 𝛼₁,𝛼₂≥1. (i)Suppose that 𝑤^1−𝑝′∈DC(𝑦). Then the operator ℛ_𝛼1,𝛼2 is bounded from 𝐿^𝑝(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ²⁺) if and only if 𝑃₁∶=sup_𝑎,𝑏>0⎛⎜⎜⎝^𝑎0^𝑏0𝑤^1−𝑝′𝑥₁,𝑥₂𝑎−𝑥₁^{(1−𝛼1)𝑝′}𝑑𝑥₁𝑑𝑥₂⎞⎟⎟⎠^1/𝑝′^∞𝑎^∞𝑏𝑣𝑥₁,𝑥₂𝑥^{(1−𝛼22)𝑞}𝑑𝑥₁𝑑𝑥₂^1/𝑞𝑃<∞,₂∶=sup_𝑎,𝑏>0^𝑎0^𝑏0𝑤^1−𝑝′𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂^1/𝑝′^∞𝑎^∞𝑏𝑣𝑥₁,𝑥₂𝑥₁−𝑎^{(1−𝛼1)𝑞}𝑥^{(1−𝛼22)𝑞}𝑑𝑥₁𝑑𝑥₂^1/𝑞<∞.(4.4) Moreover, ‖ℛ_𝛼1,𝛼2‖≈max{𝑃₁,𝑃₂}.(ii)Let 𝑤^1−𝑝∈DC(𝑥). Then the operator ℛ_𝛼1,𝛼2 is bounded from 𝐿^𝑝(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ₊) if and only if 𝑃₁∶=sup_𝑎,𝑏>0⎛⎜⎜⎝^𝑎0^𝑏0𝑤^1−𝑝′𝑥₁,𝑥₂𝑏−𝑥₂^{(1−𝛼2)𝑝′}𝑑𝑥₁𝑑𝑥₂⎞⎟⎟⎠^1/𝑝′^∞𝑎^∞𝑏𝑣𝑥₁,𝑥₂𝑥^{(1−𝛼11)𝑞}𝑑𝑥₁𝑑𝑥₂^1/𝑞𝑃<∞,₂∶=sup_𝑎,𝑏>0^𝑎0^𝑏0𝑤^1−𝑝′𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂^1/𝑝′^∞𝑎^∞𝑏𝑣𝑥₁,𝑥₂𝑥₂−𝑏^{(1−𝛼2)𝑞}𝑥^{(1−𝛼11)𝑞}𝑑𝑥₁𝑑𝑥₂^1/𝑞<∞.(4.5) Moreover, ‖ℛ_𝛼1,𝛼2‖≈max{𝑃₁,𝑃₂}.

Let us introduce the following multiple integral operators: (ℋℛ)_𝛼1,𝛼2𝑓𝑥₁,𝑥₂=𝑥^𝛼11−1^𝑥1₀^𝑥2₀𝑓𝑡₁,𝑡₂𝑑𝑡₁𝑑𝑡₂𝑥₂−𝑡₂^1−𝛼2(ℛℋ)_𝛼1,𝛼2,𝑓𝑥₁,𝑥₂=𝑥^𝛼22−1^𝑥1₀^𝑥2₀𝑓𝑡₁,𝑡₂𝑑𝑡₁𝑑𝑡₂𝑥₁−𝑡₁^1−𝛼1(ℋ𝒲)_𝛼1,𝛼2,𝑓𝑥₁,𝑥₂=𝑥^𝛼11−1^𝑥1₀^∞𝑥₂𝑓𝑡₁,𝑡₂𝑑𝑡₁𝑑𝑡₂𝑡₂−𝑥₂^1−𝛼2(𝒲ℋ)_𝛼1,𝛼2,𝑓𝑥₁,𝑥₂=𝑥^𝛼22−1^∞𝑥₁^𝑥2₀𝑓𝑡₁,𝑡₂𝑑𝑡₁𝑑𝑡₂𝑡₁−𝑥₁^1−𝛼1ℋ_,ℛ_𝛼1,𝛼2𝑓𝑥₁,𝑥₂=^∞𝑥₁^𝑥2₀𝑓𝑡₁,𝑡₂𝑑𝑡₁𝑑𝑡₂𝑡^1−𝛼1₁𝑥₂−𝑡₂^1−𝛼2ℛℋ_,_𝛼1,𝛼2𝑓𝑥₁,𝑥₂=^𝑥1₀^∞𝑥₂𝑓𝑡₁,𝑡₂𝑑𝑡₁𝑑𝑡₂𝑥₁−𝑡₁^1−𝛼1𝑡^1−𝛼2₂ℋ_,𝒲_𝛼1,𝛼2𝑓𝑥₁,𝑥₂=^∞𝑥₁^∞𝑥₂𝑓𝑡₁,𝑡₂𝑑𝑡₁𝑑𝑡₂𝑡^1−𝛼1₁𝑡₂−𝑥₂^1−𝛼2𝒲ℋ_,_𝛼1,𝛼2𝑓𝑥₁,𝑥₂=^∞𝑥₁^∞𝑥₂𝑓𝑡₁,𝑡₂𝑑𝑡₁𝑑𝑡₂𝑡₁−𝑥₁^1−𝛼1𝑡^1−𝛼2₂.(4.6)

Now we prove some auxiliary statements.

Proposition 4.3. Let 1<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 𝐿^𝑝(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ₊) if and only if 𝐼₁∶=sup_𝑎,𝑏>0⎛⎜⎜⎝^𝑎0^𝑏0𝑤^1−𝑝′𝑥₁,𝑥₂𝑎−𝑥₁^{(1−𝛼1)𝑝′}𝑑𝑥₁𝑑𝑥₂⎞⎟⎟⎠^1/𝑝′^∞𝑎^∞𝑏𝑣𝑥₁,𝑥₂𝑥^{(1−𝛼22)𝑞}𝑑𝑥₁𝑑𝑥₂^1/𝑞𝐼<∞,₂∶=sup_𝑎,𝑏>0^𝑎0^𝑏0𝑤^1−𝑝′𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂^1/𝑝′^∞𝑎^∞𝑏𝑣𝑥₁,𝑥₂𝑥₁−𝑎^{(1−𝛼1)𝑞}𝑥^{(1−𝛼22)𝑞}𝑑𝑥₁𝑑𝑥₂^1/𝑞<∞.(4.7) Moreover, ‖(ℛℋ)_𝛼1,𝛼2‖≈max{𝐼₁,𝐼₂}.(ii)The operator (𝒲ℋ)_𝛼1,𝛼2 is bounded from 𝐿^𝑝(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣ℝ₊) if and only if 𝐽₁∶=sup_𝑎,𝑏>0^𝑎0^∞𝑏𝑣𝑥₁,𝑥₂𝑎−𝑥₁^{(1−𝛼1)𝑞}𝑥^{(1−𝛼22)𝑞}𝑑𝑥₁𝑑𝑥₂^1/𝑞^∞𝑎^𝑏0𝑤^1−𝑝′𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂^1/𝑞𝐽<∞,₂∶=sup_𝑎,𝑏>0^𝑎0^∞𝑏𝑣𝑥₁,𝑥₂𝑥^{(1−𝛼22)𝑞}𝑑𝑥₁𝑑𝑥₂^1/𝑞⎛⎜⎜⎝^∞𝑎^𝑏0𝑤^1−𝑝′𝑥₁,𝑥₂𝑥₁−𝑎^{(1−𝛼1)𝑝′}𝑑𝑥₁𝑑𝑥₂⎞⎟⎟⎠^1/𝑝′<∞.(4.8) Moreover, ‖(𝒲ℋ)_𝛼1,𝛼2‖≈max{𝐽₁,𝐽₂}.(iii)The operator (ℛℋ′)_𝛼1,𝛼2 is bounded from 𝐿^𝑝(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ₊) if and only if 𝐽^1∶=sup_𝑎,𝑏>0^∞𝑎^𝑏0𝑣𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂^1/𝑞⎛⎜⎜⎝^𝑎0^∞𝑏𝑤^1−𝑝′𝑥₁,𝑥₂𝑥^{(1−𝛼2)𝑝′}₂𝑎−𝑥₁^{(1−𝛼1)𝑝′}𝑑𝑥₁𝑑𝑥₂⎞⎟⎟⎠^1/𝑝′𝐽<∞,_2∶=sup_𝑎,𝑏>0^∞𝑎^𝑏0𝑣𝑥₁,𝑥₂𝑥₁−𝑎^{(1−𝛼1)𝑞}𝑑𝑥₁𝑑𝑥₂^1/𝑞^𝑎0^∞𝑏𝑤^1−𝑝′𝑥₁,𝑥₂𝑥^{(1−𝛼2)𝑝′}₂𝑑𝑥₁𝑑𝑥₂^1/𝑝′<∞.(4.9) Moreover, ‖(ℛℋ′)_𝛼1,𝛼2‖≈max{𝐽^1,𝐽^2}.(iv)The operator (𝒲ℋ′)_𝛼1,𝛼2 is bounded from 𝐿^𝑝(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ₊) if and only if 𝐼^1∶=sup_𝑎,𝑏>0^𝑎0^𝑏0𝑣𝑥₁,𝑥₂𝑎−𝑥₁^{(1−𝛼1)𝑞}𝑑𝑥₁𝑑𝑥₂^1/𝑞^∞𝑎^∞𝑏𝑤^1−𝑝′𝑥₁,𝑥₂𝑥^{(1−𝛼2)𝑝′}₂𝑑𝑥₁𝑑𝑥₂^1/𝑝′𝐼<∞,_2∶=sup_𝑎,𝑏>0^𝑎0^𝑏0𝑣𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂^1/𝑞⎛⎜⎜⎝^∞𝑎^∞𝑏𝑤^1−𝑝′𝑥₁,𝑥₂𝑥^{(1−𝛼2)𝑝′}₂𝑥₁−𝑎^{(1−𝛼1)𝑝′}𝑑𝑥₁𝑑𝑥₂⎞⎟⎟⎠^1/𝑝′<∞.(4.10) Moreover, ‖(𝒲ℋ′)_𝛼1,𝛼2‖≈max{𝐼^1,𝐼^2}.

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 [25] (see also the proof of Theorem  1.1.6 in [13]).
Sufficiency. First suppose that 𝑆∶=∫^∞0𝑤^21−𝑝(𝑥₂)𝑑𝑥₂=∞. Let {𝑎_𝑘}^{+∞𝑘=−∞} be a sequence of positive numbers for which the equality 2^𝑘=^𝑎𝑘₀𝑤^1−𝑝′₂𝑥₂𝑑𝑥₂(4.11) holds for all 𝑘∈ℤ. It is clear that {𝑎_𝑘} is increasing and ℝ₊=∪_𝑘∈ℤ[𝑎_𝑘,𝑎_𝑘+1). Moreover, it is easy to verify that 2^𝑘=^𝑎𝑘+1_𝑎𝑘𝑤^1−𝑝′₂𝑥₂𝑑𝑥₂.(4.12) Let 𝑓≥0. We have that ‖‖(ℛℋ)_𝛼1,𝛼2𝑓‖‖^𝑞𝐿_{𝑞𝑣,ℝ2+}=_ℝ2+𝑣𝑥₁,𝑥₂(ℛℋ)_𝛼1,𝛼2𝑓^𝑞𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂≤_𝑘∈ℤ^∞0^𝑎𝑘+1_𝑎𝑘𝑣𝑥₁,𝑥₂𝑥^{(1−𝛼22)𝑞}^𝑥1₀^𝑥2₀𝑓𝑡₁,𝑡₂𝑥₁−𝑡₁^1−𝛼1𝑑𝑡₁𝑑𝑡₂^𝑞𝑑𝑥₁𝑑𝑥₂≤_𝑘∈ℤ^∞0^𝑎𝑘+1_𝑎𝑘𝑣𝑥₁,𝑥₂𝑥^{(1−𝛼22)𝑞}𝑑𝑥₂^𝑥1₀𝑥₁−𝑡₁^𝛼1−1^𝑎𝑘+1₀𝑓𝑡₁,𝑡₂𝑑𝑡₂𝑑𝑡¹^𝑞𝑑𝑥₁=_𝑘∈ℤ^∞0𝑉_𝑘𝑥₁^𝑥1₀𝑥₁−𝑡₁^(𝛼1−1)𝐹_𝑘𝑡₁𝑑𝑡₁^𝑞𝑑𝑥₁,(4.13) where 𝑉_𝑘𝑥₁∶=^𝑎𝑘+1_𝑎𝑘𝑣𝑥₁,𝑥₂𝑥^{(1−𝛼22)𝑞}𝑑𝑥₂,𝐹_𝑘𝑡₁∶=^𝑎𝑘+1₀𝑓𝑡₁,𝑡₂𝑑𝑡₂.(4.14)
It is obvious that 𝐼^𝑞1≥sup_{𝑎>0𝑗∈ℤ}^∞𝑎^𝑎𝑗+1_𝑎𝑗𝑣𝑥₁,𝑥₂𝑥₁−𝑎^{(1−𝛼1)𝑞}𝑥^{(1−𝛼22)𝑞}𝑑𝑥₁𝑑𝑥₂⎛⎜⎜⎝^𝑎0^𝑎𝑗₀𝑤^1−𝑝′𝑥₁,𝑥₂𝑎−𝑥₁^{(1−𝛼1)𝑝′}𝑑𝑥₁𝑑𝑥₂⎞⎟⎟⎠^𝑞/𝑝′𝐼_𝑞2,≥sup_{𝑎>0𝑗∈ℤ}^∞𝑎^𝑎𝑗+1_𝑎𝑗𝑣𝑥₁,𝑥₂𝑥^{(1−𝛼22)𝑞}𝑑𝑥₁𝑑𝑥₂⎛⎜⎜⎝^𝑎0^𝑎𝑗₀𝑤^1−𝑝′𝑥₁,𝑥₂𝑎−𝑥₁^{(1−𝛼1)𝑝′}𝑑𝑥₁𝑑𝑥₂⎞⎟⎟⎠^𝑞/𝑝′.(4.15) Hence, by using the two-weight criteria for the one-dimensional Riemann-Liouville operator without singularity (see [24]), we find that ‖‖(ℛℋ)_𝛼1,𝛼2𝑓‖‖^𝑞𝐿_{𝑞𝑣,ℝ2+}≤𝑐𝐼^𝑞_𝑗∈ℤ^∞0𝑤₁𝑥₁^𝑎𝑗₀𝑤^1−𝑝′₂𝑥₂𝑑𝑥₂^1−𝑝𝐹_𝑗𝑥₁^𝑝𝑑𝑥₁^𝑞/𝑝≤𝑐𝐼^𝑞^∞0𝑤₁𝑥₁_𝑗∈𝑍^𝑎𝑗₀𝑤^1−𝑝′₂𝑥₂𝑑𝑥₂^1−𝑝^𝑗_𝑘=−∞^𝑎𝑘+1_𝑎𝑘𝑓𝑥₁,𝑡₂𝑑𝑡₂^𝑝𝑑𝑥₁^𝑞/𝑝,(4.16) where 𝐼=max{𝐼₁,𝐼₂}.
On the other hand, (4.11) yields ^+∞_𝑘=𝑛^𝑎𝑘₀𝑤^1−𝑝′₂𝑥₂𝑑𝑥₂^1−𝑝^𝑛_𝑘=−∞^𝑎𝑘+1_𝑎𝑘𝑤^1−𝑝′₂𝑥₂𝑑𝑥₂^𝑝−1=^+∞_𝑘=𝑛^𝑎𝑘₀𝑤^1−𝑝′₂𝑥₂𝑑𝑥₂^1−𝑝^𝑎𝑛+1₀𝑤^1−𝑝′₂𝑥₂𝑑𝑥₂^𝑝−1=^+∞_𝑘=𝑛2^{𝑘(1−𝑝)}2^{(𝑛+1)(𝑝−1)}≤𝑐(4.17) for all 𝑛∈ℤ. Hence by Hardy’s inequality in discrete case (see, for example, [25, 26]) and Hölder’s inequality we have that ‖‖(ℛℋ)_𝛼1,𝛼2𝑓‖‖^𝑞𝐿_{𝑞𝑣,ℝ2+}≤𝑐𝐼^𝑞⎡⎢⎢⎣^∞0𝑤₁𝑥₁_𝑗∈ℤ^𝑎𝑗+1_𝑎𝑗𝑤^1−𝑝′₂𝑥₂𝑑𝑥₂^1−𝑝^𝑎𝑗+1_𝑎𝑗𝑓𝑥₁,𝑡₂𝑑𝑡₂^𝑝𝑑𝑥₁⎤⎥⎥⎦^𝑞/𝑝≤𝑐𝐼^𝑞^∞0𝑤₁𝑥₁_𝑗∈ℤ^𝑎𝑗+1_𝑎𝑗𝑤₂𝑡₂𝑓^𝑝𝑥₁,𝑡₂𝑑𝑡₂𝑑𝑥¹^𝑞/𝑝=𝑐𝐼^𝑞‖𝑓‖^𝑞𝐿_{𝑝𝑤,ℝ2+}.(4.18)
If 𝑆<∞, then without loss of generality we can assume that 𝑆=1. In this case we choose the sequence {𝑎_𝑘}^0𝑘=−∞ 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 {𝑥_𝑘} so that ∫^∞𝑥_𝑘𝑤^21−𝑝(𝑥)𝑑𝑥=2^𝑘 (notice that 𝑥_𝑘 is decreasing) and argue as in the proof of (i).

Proposition 4.4. Let 1<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 𝐿^𝑝(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ²⁺) if and only if 𝐼₁∶=sup_𝑎,𝑏>0⎛⎜⎜⎝^𝑎0^𝑏0𝑤^1−𝑝′𝑥₁,𝑥₂𝑏−𝑥₂^{(1−𝛼2)𝑝′}𝑑𝑥₁𝑑𝑥₂⎞⎟⎟⎠^1/𝑝′^∞𝑎^∞𝑏𝑣𝑥₁,𝑥₂𝑥^{(1−𝛼11)𝑞}𝑑𝑥₁𝑑𝑥₂^1/𝑞𝐼<∞,₂∶=sup_𝑎,𝑏>0^𝑎0^𝑏0𝑤^1−𝑝′𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂^1/𝑝′^∞𝑎^∞𝑏𝑣𝑥₁,𝑥₂𝑥^{(1−𝛼11)𝑞}𝑥₂−𝑏^{(1−𝛼2)𝑞}𝑑𝑥₁𝑑𝑥₂^1/𝑞<∞.(4.19) Moreover, ‖(ℋℛ)_𝛼1,𝛼2‖≈max{𝐼₁,𝐼₂}.(ii)The operator (ℋ𝒲)_𝛼1,𝛼2 is bounded from 𝐿^𝑝(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ₊) if and only if 𝐽₁∶=sup_𝑎,𝑏>0^∞𝑎^𝑏0𝑣𝑥₁,𝑥₂𝑥^{(1−𝛼11)𝑞}𝑏−𝑥₂^{(1−𝛼2)𝑞}𝑑𝑥₁𝑑𝑥₂^1/𝑞^𝑎0^∞𝑏𝑤^1−𝑝′𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂^1/𝑝′𝐽<∞,₂∶=sup_𝑎,𝑏>0^∞𝑎^𝑏0𝑣𝑥₁,𝑥₂𝑥^{(𝛼11−1)𝑞}𝑑𝑥₁𝑑𝑥₂^1/𝑞⎛⎜⎜⎝^𝑎0^∞𝑏𝑤^1−𝑝′𝑥₁,𝑥₂𝑥₂−𝑏^{(1−𝛼2)𝑝′}𝑑𝑥₁𝑑𝑥₂⎞⎟⎟⎠^1/𝑝′<∞.(4.20) Moreover, ‖(ℋ𝒲)_𝛼1,𝛼2‖≈max{𝐽₁,𝐽₂}.(iii)The operator (ℋ′ℛ)_𝛼1,𝛼2 is bounded from 𝐿^𝑝(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ₊) if and only if 𝐽^1∶=sup_𝑎,𝑏>0^𝑎0^∞𝑏𝑣𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂^1/𝑞⎛⎜⎜⎝^∞𝑎^𝑏0𝑤^1−𝑝′𝑥₁,𝑥₂𝑥^{(1−𝛼1)𝑝′}₁𝑏−𝑥₂^{(1−𝛼2)𝑝′}𝑑𝑥₁𝑑𝑥₂⎞⎟⎟⎠^1/𝑝′𝐽<∞,_2∶=sup_𝑎,𝑏>0^𝑎0^∞𝑏𝑣𝑥₁,𝑥₂𝑥₂−𝑏^{(1−𝛼2)𝑞}𝑑𝑥₁𝑑𝑥₂^1/𝑞^∞𝑎^𝑏0𝑤^1−𝑝′𝑥₁,𝑥₂𝑥^{(1−𝛼1)𝑝′}₁𝑑𝑥₁𝑑𝑥₂^1/𝑝′<∞.(4.21) Moreover, ‖(ℋ′ℛ)_𝛼1,𝛼2‖≈max{𝐽^1,𝐽^2}.(iv)The operator (ℋ′𝒲)_𝛼1,𝛼2 is bounded from 𝐿^𝑝(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ₊) if and only if 𝐼^1∶=sup_𝑎,𝑏>0^𝑎0^𝑏0𝑣𝑥₁,𝑥₂𝑏−𝑥₂^{(1−𝛼2)𝑞}𝑑𝑥₁𝑑𝑥₂^1/𝑞⎛⎜⎜⎝^∞𝑎^∞𝑏𝑤^1−𝑝′𝑥₁,𝑥₂x^{(1−𝛼1)𝑝′}₁𝑑𝑥₁𝑑𝑥₂⎞⎟⎟⎠^1/𝑝′𝐼<∞,_2∶=sup_𝑎,𝑏>0^𝑎0^𝑏0𝑣𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂^1/𝑞⎛⎜⎜⎝^∞𝑎^∞𝑏𝑤^1−𝑝′𝑥₁,𝑥₂𝑥^{(1−𝛼1)𝑝′}₁𝑥₂−𝑏^{(1−𝛼2)𝑝′}𝑑𝑥₁𝑑𝑥₂⎞⎟⎟⎠^1/𝑝′<∞.(4.22) Moreover, ‖(ℋ′𝒲)_𝛼1,𝛼2‖≈max{𝐼^1,𝐼^2}.

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

Theorem 4.5. Let 1<𝑝≤𝑞<∞, and let 0<𝛼₁,𝛼₂≤1. Suppose that the weight function 𝑤 on ℝ²⁺ is of product type, that is, 𝑤(𝑥₁,𝑥₂)=𝑤₁(𝑥₁)𝑤₂(𝑥₂). Suppose also that 𝑊₁(∞)=𝑊₂(∞)=∞.
(i)If 𝑣∈DC(𝑦), then 𝒲_𝛼1,𝛼2 is bounded from 𝐿^𝑝_dec(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ²⁺) if and only if 𝐴₁∶=sup_𝑎,𝑏>0^𝑎0^𝑏0𝑣𝑥₁,𝑥₂𝑎−𝑥₁^𝛼1𝑞𝑑𝑥₁𝑑𝑥₂^1/𝑞×^𝑎0𝑤₁𝑥₁𝑑𝑥₁^−1/𝑝^∞𝑏𝑊^−𝑝′₂𝑥₂𝑤₂𝑥₂𝑥^𝛼2𝑝′₂𝑑𝑥₂^1/𝑝′𝐴<∞,(4.23)₂∶=sup_𝑎,𝑏>0^𝑎0^𝑏0𝑣𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂^1/𝑞×^∞𝑎^∞𝑏𝑊^−𝑝′𝑥₁,𝑥₂𝑤𝑥₁,𝑥₂𝑥₁−𝑎^𝛼1𝑝′𝑥^𝛼2𝑝′₂𝑑𝑥₁𝑑𝑥₂^1/𝑝′<∞.(4.24)(ii)If 𝑣∈DC(𝑥), then 𝒲_𝛼1,𝛼2 is bounded from 𝐿^{𝑝𝑑𝑒𝑐}(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ²⁺) if and only if 𝐵₁∶=sup_𝑎,𝑏>0^𝑎0^𝑏0𝑣𝑥₁,𝑥₂𝑏−𝑥₂^𝛼2𝑞𝑑𝑥₁𝑑𝑥₂^1/𝑞×^∞𝑎𝑊^−𝑝′₁𝑥₁𝑤₁𝑥₁𝑥^𝛼1𝑝′₁𝑑𝑥₁^1/𝑝′^𝑏0𝑤₂𝑥₂𝑑𝑥₂^−1/𝑝𝐵<∞,₂∶=sup_𝑎,𝑏>0^𝑎0^𝑏0𝑣𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂^1/𝑞×^∞𝑎^∞𝑏𝑊^−𝑝′𝑥₁,𝑥₂𝑤𝑥₁,𝑥₂𝑥₂−𝑏^𝛼2𝑝′𝑥^𝛼1𝑝′₁𝑑𝑥₁𝑑𝑥₂^1/𝑝′<∞.(4.25)(iii)If 𝑣∈DC(𝑥)∩DC(𝑦), then 𝒲_𝛼1,𝛼2 is bounded from 𝐿^𝑝_dec(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ²⁺) if and only if 𝐶₁∶=sup_𝑎,𝑏>0^∞𝑎^∞𝑏𝑊^−𝑝′𝑥₁,𝑥₂𝑤𝑥₁,𝑥₂𝑥^𝛼2𝑝′₂𝑥^𝛼1𝑝′₁𝑑𝑥₁𝑑𝑥₂^1/𝑝′×^𝑎0^𝑏0𝑣𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂^1/𝑞<∞.(4.26)

Proof. By using Proposition A we see that the operator 𝒲_𝛼1,𝛼2 is bounded from 𝐿^𝑝_dec(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ²⁺) if and only if the inequality ⎛⎜⎜⎝_ℝ2+^𝑥1₀^𝑥2₀^𝜏1₀^𝜏2₀𝑔𝑡₁,𝑡₂𝑑𝑡₁𝑑𝑡₂𝜏₁−𝑡₁^1−𝛼1𝜏₂−𝑡₂^1−𝛼2𝑑𝜏¹𝑑𝜏₂^𝑝′×𝑊^−𝑝′𝑥₁,𝑥₂𝑤𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂⎞⎟⎟⎠^1/𝑝′≤𝑐_ℝ2+𝑔^𝑞′𝑣^1−𝑞′^1/𝑞′(4.27) holds for all 𝑔≥0. Further, it is easy to see that ^𝑥1₀^𝑥2₀^𝜏1₀^𝜏2₀𝑔t₁,𝑡₂𝑑𝑡₁𝑑𝑡₂𝜏₁−𝑡₁^1−𝛼1𝜏₂−𝑡₂^1−𝛼2𝑑𝜏¹𝑑𝜏₂=^𝑥1₀^𝑥2₀𝑔𝑡₁,𝑡₂^𝑥1_𝑡1^𝑥2_𝑡2𝑑𝜏₁𝑑𝜏₂𝜏₁−𝑡₁^1−𝛼1𝜏₂−𝑡₂^1−𝛼2𝑑𝑡¹𝑑𝑡₂=𝑐_𝛼1,𝛼2^𝑥1₀^𝑥2₀𝑔𝑡₁,𝑡₂𝑥₁−𝑡₁^𝛼1𝑥₂−𝑡₂^𝛼2𝑑𝑡₁𝑑𝑡₂.(4.28) Hence 𝒲_𝛼1,𝛼2 is bounded from 𝐿^𝑝_dec(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ²⁺) if and only if ℛ_{𝛼1+1,𝛼2+1} is bounded from 𝐿^𝑞′(𝑣^1−𝑞,ℝ²⁺) to 𝐿^𝑝′(𝑊^−𝑝′𝑤,ℝ²⁺).
By using Theorem D, (i) and (ii) follow immediately.
To prove (iii) we show that if 𝑣∈DC(𝑥)∩DC(𝑦), then (4.26) implies (4.23) and (4.24). Let 𝑎,𝑏>0. Then 𝑎∈[2^𝑚0,2^𝑚0+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 ^𝑎0^𝑏0𝑣𝑥₁,𝑥₂𝑎−𝑥₁^𝛼1q𝑑𝑥₁𝑑𝑥₂^𝑝′/𝑞^𝑎0𝑤₁𝑥₁𝑑𝑥₁^{−𝑝′/𝑝}=𝑐^𝑎0^𝑏0𝑣𝑥₁,𝑥₂𝑎−𝑥₁^𝛼1𝑞𝑑𝑥₁𝑑𝑥₂^𝑝′/𝑞^∞𝑎𝑊^−𝑝′₁𝑥₁𝑤₁𝑥₁𝑑𝑥₁≤𝑐_∞_𝑘_=𝑚0^2𝑘+1₂^𝑘𝑊^−𝑝′₁𝑥₁𝑤₁𝑥₁𝑑𝑥₁2^{(𝑚0+1)𝛼1𝑝′}^2𝑚0₀^𝑏0𝑣𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂^𝑝′/𝑞≤𝑐^∞_𝑘_=𝑚0^2𝑘+1₂^𝑘𝑥^𝛼1𝑝′₁𝑊^−𝑝′₁𝑥₁𝑤₁𝑥₁𝑑𝑥₁𝑐^{(𝑚0−𝑘)(𝑝′1/𝑞)}^2𝑘₀^𝑏0𝑣𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂^𝑝′/𝑞≤𝑐𝐶^𝑝′₁^∞𝑏𝑊^−𝑝′₂𝑥₂𝑤₂𝑥₂𝑥^𝛼2𝑝′₂𝑑𝑥₂⁻¹.(4.29) 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∈[2^𝑚0,2^𝑚0+1), by using the doubling condition for 𝑣 with respect to the first variable and Remark 4.2, we have ^𝑎0^𝑏0𝑣𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂^𝑝′/𝑞^∞𝑎𝑊^−𝑝′₁𝑥₁𝑤𝑥₁𝑥^𝛼1𝑝′₁𝑑𝑥₁≤𝑐_∞_𝑘_=𝑚0^2𝑘+1₂^𝑘𝑊^−𝑝′₁𝑥₁𝑤𝑥₁𝑑𝑥₁2^{𝑘𝛼1𝑝′}^2𝑚0+1₀^𝑏0𝑣𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂^𝑝′/𝑞≤𝑐^∞_𝑘_=𝑚0^2𝑘+1₂^𝑘𝑊^−𝑝′₁𝑥₁𝑤𝑥₁𝑑𝑥₁𝑐^{(𝑚0−𝑘+2)(𝑝′1/𝑞)}^2𝑘−1₀^𝑏02^𝑘−𝑥₁^𝛼1𝑞𝑣𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂^𝑝′/𝑞≤𝑐𝐴^𝑝′₁^∞𝑏𝑊^−𝑝′₂𝑥₂𝑤₂𝑥₂𝑥^𝛼2𝑝′₂𝑑𝑥₂⁻¹.(4.30) 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. Let 1<𝑝≤𝑞<∞, and let 0<𝛼₁,𝛼₂≤1. Suppose that the weight function 𝑤 on ℝ²⁺ 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 𝐿^𝑝_dec(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ²⁺) if and only if sup_𝑎,𝑏>0^𝑎0^𝑏0𝑥^𝛼1𝑞1𝑣𝑥₁,𝑥₂(𝑏−𝑥₂)^−𝛼2𝑞𝑑𝑥₁𝑑𝑥₂^1/𝑞^𝑎0^𝑏0𝑤₁𝑥₁𝑤₂𝑥₂𝑑𝑥₁𝑑𝑥₂^−1/𝑝sup_𝑎,𝑏>0<∞,(4.31)^𝑎0^𝑏0𝑥^𝛼1𝑞1𝑣𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂^1/𝑞^𝑎0𝑤₁𝑥₁𝑑𝑥₁^−1/𝑝×^∞𝑏𝑊^−𝑝′₂𝑥₂𝑤₂𝑥₂𝑥₂−𝑏^𝛼2𝑝′𝑑𝑥₂^1/𝑝′sup_𝑎,𝑏>0<∞,(4.32)^∞𝑎^𝑏0𝑣𝑥₁,𝑥₂𝑥^{(1−𝛼11)𝑞}𝑏−𝑥₂^−𝛼2𝑞𝑑x₁𝑑𝑥₂^1/𝑞^𝑎0𝑥^𝑝′₁𝑊^−𝑝′₁𝑥₁𝑤₁𝑥₁𝑑𝑥₁^1/𝑝′×^𝑏0𝑤₂𝑥₂𝑑𝑥₂^−1/𝑝sup_𝑎,𝑏>0<∞,(4.33)^∞𝑎^𝑏0𝑥^{(𝛼11−1)𝑞}𝑣𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂^1/𝑞⎛⎜⎜⎝^𝑎0^∞𝑏𝑊^−𝑝′𝑥₁,𝑥₂𝑤𝑥₁,𝑥₂𝑥^𝑝′₁𝑥₂−𝑏^{−𝛼2𝑝′}𝑑𝑥₁𝑑𝑥₂⎞⎟⎟⎠^1/𝑝′<∞.(4.34)(ii)The operator (𝒲ℛ)_𝛼1,𝛼2 is bounded from 𝐿^𝑝_dec(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ²⁺) if and only if sup_𝑎,𝑏>0^𝑎0^𝑏0𝑥^𝛼2𝑞2𝑣𝑥₁,𝑥₂𝑎−𝑥₁^−𝛼1𝑞𝑑𝑥₁𝑑𝑥₂^1/𝑞^𝑎0^𝑏0𝑤₁𝑥₁𝑤₂𝑥₂𝑑𝑥₁𝑑𝑥₂^−1/𝑝sup_𝑎,𝑏>0<∞,(4.35)^𝑎0^𝑏0𝑥^𝛼2𝑞2𝑣𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂^1/𝑞^𝑏0𝑤₂𝑥₂𝑑𝑥₂^−1/𝑝×^∞𝑎𝑊^−𝑝′₁𝑥₁𝑤₁𝑥₁𝑥₁−𝑎^𝛼1𝑝′𝑑𝑥₁^1/𝑝′sup_𝑎,𝑏>0<∞,(4.36)^𝑎0^∞𝑏𝑣𝑥₁,𝑥₂𝑥^{(1−𝛼22)𝑞}𝑎−𝑥₁^−𝛼1𝑞𝑑𝑥₁𝑑𝑥₂^1/𝑞^𝑎0𝑤₁𝑥₁𝑑𝑥₁^−1/𝑝×^𝑏0𝑥^𝑝′₂𝑊^−𝑝′₂𝑥₂𝑤₂𝑥₂𝑑𝑥₂^1/𝑝′sup_𝑎,𝑏>0<∞,(4.37)^𝑎0^∞𝑏𝑥^{(𝛼22−1)𝑞}𝑣𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂^1/𝑞⎛⎜⎜⎝^∞𝑎^𝑏0𝑊^−𝑝′𝑥₁,𝑥₂𝑤𝑥₁,𝑥₂𝑥^𝑝′₂𝑥₁−𝑎^{−𝛼1𝑝′}𝑑𝑥₁𝑑𝑥₂⎞⎟⎟⎠^1/𝑝′<∞.(4.38)

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 (ℛ𝒲)_𝛼1,𝛼2𝑓≈(ℋ𝒲)_𝛼1,𝛼2𝑓, 𝑓↓, holds. Indeed, by using the fact that 𝑓 is nonincreasing in the first variable, we find that (ℛ𝒲=)_𝛼1,𝛼2𝑓_𝑥10/2𝑥₁,𝑥₂^∞𝑥₂(⋯)+^𝑥1_𝑥1/2^∞𝑥₂≤𝑐_𝛼1(⋯)𝑥^𝛼11−1^𝑥10/2^∞𝑥₂𝑓𝑡₁,𝑡₂𝑡₂−𝑥₂^1−𝛼2𝑑𝑡₁𝑑𝑡₂+𝑐^𝛼₁𝑥^𝛼11−1^𝑥10/2^∞𝑥₂𝑓𝑡₁,𝑡₂𝑡₂−𝑥₂^1−𝛼2𝑑𝑡₁𝑑𝑡₂≤𝑐_𝛼1,𝛼2(ℋ𝒲)_𝛼1,𝛼2𝑓𝑥₁,𝑥₂.(4.39) The inequality (ℋ𝒲)_𝛼1,𝛼2𝑓𝑥₁,𝑥₂≤(ℛ𝒲)_𝛼1,𝛼2𝑓𝑥₁,𝑥₂(4.40) is obvious because 𝑥₁−𝑡₁≤𝑥₁ for 0<𝑡₁≤𝑥₁.
Further, it is easy to check that ^𝑥1₀^𝑥2₀^∞𝜏₁^𝜏2₀𝑔𝑡₁,𝑡₂𝑡^1−𝛼1₁𝜏₂−𝑡₂^1−𝛼2𝑑𝑡₁𝑑𝑡₂𝑑𝜏¹𝑑𝜏₂=^𝑥1₀^𝑥2₀^𝑥1_𝜏1^𝜏2₀𝑔𝑡₁,𝑡₂𝑡^1−𝛼1₁𝜏₂−𝑡₂^1−𝛼2𝑑𝑡₁𝑑𝑡₂𝑑𝜏¹𝑑𝜏₂+^𝑥1₀^𝑥2₀^∞𝑥₁^𝜏2₀𝑔𝑡₁,𝑡₂𝑡^1−𝛼1₁𝜏₂−𝑡₂^1−𝛼2𝑑𝑡₁𝑑𝑡₂𝑑𝜏¹𝑑𝜏₂=^𝑥1₀^𝑥2₀𝑔𝑡₁,𝑡₂𝑡^𝛼11−1^𝑡1₀^𝑥2_𝑡2𝜏₂−𝑡₂^𝛼2−1𝑑𝜏₁𝑑𝜏₂𝑑𝑡¹𝑑𝑡₂+^∞𝑥₁^𝑥2₀𝑔𝑡₁,𝑡₂𝑡^𝛼11−1^𝑥1₀^𝑥2_𝑡2𝜏₂−𝑡₂^𝛼2−1𝑑𝜏₁𝑑𝜏₂𝑑𝑡¹𝑑𝑡₂=𝑐^𝑥1₀^𝑥2₀g𝑡₁,𝑡₂𝑡^𝛼1₁𝑥₂−𝑡₂^𝛼2𝑑𝑡₁𝑑𝑡₂+𝑐𝑥₁^∞𝑥₁^𝑥2₀𝑔𝑡₁,𝑡₂𝑡^𝛼11−1𝑥₂−𝑡₂^𝛼2𝑑𝑡₁𝑑𝑡₂.(4.41)
Hence, since the boundedness of (ℋ𝒲)_𝛼1,𝛼2 from 𝐿^𝑝_dec(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ²⁺) is equivalent to the inequality (see also [4]) ⎛⎜⎜⎝_ℝ2+^𝑥1₀^𝑥2₀^∞𝜏₁^𝜏2₀𝑔𝑡₁,𝑡₂𝑑𝑡₁𝑑𝑡₂𝑡^1−𝛼1₁𝜏₂−𝑡₂^1−𝛼2𝑑𝜏¹𝑑𝜏₂^𝑝′𝑊^−𝑝′𝑥₁,𝑥₂𝑤𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂⎞⎟⎟⎠^1/𝑝′≤𝑐_ℝ2+𝑔^𝑞′𝑣^1−𝑞′^1/𝑞′,(4.42) we can conclude that Proposition 4.4 yields the desired result.

Proposition 4.7. Let the conditions of Theorem 4.6 be satisfied. Then(i)if 𝑣∈DC(𝑥), then (ℛ𝒲)_𝛼1,𝛼2 is bounded from 𝐿^{𝑝𝑑𝑒𝑐}(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ²⁺) if and only if (4.33) and (4.34) hold;(ii)if 𝑣∈DC(𝑦), then (ℛ𝒲)_𝛼1,𝛼2 is bounded from 𝐿^{𝑝𝑑𝑒𝑐}(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ²⁺) if and only if (4.32) and (4.34) are satisfied;(iii)if 𝑣∈DC(𝑥)∩DC(𝑦), then (ℛ𝒲)_𝛼1,𝛼2 is bounded from 𝐿^{𝑝𝑑𝑒𝑐}(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ²⁺) 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∈[2^𝑚0,2^𝑚0+1) for some integer 𝑚₀. By using the doubling condition for 𝑣 with respect to the second variable, we have ^𝑎0^𝑏0𝑥^𝛼1𝑞1𝑣𝑥₁,𝑥₂𝑏−𝑥₂^𝛼2𝑞𝑑𝑥₁𝑑𝑥₂^𝑝′/𝑞^𝑏0𝑤₂𝑥₂𝑑𝑥₂^{−𝑝′/𝑞}≤𝑐^𝑎0^2𝑚0+1₀𝑥^𝛼1𝑞1𝑣𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂^𝑝′/𝑞^∞2_𝑚0𝑊^−𝑝′₂𝑥₂𝑤₂𝑥₂𝑑𝑥₂2^{(𝑚0+1)𝛼2𝑝′}≤𝑐_𝑘≥𝑚0^2𝑘+1₂^𝑘𝑊^−𝑝′₂𝑥₂𝑤₂𝑥₂𝑑𝑥₂^𝑎0^2𝑘−1₀𝑥^𝛼1𝑞1𝑣𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂^𝑝′/𝑞×𝑐^{(𝑚0−𝑘)𝑝′1/𝑞}2^{(𝑚0+1)𝛼2𝑝′}≤𝑐_𝑘≥𝑚0^2𝑘+1₂^𝑘𝑊^−𝑝′₂𝑥₂𝑤₂𝑥₂𝑥₂−2^𝑘−1^𝛼2𝑝′𝑑𝑥₂^𝑎0^2𝑘−1₀𝑥^𝛼1𝑞1𝑣𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂^𝑝′/𝑞×𝑐^{(𝑚0−𝑘)𝑝′1/𝑞}≤𝑐^𝑎0𝑤₁𝑥₁𝑑𝑥₁^1/𝑝.(4.43) 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(i)if 𝑣∈DC(𝑥), then (𝒲ℛ)_𝛼1,𝛼2 is bounded from 𝐿^{𝑝𝑑𝑒𝑐}(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ²⁺) if and only if (4.36) and (4.38) hold;(ii)if 𝑣∈DC(𝑦), then (𝒲ℛ)_𝛼1,𝛼2 is bounded from 𝐿^𝑝_dec(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ²⁺) if and only if (4.37) and (4.38) are satisfied;(iii)if 𝑣∈DC(𝑥)∩DC(𝑦), then (𝒲ℛ)_𝛼1,𝛼2 is bounded from 𝐿^𝑝_dec(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ²⁺) if and only if (4.38) holds.

Now we are ready to discuss the operators ℐ_𝛼1,𝛼2 on the cone of nonincreasing functions.

Theorem 4.9. Let 1<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 ℐ_𝛼1,𝛼2 is bounded from 𝐿^𝑝_dec(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ²⁺) 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<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 ℐ_𝛼1,𝛼2 is bounded from 𝐿^𝑝_dec(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ²⁺) 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<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 ℐ_𝛼1,𝛼2 is bounded from 𝐿^𝑝_dec(𝑤,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ²⁺) 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 ℐ_𝛼1,𝛼2𝑓=ℛ_𝛼1,𝛼2𝑓+𝒲_𝛼1,𝛼2𝑓+ℛ𝒲_𝛼1,𝛼2𝑓+𝒲ℛ_𝛼1,𝛼2𝑓.(4.44) Corollary B, Theorem 4.5, and Propositions 4.7 and 4.8.

The next statement shows that the two-weight inequality for ℐ_𝛼1,𝛼2 can be characterized by one condition when 𝑤≈1.

Corollary 4.12. Let 1<𝑝≤𝑞<∞, and let 0<𝛼₁,𝛼₂<1/𝑝. Suppose that 𝑣∈DC(𝑥)∪DC(𝑦). Then the operator ℐ_𝛼1,𝛼2 is bounded from 𝐿^𝑝_dec(1,ℝ²⁺) to 𝐿^𝑞(𝑣,ℝ²⁺) if and only if 𝐷∶=sup_𝑎,𝑏>0𝑎^{(𝛼1−(1/𝑝))}𝑏^{(𝛼2−(1/𝑝))}^𝑎0^𝑏0𝑣(𝑡,𝜏)𝑑𝑡𝑑𝜏^1/𝑞<∞.(4.45)

Proof. Necessity can be derived by substituting the test function 𝑓_𝑎,𝑏(𝑥)=𝜒_{(0,𝑎)×(0,𝑏)}(𝑥) in the two-weight inequality for ℐ_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.