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Journal of Function Spaces and Applications
VolumeΒ 2012Β (2012), Article IDΒ 474681, 26 pages
http://dx.doi.org/10.1155/2012/474681
Research Article

Potential Operators on Cones of Nonincreasing Functions

1A. Razmadze Mathematical Institute, Ivane Javakhishvili Tbilisi State University, 2 University Street, 0143 Tbilisi, Georgia
2Faculty of Informatics and Control Systems, Georgian Technical University, 77 Kostava Street, Tbilisi, Georgia
3Abdus Salam School of Mathematical Sciences, GC University, 68-B New Muslim Town, Lahore, Pakistan

Received 16 May 2011; Accepted 15 June 2011

Academic Editor: V. M.Β Kokilashvili

Copyright Β© 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.

Abstract

Necessary and sufficient conditions on weight pairs guaranteeing the two-weight inequalities for the potential operators (πΌπ›Όβˆ«π‘“)(π‘₯)=∞0(𝑓(𝑑)/|π‘₯βˆ’π‘‘|1βˆ’π›Ό)𝑑𝑑 and (ℐ𝛼1,𝛼2βˆ«π‘“)(π‘₯,𝑦)=∞0∫∞0(𝑓(𝑑,𝜏)/|π‘₯βˆ’π‘‘|1βˆ’π›Ό1|π‘¦βˆ’πœ|1βˆ’π›Ό2)π‘‘π‘‘π‘‘πœ 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 ∞.

1. Introduction

Our aim is to derive necessary and sufficient conditions on weight pairs governing the boundedness of the following potential operators: ξ€·πΌπ›Όπ‘“ξ€Έξ€œ(π‘₯)=∞0𝑓(𝑑)|π‘₯βˆ’π‘‘|1βˆ’π›Όξ€·β„π‘‘π‘‘,0<𝛼<1,𝛼1,𝛼2𝑓(π‘₯,𝑦)=∞0𝑓(𝑑,𝜏)|π‘₯βˆ’π‘‘|1βˆ’π›Ό1||||π‘¦βˆ’πœ1βˆ’π›Ό2π‘‘π‘‘π‘‘πœ,0<𝛼1,𝛼2<1,(1.1) from 𝐿𝑝dec to πΏπ‘ž, where 1<𝑝,π‘ž<∞.

Historically, necessary and sufficient condition on a weight function 𝑒, for which the boundedness of the one-dimensional Hardy transform 1(𝐻𝑓)(π‘₯)=π‘₯ξ€œπ‘₯0𝑓(𝑑)𝑑𝑑(1.2) from 𝐿𝑝dec(𝑒,ℝ+) to 𝐿𝑝(𝑒,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 𝑇𝛼𝑓(ξ€œπ‘₯)=ℝ𝑛||||𝑓(𝑦)π‘₯βˆ’π‘¦π›Όβˆ’π‘›π‘‘π‘¦,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 ℛ𝛼1,𝛼2π‘“ξ€Έξ€œ(π‘₯,𝑦)=π‘₯0ξ€œπ‘¦0𝑓(𝑑,𝜏)(π‘₯βˆ’π‘‘)1βˆ’π›Ό1(π‘¦βˆ’πœ)1βˆ’π›Ό2π‘‘π‘‘π‘‘πœ,0<𝛼1,𝛼2<1,(1.4) from 𝐿𝑝dec(𝑀,ℝ2+) to 𝐿𝑝(𝑣,ℝ2+) were derived, provided that 𝑀 is a product of one-dimensional weights. Earlier, the problem of the boundedness of the two-dimensional Hardy transform 𝐻2=β„›1,1 from 𝐿𝑝dec(𝑀,ℝ2+) to 𝐿𝑝(𝑣,ℝ2+) was studied in [4] under the condition that 𝑀 and 𝑣 have the following form: 𝑀(π‘₯,𝑦)=𝑀1(π‘₯)𝑀2(𝑦),𝑣(π‘₯,𝑦)=𝑣1(π‘₯)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 𝐢; by the symbol π‘‡π‘“β‰ˆπΎπ‘“, 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 π‘₯ such that (𝑇𝑓)(π‘₯)≀𝑐1(𝐾𝑓)(π‘₯)≀𝑐2(𝑇𝑓)(π‘₯); ℝ+ denotes the interval (0,∞) and 𝑝′ means the number 𝑝/(π‘βˆ’1) for 1<𝑝<∞; βˆ«π‘Š(π‘₯)∢=π‘₯0𝑀(𝑑)𝑑𝑑; π‘Šπ‘—(π‘₯π‘—βˆ«)∢=π‘₯𝑗0𝑀𝑗(𝑑)𝑑𝑑; π‘Š(𝑑1,…,𝑑𝑛)∢=Π𝑛𝑖=1π‘Šπ‘–(𝑑𝑖).

2. Preliminaries

We say that a function π‘“βˆΆβ„π‘›+→ℝ+ is nonincreasing if 𝑓 is nonincreasing in each variable separately.

Let π’Ÿ be the class of all nonnegative nonincreasing functions on ℝ𝑛+. Suppose that 𝑒 is measurable a.e. positive function (weight) on ℝ𝑛+. We denote by 𝐿𝑝(𝑒,ℝ𝑛+), 0<𝑝<∞, the class of all nonnegative functions on ℝ𝑛+ for which ‖𝑓‖𝐿𝑝(𝑒,ℝ𝑛+)ξƒ©ξ€œβˆΆ=ℝ𝑛+𝑓𝑝π‘₯1,…,π‘₯𝑛𝑒π‘₯1,…,π‘₯𝑛𝑑π‘₯1⋯𝑑π‘₯𝑛ξƒͺ1/𝑝=ξƒ©ξ€œβ„π‘›+𝑓𝑝ξƒͺ(π‘₯)𝑒(π‘₯)𝑑π‘₯1/𝑝<∞.(2.1) By the symbol 𝐿𝑝dec(𝑒,ℝ𝑛+) we mean the class 𝐿𝑝(𝑒,ℝ𝑛+)βˆ©π’Ÿ.

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 𝑀 be weight functions on ℝ+, and let π‘Š(∞)=∞. (i)Suppose that 1<π‘β‰€π‘ž<∞. Then the inequality ξ‚Έξ€œβˆž0(𝐻𝑓(π‘₯))π‘žξ‚Ήπ‘£(π‘₯)𝑑π‘₯1/π‘žξ‚Έξ€œβ‰€πΆβˆž0(𝑓(π‘₯))𝑝𝑀(π‘₯)𝑑π‘₯1/𝑝,π‘“βˆˆπΏπ‘dec𝑀,ℝ+ξ€Έ,(2.2) holds if and only if the following two conditions are satisfied: supπ‘Ž>0ξ‚΅ξ€œπ‘Ž0𝑣(π‘₯)𝑑π‘₯1/π‘žξ‚΅ξ€œπ‘Ž0𝑀(π‘₯)𝑑π‘₯βˆ’1/𝑝<∞,supπ‘Ž>0ξ‚΅ξ€œβˆžπ‘Žπ‘£(π‘₯)π‘₯π‘žξ‚Άπ‘‘π‘₯1/π‘žξ‚΅ξ€œπ‘Ž0π‘Šβˆ’π‘β€²(π‘₯)π‘₯𝑝′𝑀(π‘₯)𝑑π‘₯1/𝑝′<∞.(2.3)(ii)Let 1<π‘ž<𝑝<∞. Then 𝐻 is bounded from 𝐿𝑝dec(𝑀,ℝ+) to πΏπ‘ž(𝑣,ℝ+) if and only if the following two conditions are satisfied: ξƒ¬ξ€œβˆž0ξƒ¬ξ‚΅ξ€œπ‘‘0𝑣(π‘₯)𝑑π‘₯1/π‘π‘Šβˆ’1/𝑝(𝑑)π‘Ÿξƒ­π‘£(𝑑)𝑑𝑑1/π‘Ÿξƒ¬ξ€œ<∞,∞0ξƒ¬ξ‚΅ξ€œβˆžπ‘‘π‘₯βˆ’π‘žπ‘£ξ‚Ά(π‘₯)𝑑π‘₯1/π‘ξ‚΅ξ€œπ‘‘0π‘₯π‘β€²π‘Šβˆ’π‘β€²ξ‚Ά(π‘₯)𝑀(π‘₯)𝑑π‘₯1/π‘β€²ξƒ­π‘Ÿπ‘‘π‘β€²π‘Šβˆ’π‘β€²ξƒ­(𝑑)𝑀(𝑑)𝑑𝑑1/π‘Ÿ<∞,(2.4) where π‘Ÿ=π‘π‘ž/(π‘βˆ’π‘ž).

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

Proposition A. Let 1<𝑝,π‘ž<∞. Suppose that 𝑇 is a positive integral operator defined on functions π‘“βˆΆβ„π‘›+→ℝ+, which are nonincreasing in each variable separately. Suppose that π‘‡βˆ— is its formal adjoint. Let 𝑀(π‘₯1,…,π‘₯𝑛)=𝑀1(π‘₯1)⋯𝑀𝑛(π‘₯𝑛) be a product weight such that π‘Šπ‘–(∞)=∞, 𝑖=1,…,𝑛. Let 𝑣 be a general weight on ℝ𝑛+. Then the operator 𝑇 is bounded from 𝐿𝑝dec(𝑀,ℝ𝑛+) to 𝐿𝑝(𝑣,ℝ𝑛+) if and only if the inequality βŽ›βŽœβŽœβŽξ€œβ„π‘›+ξ‚΅ξ€œπ‘₯10β‹―ξ€œπ‘₯𝑛0π‘‡βˆ—π‘”ξ‚Άπ‘β€²π‘Šβˆ’π‘β€²ξ€·π‘₯1,…,π‘₯𝑛𝑀π‘₯1,…,π‘₯𝑛𝑑π‘₯1⋯𝑑π‘₯π‘›βŽžβŽŸβŽŸβŽ 1/π‘β€²ξƒ©ξ€œβ‰€π‘β„π‘›+𝑔(π‘₯)π‘žβ€²π‘£1βˆ’π‘žβ€²ξƒͺ(π‘₯)𝑑π‘₯1/π‘žβ€²(2.5) holds for all 𝑔β‰₯0.

Let 𝑅𝛼 be the Riemann-Liouville transform with single kernel ξ€·π‘…π›Όπ‘“ξ€Έξ€œ(π‘₯)=π‘₯0𝑓(𝑑)(π‘₯βˆ’π‘‘)1βˆ’π›Όπ‘‘π‘‘,π‘₯βˆˆβ„+,𝛼>0.(2.6)

If 𝛼=1, then 𝑅𝛼 is the Hardy transform. The 𝐿𝑝(𝑀,ℝ+)β†’πΏπ‘ž(𝑣,ℝ+) boundedness for 𝑅1 was characterized by Muckenhoupt ([20]) for 𝑝=π‘ž, and by Kokilashvili [21] and Bradley [22] for 𝑝<π‘ž (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 𝑅𝛼 is bounded from 𝐿𝑝(𝑀,ℝ+) to πΏπ‘ž(𝑣,ℝ+) if and only if sup𝑑>0ξ‚΅ξ€œβˆžπ‘‘(π‘₯βˆ’π‘‘)(π›Όβˆ’1)π‘žξ‚Άπ‘£(π‘₯)𝑑π‘₯1/π‘žξ‚΅ξ€œπ‘‘0𝑀1βˆ’π‘β€²ξ‚Ά(𝑦)𝑑𝑦1/𝑝′<∞,sup𝑑>0ξ‚΅ξ€œβˆžπ‘‘ξ‚Άπ‘£(π‘₯)𝑑π‘₯1/π‘žξ‚΅ξ€œπ‘‘0(π‘‘βˆ’π‘₯)(π›Όβˆ’1)𝑝′𝑀1βˆ’π‘β€²ξ‚Ά(𝑦)𝑑𝑦1/𝑝′<∞,(2.7) for 1<π‘β‰€π‘ž<∞ and ξƒ―ξ€œβˆž0ξ‚΅ξ€œβˆžπ‘‘(π‘₯βˆ’π‘‘)(π›Όβˆ’1)π‘žξ‚Άπ‘£(π‘₯)𝑑π‘₯π‘Ÿ/π‘žξ‚΅ξ€œπ‘‘0𝑀1βˆ’π‘β€²ξ‚Ά(𝑦)π‘‘π‘¦π‘Ÿ/π‘žβ€²π‘€1βˆ’π‘β€²ξƒ°(𝑑)𝑑𝑑1/π‘Ÿξƒ―ξ€œ<∞,∞0ξ‚΅ξ€œβˆžπ‘‘ξ‚Άπ‘£(π‘₯)𝑑π‘₯π‘Ÿ/π‘ξ‚΅ξ€œπ‘‘0(π‘‘βˆ’π‘¦)(π›Όβˆ’1)𝑝′𝑀1βˆ’π‘β€²ξ‚Ά(𝑦)π‘‘π‘¦π‘Ÿ/𝑝′𝑣(𝑑)𝑑𝑑1/π‘Ÿ<∞,(2.8) for 1<π‘ž<𝑝<∞, where π‘Ÿ is defined as follows: 1/π‘Ÿ=1/π‘žβˆ’1/𝑝.

Theorem C (see [10]). Let 1<π‘β‰€π‘ž<∞, and let 0<𝛼𝑖<1, 𝑖=1,2. Assume that 𝑣 and 𝑀 are weights on ℝ2+. Suppose also that 𝑀(π‘₯1,π‘₯2)=𝑀1(π‘₯1)𝑀2(π‘₯2) for some one-dimensional weights 𝑀1 and 𝑀2 and that π‘Šπ‘–(∞)=∞, 𝑖=1,2. Then the following conditions are equivalent: (a)ℛ𝛼1,𝛼2 is bounded from 𝐿𝑝dec(𝑀,ℝ2+) to πΏπ‘ž(𝑣,ℝ2+);(b)the following four conditions hold simultaneously:(i)supπ‘Ž1,π‘Ž2>0ξ‚΅ξ€œπ‘Ž10ξ€œπ‘Ž20𝑀𝑑1,𝑑2𝑑𝑑1𝑑𝑑2ξ‚Άβˆ’1/π‘ξ‚΅ξ€œπ‘Ž10ξ€œπ‘Ž20𝑑𝛼11𝑑𝛼22ξ€Έπ‘žπ‘£ξ€·π‘‘1,𝑑2𝑑𝑑1𝑑𝑑2ξ‚Ά1/π‘ž<∞,(2.9)(ii)supπ‘Ž1,π‘Ž2>0ξ‚΅ξ€œπ‘Ž10ξ€œπ‘Ž20𝑑1𝑑2ξ€Έπ‘β€²π‘Šβˆ’π‘β€²ξ€·π‘‘1,𝑑2𝑀𝑑1,𝑑2𝑑𝑑1𝑑𝑑2ξ‚Ά1/π‘β€²Γ—ξ‚΅ξ€œβˆžπ‘Ž1ξ€œβˆžπ‘Ž2𝑑𝛼11βˆ’1𝑑𝛼22βˆ’1ξ‚π‘žπ‘£ξ€·π‘‘1,𝑑2𝑑𝑑1𝑑𝑑2ξ‚Ά1/π‘ž<∞,(2.10)(iii)supπ‘Ž1,π‘Ž2>0ξ‚΅ξ€œπ‘Ž10𝑀1𝑑1𝑑𝑑1ξ‚Άβˆ’1/π‘ξ‚΅ξ€œπ‘Ž20𝑑𝑝′2π‘Šβˆ’π‘β€²2𝑑2𝑀2𝑑2𝑑𝑑2ξ‚Ά1/π‘β€²Γ—ξ‚΅ξ€œπ‘Ž10ξ€œβˆžπ‘Ž2π‘‘π‘žπ›Ό11π‘‘π‘ž(𝛼22βˆ’1)𝑣𝑑1,𝑑2𝑑𝑑1𝑑𝑑2ξ‚Ά1/π‘ž<∞,(2.11)(iv)supπ‘Ž1,π‘Ž2>0ξ‚΅ξ€œπ‘Ž10𝑑𝑝′1π‘Šβˆ’π‘β€²1𝑑1𝑀1𝑑1𝑑𝑑1ξ‚Ά1/π‘β€²ξ‚΅ξ€œπ‘Ž20𝑀2𝑑2𝑑𝑑2ξ‚Άβˆ’1/π‘Γ—ξ‚΅ξ€œβˆžπ‘Ž1ξ€œπ‘Ž20π‘‘π‘ž(𝛼11βˆ’1)π‘‘π‘žπ›Ό22v𝑑1,𝑑2𝑑𝑑1𝑑𝑑2ξ‚Ά1/π‘ž<∞.(2.12)

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

Corollary A (see [10]). Let 1<π‘β‰€π‘ž<∞, and let 0<𝛼𝑖<1, 𝑖=1,2. Then the following conditions are equivalent:(a)the boundedness of ℛ𝛼1,𝛼2 from 𝐿𝑝dec(𝑀,ℝ2+) to πΏπ‘ž(𝑣,ℝ2+) holds for 𝑀≑1;(b)𝐡1∢=supπ‘Ž1,π‘Ž2>0𝐡1ξ€·π‘Ž1,π‘Ž2ξ€ΈβˆΆ=supπ‘Ž1,π‘Ž2>0ξ€·π‘Ž1π‘Ž2ξ€Έβˆ’1/π‘ξ‚΅ξ€œπ‘Ž10ξ€œπ‘Ž20π‘₯π‘žπ›Ό11π‘₯π‘žπ›Ό22𝑣π‘₯1,π‘₯2𝑑π‘₯1𝑑π‘₯2ξ‚Ά1/π‘ž<∞;(2.13)(c)𝐡2∢=supπ‘Ž1,π‘Ž2>0𝐡2ξ€·π‘Ž1,π‘Ž2ξ€ΈβˆΆ=supπ‘Ž1,π‘Ž2>0ξ€·π‘Ž1π‘Ž2ξ€Έ1/π‘β€²ξ‚΅ξ€œβˆžπ‘Ž1ξ€œβˆžπ‘Ž2π‘₯π‘ž(𝛼11βˆ’1)π‘₯π‘ž(𝛼22βˆ’1)𝑣π‘₯1,π‘₯2𝑑π‘₯1𝑑π‘₯2ξ‚Ά1/π‘ž<∞;(2.14)(d)𝐡3∢=supπ‘Ž1,π‘Ž2>0𝐡3ξ€·π‘Ž1,π‘Ž2ξ€ΈβˆΆ=supπ‘Ž1,π‘Ž2>0π‘Ž1βˆ’1/π‘π‘Ž1/𝑝′2ξ‚΅ξ€œπ‘Ž10ξ€œβˆžπ‘Ž2π‘₯π‘žπ›Ό11π‘₯π‘ž(𝛼22βˆ’1)𝑣π‘₯1,π‘₯2𝑑π‘₯1𝑑π‘₯2ξ‚Ά1/π‘ž<∞;(2.15)(e)𝐡4∢=supπ‘Ž1,π‘Ž2>0𝐡4ξ€·π‘Ž1,π‘Ž2ξ€ΈβˆΆ=supπ‘Ž1,π‘Ž2>0π‘Ž1/𝑝′1π‘Ž2βˆ’1/π‘ξ‚΅ξ€œβˆžπ‘Ž1ξ€œπ‘Ž20π‘₯π‘ž(𝛼11βˆ’1)π‘₯π‘žπ›Ό22𝑣π‘₯1,π‘₯2𝑑π‘₯1𝑑π‘₯2ξ‚Ά1/π‘ž<∞.(2.16)

3. Potentials on ℝ+

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(π‘₯)β‰ˆπ‘₯𝛼𝐻𝑓(π‘₯),(3.1) where 𝐻 is the Hardy operator defined above.

Proof. We follow the proof of Proposition  3.1 of [10]. We have ξ€·π‘…π›Όπ‘“ξ€Έξ€œ(π‘₯)=0π‘₯/2𝑓(𝑑)(π‘₯βˆ’π‘‘)1βˆ’π›Όξ€œπ‘‘π‘‘+π‘₯π‘₯/2𝑓(𝑑)(π‘₯βˆ’π‘‘)1βˆ’π›Όπ‘‘π‘‘βˆΆ=𝐽1(π‘₯)+𝐽2(π‘₯).(3.2)
Observe that if 0<𝑑<π‘₯/2, then (π‘₯βˆ’π‘‘)π›Όβˆ’1≀21βˆ’π›Όπ‘₯π›Όβˆ’1. Hence, 𝐽1(π‘₯)≀21βˆ’π›Όπ‘₯π›Όβˆ’1ξ€œπ‘₯0𝑓(𝑑)𝑑𝑑=21βˆ’π›Όπ‘₯𝛼(𝐻𝑓)(π‘₯).(3.3) Further, since 𝑓 is nonincreasing, we have that 𝐽2(π‘₯)β‰€π›Όβˆ’1ξ‚€π‘₯2𝛼𝑓π‘₯2≀𝑐𝛼π‘₯𝛼(𝐻𝑓)(π‘₯).(3.4)
Finally we have the upper estimate for 𝑅𝛼.
The lower estimate is obvious because (π‘₯βˆ’π‘‘)π›Όβˆ’1β‰₯π‘₯π›Όβˆ’1 for 𝑑≀π‘₯.

In the next statement we assume that π‘Šπ›Ό is the operator given by ξ€·π‘Šπ›Όπ‘“ξ€Έξ€œ(π‘₯)=∞π‘₯𝑓(𝑑)(π‘‘βˆ’π‘₯)1βˆ’π›Όπ‘‘π‘‘,𝛼>0.(3.5)

Lemma 3.2. Let 1<π‘β‰€π‘ž<∞, and let 𝛼>0. Suppose that π‘Š(∞)=∞. Then the operator π‘Šπ›Ό is bounded from 𝐿𝑝dec(𝑀,ℝ+) to πΏπ‘ž(𝑣,ℝ+) if and only if βŽ›βŽœβŽœβŽξ€œβˆž0ξ‚΅ξ€œπ‘₯0𝑔(𝑑)(π‘₯βˆ’π‘‘)βˆ’π›Όξ‚Άπ‘‘π‘‘π‘β€²π‘Šβˆ’π‘β€²βŽžβŽŸβŽŸβŽ (π‘₯)𝑀(π‘₯)𝑑π‘₯1/π‘β€²ξ‚΅ξ€œβ‰€π‘βˆž0𝑔(𝑑)π‘žβ€²π‘£1βˆ’π‘žβ€²ξ‚Ά(𝑑)𝑑𝑑1/π‘žβ€²,𝑔β‰₯0.(3.6)

Proof. Taking Proposition A into account (for 𝑛=1), an integral operator ξ€œ(𝑇𝑓)(π‘₯)=∞0π‘˜(π‘₯,𝑦)𝑓(𝑦)𝑑𝑦(3.7) is bounded from 𝐿𝑝dec(𝑀,ℝ+) to πΏπ‘ž(𝑣,ℝ+) if and only if βŽ›βŽœβŽœβŽξ€œβˆž0ξ‚΅ξ€œπ‘₯0ξ€·π‘‡βˆ—π‘“ξ€Έξ‚Ά(𝜏)π‘‘πœπ‘β€²π‘Šβˆ’π‘β€²βŽžβŽŸβŽŸβŽ (π‘₯)𝑀(π‘₯)𝑑π‘₯1/π‘β€²ξ‚΅ξ€œβ‰€π‘βˆž0𝑓(𝑑)π‘žβ€²π‘£1βˆ’π‘žβ€²ξ‚Ά(𝑑)𝑑𝑑1/π‘žβ€²,𝑓β‰₯0,(3.8) where π‘‡βˆ— is a formal adjoint to 𝑇.
We have ξ€œπ‘₯0𝑅𝛼𝑓(ξ€œπ‘‘)𝑑𝑑=π‘₯0ξ‚΅ξ€œπ‘‘0𝑓(𝜏)(π‘‘βˆ’πœ)1βˆ’π›Όξ‚Άξ€œπ‘‘πœπ‘‘π‘‘=π‘₯0ξ‚΅ξ€œπ‘“(𝜏)0π‘₯βˆ’πœπ‘‘π‘’π‘’1βˆ’π›Όξ‚Ά1π‘‘πœ=π›Όξ€œπ‘₯0𝑓(𝜏)(π‘₯βˆ’πœ)π›Όπ‘‘πœ.(3.9) Taking 𝑇=π‘Šπ›Ό and π‘‡βˆ—=𝑅𝛼, we derive the desired result.

Now we formulate the main results of this section.

Theorem 3.3. Let 1<π‘β‰€π‘ž<∞, and let 0<𝛼<1. Suppose that π‘Š(∞)=∞. Then 𝐼𝛼 is bounded from 𝐿𝑝dec(𝑀,ℝ+) to πΏπ‘ž(𝑣,ℝ+) if and only if supπ‘Ž>0𝐴1(π‘Ž,𝑣,𝑀)∢=supπ‘Ž>0ξ‚΅ξ€œπ‘Ž0𝑀(𝑑)π‘‘π‘‘βˆ’1/π‘ξ‚΅ξ€œπ‘Ž0π‘‘π›Όπ‘žξ‚Άπ‘£(𝑑)𝑑𝑑1/π‘ž<∞,(3.10)supπ‘Ž>0𝐴2(π‘Ž,𝑣,𝑀)∢=supπ‘Ž>0ξ‚΅ξ€œπ‘Ž0π‘‘π‘β€²π‘Šβˆ’π‘β€²ξ‚Ά(𝑑)𝑀(t)𝑑𝑑1/π‘β€²ξ‚΅ξ€œβˆžπ‘Žπ‘‘(π›Όβˆ’1)π‘žξ‚Άπ‘£(𝑑)𝑑𝑑1/π‘ž<∞,(3.11)supπ‘Ž>0𝐴3(π‘Ž,𝑣,𝑀)∢=supπ‘Ž>0ξ‚΅ξ€œβˆžπ‘Žπ‘Šβˆ’π‘β€²(π‘₯)𝑀(π‘₯)(π‘₯βˆ’π‘Ž)𝛼𝑝′𝑑π‘₯1/π‘β€²ξ‚΅ξ€œπ‘Ž0𝑣(π‘₯)𝑑π‘₯1/π‘ž<∞,(3.12)supπ‘Ž>0𝐴4(π‘Ž,𝑣,𝑀)∢=supπ‘Ž>0ξ‚΅ξ€œπ‘Ž0𝑀(π‘₯)𝑑π‘₯βˆ’1/π‘ξ‚΅ξ€œπ‘Ž0𝑣(π‘₯)(π‘Žβˆ’π‘₯)π›Όπ‘žξ‚Άπ‘‘π‘₯1/π‘ž<∞.(3.13)

Theorem 3.4. Let 1<π‘ž<𝑝<∞, and let 0<𝛼<1. π‘Š(∞)=∞. Then 𝐼𝛼 is bounded from 𝐿𝑝dec(𝑀,ℝ+) to πΏπ‘ž(𝑣,ℝ+) if and only if ξƒ¬ξ€œβ„+ξƒ¬ξ‚΅ξ€œπ‘‘0π‘₯π›Όπ‘žξ‚Άπ‘£(π‘₯)𝑑π‘₯1/π‘π‘Šβˆ’1/𝑝(𝑑)π‘Ÿπ‘‘π›Όπ‘žξƒ­π‘£(𝑑)𝑑𝑑1/π‘ŸβŽ‘βŽ’βŽ’βŽ£ξ€œ<∞,ℝ+βŽ‘βŽ’βŽ’βŽ£ξ‚΅ξ€œβˆžπ‘‘π‘£(π‘₯)π‘₯(1βˆ’π›Ό)π‘žξ‚Άπ‘‘π‘₯1/π‘ξƒ©ξ€œπ‘‘0π‘Šβˆ’π‘β€²(π‘₯)𝑀(π‘₯)π‘₯βˆ’π‘β€²ξƒͺ1/π‘β€²βŽ€βŽ₯βŽ₯βŽ¦π‘Ÿπ‘‘π‘β€²π‘Šβˆ’π‘β€²βŽ€βŽ₯βŽ₯⎦(𝑑)𝑀(𝑑)𝑑𝑑1/π‘ŸβŽ‘βŽ’βŽ’βŽ£ξ€œ<∞,ℝ+βŽ‘βŽ’βŽ’βŽ£ξƒ©ξ€œβˆžπ‘‘π‘Šβˆ’π‘β€²(π‘₯)𝑀(π‘₯)(π‘₯βˆ’π‘‘)βˆ’π›Όπ‘β€²ξƒͺ1/π‘β€²ξ‚΅ξ€œπ‘‘0𝑣(π‘₯)𝑑π‘₯1/π‘βŽ€βŽ₯βŽ₯βŽ¦π‘ŸβŽ€βŽ₯βŽ₯βŽ¦π‘£(𝑑)𝑑𝑑1/π‘Ÿξƒ¬ξ€œ<∞,ℝ+ξ‚Έπ‘Šβˆ’1ξ€œ(𝑑)𝑑0𝑣(π‘₯)(π‘‘βˆ’π‘₯)βˆ’π›Όπ‘žξ‚Ήπ‘‘π‘₯π‘Ÿ/π‘žξƒ­π‘€(𝑑)𝑑𝑑1/π‘Ÿ<∞,(3.14) where 1/π‘Ÿ=1/π‘žβˆ’1/𝑝.

Proof of Theorems 3.3 and 3.4. By using the representation 𝐼𝛼𝑓𝑅(π‘₯)=π›Όπ‘“ξ€Έξ€·π‘Š(π‘₯)+𝛼𝑓(π‘₯),π‘₯>0,(3.15) the obvious equality ξ€œβˆžπ‘‘π‘Šβˆ’π‘β€²(π‘₯)𝑀(π‘₯)𝑑π‘₯=π‘π‘π‘Š1βˆ’π‘β€²(𝑑).(3.16) Theorems A and B and Lemmas 3.1 and 3.2, we have the desired results.

Corollary 3.5. Let 1<π‘β‰€π‘ž<∞, and let 0<𝛼<1/𝑝. Then the operator 𝐼𝛼 is bounded from 𝐿𝑝dec(1,ℝ+) to πΏπ‘ž(𝑣,ℝ+) if and only if 𝐡∢=supπ‘Ž>0π‘Ž(π›Όβˆ’1/𝑝)ξ‚΅ξ€œπ‘Ž0𝑣(𝑑)𝑑𝑑1/π‘ž<∞.(3.17)

Proof. Necessity follows immediately taking the test function π‘“π‘Ž(π‘₯)=πœ’(0,π‘Ž)(π‘₯) in the two-weight inequality ξ‚΅ξ€œβˆž0𝐼𝑣(π‘₯)𝛼𝑓(π‘₯)π‘žξ‚Άπ‘‘x1/π‘žξ‚΅ξ€œβ‰€π‘βˆž0(𝑓(π‘₯))𝑝𝑑π‘₯1/𝑝(3.18) and observing that πΌπ›Όπ‘“π‘Žβˆ«(π‘₯)β‰₯π‘Ž0(𝑑𝑑/|π‘₯βˆ’π‘‘|1βˆ’π›Ό)β‰₯π‘Žπ›Ό for π‘₯∈(0,π‘Ž).
Sufficiency. By Theorem 3.3, it is enough to show that 𝐴max1,𝐴2,𝐴3,𝐴4≀𝑐𝐡,(3.19) where π΄π‘–βˆΆ=supπ‘Ž>0𝐴𝑖(π‘Ž,𝑣,1), 𝑖=1,2,3,4 (see Theorem 3.3 for the definition of 𝐴𝑖(π‘Ž,𝑣,𝑀)).
The estimates 𝐴𝑖≀𝑐𝐡, 𝑖=1,4, are obvious. We show that 𝐴𝑖≀𝑐𝐡 for 𝑖=2,3. We have π΄π‘ž2(π‘Ž,𝑣,1)=π‘Žπ‘ž/π‘β€²βˆžξ“π‘˜=0ξ€œ2π‘˜+1π‘Ž2π‘˜π‘Žπ‘‘(π›Όβˆ’1)π‘žπ‘£(𝑑)π‘‘π‘‘β‰€π‘Žπ‘ž/π‘β€²βˆžξ“π‘˜=0ξ€·2π‘˜π‘Žξ€Έ(π›Όβˆ’1)π‘žξƒ©ξ€œ2π‘˜+1π‘Ž2π‘˜π‘Žξƒͺ𝑣(𝑑)π‘‘π‘‘β‰€π‘π΅π‘žπ‘Žπ‘ž/π‘β€²βˆžξ“π‘˜=0ξ€·2π‘˜π‘Žξ€Έ(π›Όβˆ’1)π‘žξ€·2π‘˜+1π‘Žξ€Έ(1/π‘βˆ’π›Ό)π‘ž=π‘π΅π‘žπ‘Žπ‘ž/π‘β€²ξƒ©βˆžξ“π‘˜=02βˆ’π‘˜π‘ž/𝑝′ξƒͺπ‘Žβˆ’π‘ž/π‘β€²β‰€π‘π΅π‘ž.(3.20) Further, by the condition 0<𝛼<1/𝑝, we have that π΄π‘ž3ξ‚΅ξ€œ(π‘Ž,𝑣,1)β‰€βˆžπ‘Žπ‘₯(π›Όβˆ’1)𝑝′𝑑π‘₯1/π‘β€²ξ‚΅ξ€œπ‘Ž0𝑣(𝑑)𝑑𝑑1/π‘ž=𝑐𝛼,π‘π‘Žπ›Όβˆ’1/π‘ξ‚΅ξ€œπ‘Ž0𝑣(𝑑)𝑑𝑑1/π‘žβ‰€π‘π΅.(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 𝑑>0 the following inequality holds: ξ€œ02π‘‘ξ‚»ξ€œπœŒ(π‘₯)𝑑π‘₯≀𝑏min𝑑0ξ€œπœŒ(π‘₯)𝑑π‘₯,𝑑2𝑑.𝜌(π‘₯)𝑑π‘₯(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 ξ€œ02π‘‘πœŒ(π‘₯)𝑑π‘₯β‰₯𝑏1ξ‚»ξ€œmax𝑑0ξ€œπœŒ(π‘₯)𝑑π‘₯,𝑑2𝑑.𝜌(π‘₯)𝑑π‘₯(3.23) Indeed by (3.22) we have ξ€œ02𝑑1𝜌(π‘₯)𝑑π‘₯β‰₯π‘ξ€œ02π‘‘ξ€œπœŒ(π‘₯)𝑑π‘₯+𝑑2π‘‘πœŒ(π‘₯)𝑑π‘₯.(3.24) Then ξ€œ02π‘‘π‘πœŒ(π‘₯)𝑑π‘₯β‰₯ξ€œπ‘βˆ’1𝑑2π‘‘πœŒ(π‘₯)𝑑π‘₯.(3.25) Analogously, ξ€œ02π‘‘π‘πœŒ(π‘₯)𝑑π‘₯β‰₯ξ€œπ‘βˆ’1𝑑0𝜌(π‘₯)𝑑π‘₯.(3.26) Finally, we have (3.23).

Corollary 3.8. Let 1<π‘β‰€π‘ž<∞, and let 0<𝛼<1. Suppose that π‘Š(∞)=∞. Suppose also that π‘£βˆˆDC(ℝ+). Then 𝐼𝛼 is bounded from 𝐿𝑝dec(𝑀,ℝ+) to πΏπ‘ž(𝑣,ℝ+) if and only if condition (3.11) is satisfied.

Proof. Observe that by Remark 3.7, for π‘š0βˆˆβ„€, the inequality ξ€œ2π‘š00𝑣(π‘₯)𝑑π‘₯β‰€π‘π‘š01βˆ’π‘˜ξ€œ2π‘˜0𝑣(π‘₯)𝑑π‘₯(3.27) holds for all π‘˜>π‘š0, where 𝑏1 is defined in (3.23).
Let π‘Ž>0. Then there is π‘š0βˆˆβ„€ such that π‘Žβˆˆ[2π‘š0,2π‘š0+1). By applying (3.27) and the doubling condition for 𝑣, we find that ξ‚΅ξ€œπ‘Ž0𝑀(𝑑)π‘‘π‘‘βˆ’π‘β€²/π‘ξ‚΅ξ€œπ‘Ž0π‘‘π›Όπ‘žξ‚Άπ‘£(𝑑)𝑑𝑑𝑝′/π‘žξ‚΅ξ€œ=π‘βˆžπ‘Žπ‘Šβˆ’π‘β€²ξ€œ(𝑑)𝑀(𝑑)π‘‘π‘‘ξ‚Άξ‚΅π‘Ž0π‘‘π›Όπ‘žξ‚Άπ‘£(𝑑)𝑑𝑑𝑝′/π‘žξ‚΅ξ€œβ‰€π‘βˆž2π‘š0π‘Šβˆ’π‘β€²(ξ‚Άξƒ©ξ€œπ‘‘)𝑀(𝑑)𝑑𝑑2π‘š0+10π‘‘π›Όπ‘žξƒͺ𝑣(𝑑)𝑑𝑑𝑝′/π‘žβ‰€π‘βˆžξ“π‘˜=π‘š0ξƒ©ξ€œ2π‘˜+12π‘˜π‘Šβˆ’π‘β€²(ξ€œπ‘‘)𝑀(𝑑)𝑑𝑑ξƒͺ2π‘š0+10ξƒͺ𝑣(𝑑)𝑑𝑑𝑝′/π‘ž2π‘š0π›Όπ‘β€²β‰€π‘βˆžξ“π‘˜=π‘š0ξƒ©ξ€œ2π‘˜+12π‘˜π‘Šβˆ’π‘β€²ξƒͺ𝑏(𝑑)𝑀(𝑑)π‘‘π‘‘π‘š01βˆ’π‘˜βˆ’1ξƒ©ξ€œ2π‘˜+20𝑣ξƒͺ(𝑑)𝑑𝑑𝑝′/π‘ž2π‘š0π›Όπ‘β€²β‰€π‘βˆžξ“π‘˜=π‘š0π‘π‘š01βˆ’π‘˜βˆ’1ξƒ©ξ€œ2π‘˜+12π‘˜π‘Šβˆ’π‘β€²(ξ€œπ‘‘)𝑀(𝑑)𝑑𝑑ξƒͺ2π‘˜+22π‘˜+1ξƒͺ𝑣(𝑑)𝑑𝑑𝑝′/π‘ž2π‘˜(π›Όβˆ’1)𝑝′2π‘˜π‘β€²β‰€π‘βˆžξ“π‘˜=π‘š0π‘π‘š01βˆ’π‘˜βˆ’1ξƒ©ξ€œ2π‘˜+12π‘˜π‘‘π‘β€²π‘Šβˆ’π‘β€²ξ€œ(𝑑)𝑀(𝑑)𝑑𝑑ξƒͺ2π‘˜+22π‘˜+1𝑣(𝑑)𝑑(π›Όβˆ’1)π‘žξƒͺ𝑑𝑑𝑝′/π‘žξ‚΅β‰€π‘supπ‘Ž>0𝐴2ξ‚Ά(π‘Ž,𝑣,𝑀)π‘β€²βˆžξ“π‘˜=π‘š0π‘π‘š01βˆ’π‘˜βˆ’1≀𝑐supπ‘Ž>0𝐴2ξ‚Ά(π‘Ž,𝑣,𝑀)𝑝′.(3.28) So, we have seen that (3.11)β‡’(3.10). Let us check now that (3.13)β‡’(3.12).
Indeed, for π‘Ž>0, we choose π‘š0 so that π‘Žβˆˆ[2π‘š0,2π‘š0+1). Then, by using the condition π‘£βˆˆDC(ℝ+) and Remark 3.7, ξ‚΅ξ€œβˆžπ‘Žπ‘Šβˆ’π‘β€²(π‘₯)𝑀(π‘₯)(π‘₯βˆ’π‘Ž)π›Όπ‘β€²ξ€œπ‘‘π‘₯ξ‚Άξ‚΅π‘Ž0𝑣(π‘₯)𝑑π‘₯𝑝′/π‘žβ‰€ξ‚΅ξ€œβˆž2π‘š0π‘Šβˆ’π‘β€²(π‘₯)𝑀(π‘₯)π‘₯π›Όπ‘β€²ξ‚Άξƒ©ξ€œπ‘‘π‘₯2π‘š0+10ξƒͺ𝑣(π‘₯)𝑑π‘₯𝑝′/π‘žβ‰€π‘βˆžξ“π‘˜=π‘š02π‘˜π›Όπ‘β€²ξƒ©ξ€œ2π‘˜+12π‘˜π‘Šβˆ’π‘β€²ξ€œ(π‘₯)𝑀(π‘₯)𝑑π‘₯ξƒͺ2π‘š0+10ξƒͺ𝑣(π‘₯)𝑑π‘₯𝑝′/π‘žβ‰€π‘βˆžξ“π‘˜=π‘š02π‘˜π›Όπ‘β€²ξƒ©ξ€œ2π‘˜+12π‘˜π‘Šβˆ’π‘β€²ξƒͺ𝑏(π‘₯)𝑀(π‘₯)𝑑π‘₯π‘š01βˆ’π‘˜+2ξƒ©ξ€œ2π‘˜βˆ’10ξƒͺ𝑣(π‘₯)𝑑π‘₯𝑝′/π‘žβ‰€π‘βˆžξ“π‘˜=π‘š0π‘π‘š01βˆ’π‘˜+2ξ‚΅ξ€œβˆž2π‘˜π‘Šβˆ’π‘β€²ξ‚Άξƒ©ξ€œ(π‘₯)𝑀(π‘₯)𝑑π‘₯2π‘˜0ξ€·2𝑣(π‘₯)π‘˜ξ€Έβˆ’π‘₯π›Όπ‘žξƒͺ𝑑π‘₯𝑝′/π‘žξ‚΅β‰€π‘supπ‘Ž>0𝐴4ξ‚Ά(π‘Ž,𝑣,𝑀)π‘β€²βˆžξ“π‘˜=π‘š0π‘π‘š01βˆ’π‘˜+2≀𝑐supπ‘Ž>0𝐴4ξ‚Ά(π‘Ž,𝑣,𝑀)𝑝′.(3.29) Hence, (3.13)β‡’(3.12) follows. Implication (3.11)β‡’(3.13) follows in the same way as in the case of implication (3.11)β‡’(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 ℐ𝛼1,𝛼2.

To derive the main results, we introduce the following multiple potential operators: 𝒲𝛼1,𝛼2𝑓π‘₯1,π‘₯2ξ€Έ=ξ€œβˆžπ‘₯1ξ€œβˆžπ‘₯2𝑓𝑑1,𝑑2𝑑𝑑1𝑑𝑑2𝑑1βˆ’π‘₯1ξ€Έ1βˆ’π›Ό1𝑑2βˆ’π‘₯2ξ€Έ1βˆ’π›Ό2,(ℛ𝒲)𝛼1,𝛼2𝑓π‘₯1,π‘₯2ξ€Έ=ξ€œπ‘₯10ξ€œβˆžπ‘₯2𝑓𝑑1,𝑑2𝑑𝑑1𝑑𝑑2ξ€·π‘₯1βˆ’π‘‘1ξ€Έ1βˆ’π›Ό1𝑑2βˆ’π‘₯2ξ€Έ1βˆ’π›Ό2,(𝒲ℛ)𝛼1,𝛼2𝑓π‘₯1,π‘₯2ξ€Έ=ξ€œβˆžπ‘₯1ξ€œπ‘₯20𝑓𝑑1,𝑑2𝑑𝑑1𝑑𝑑2𝑑1βˆ’π‘₯1ξ€Έ1βˆ’π›Ό1ξ€·π‘₯2βˆ’π‘‘2ξ€Έ1βˆ’π›Ό2,(4.1) where π‘₯1,π‘₯2βˆˆβ„+, 𝑓β‰₯0, and 0<𝛼𝑖<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 𝑑>0 and almost every π‘₯>0 the following inequality holds: ξ€œ02π‘‘ξ‚»ξ€œπœŒ(π‘₯,𝑦)𝑑𝑦≀𝑐min𝑑0ξ€œπœŒ(π‘₯,𝑦)𝑑𝑦,𝑑2𝑑.𝜌(π‘₯,𝑦)𝑑𝑦(4.2)

Analogously is defined the class of weights DC(π‘₯).

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 ξ€œ02π‘‘πœŒ(π‘₯,𝑦)𝑑𝑦β‰₯𝑐1ξ‚»ξ€œmax𝑑0ξ€œπœŒ(π‘₯,𝑦)𝑑𝑦,𝑑2𝑑.𝜌(π‘₯,𝑦)𝑑𝑦(4.3)

Analogously, 𝜌∈DC(π‘₯)β‡’πœŒβˆˆRDC(π‘₯). 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 π‘£βˆˆDC(π‘₯), then for the boundedness of ℛ𝛼1,𝛼2 from 𝐿𝑝dec(𝑀,ℝ2+) to πΏπ‘ž(𝑣,ℝ2+), it is necessary and sufficient that conditions (2.10) and (2.12) are satisfied.(ii)If π‘£βˆˆDC(𝑦), then ℛ𝛼1,𝛼2 is bounded from 𝐿𝑝dec(𝑀,ℝ2+) to πΏπ‘ž(𝑣,ℝ2+) if and only If conditions (2.10) and (2.11) are satisfied.(iii)If π‘£βˆˆDC(π‘₯)∩DC(𝑦), then ℛ𝛼1,𝛼2 is bounded from 𝐿𝑝dec(𝑀,ℝ2+) to πΏπ‘ž(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 ℛ𝛼1,𝛼2 with 𝛼1,𝛼2>1 (see [25], [13, Section  1.6]).

Theorem D. Let 1<π‘β‰€π‘ž<∞, and let 𝛼1,𝛼2β‰₯1. (i)Suppose that 𝑀1βˆ’π‘β€²βˆˆDC(𝑦). Then the operator ℛ𝛼1,𝛼2 is bounded from 𝐿𝑝(𝑀,ℝ2+) to πΏπ‘ž(𝑣,ℝ2+) if and only if 𝑃1∢=supπ‘Ž,𝑏>0βŽ›βŽœβŽœβŽξ€œπ‘Ž0ξ€œπ‘0𝑀1βˆ’π‘β€²ξ€·π‘₯1,π‘₯2ξ€Έξ€·π‘Žβˆ’π‘₯1ξ€Έ(1βˆ’π›Ό1)𝑝′𝑑π‘₯1𝑑π‘₯2⎞⎟⎟⎠1/π‘β€²ξƒ©ξ€œβˆžπ‘Žξ€œβˆžπ‘π‘£ξ€·π‘₯1,π‘₯2ξ€Έπ‘₯(1βˆ’π›Ό22)π‘žπ‘‘π‘₯1𝑑π‘₯2ξƒͺ1/π‘žπ‘ƒ<∞,2∢=supπ‘Ž,𝑏>0ξ‚΅ξ€œπ‘Ž0ξ€œπ‘0𝑀1βˆ’π‘β€²ξ€·π‘₯1,π‘₯2𝑑π‘₯1𝑑π‘₯2ξ‚Ά1/π‘β€²ξƒ©ξ€œβˆžπ‘Žξ€œβˆžπ‘π‘£ξ€·π‘₯1,π‘₯2ξ€Έξ€·π‘₯1ξ€Έβˆ’π‘Ž(1βˆ’π›Ό1)π‘žπ‘₯(1βˆ’π›Ό22)π‘žπ‘‘π‘₯1𝑑π‘₯2ξƒͺ1/π‘ž<∞.(4.4) Moreover, ‖ℛ𝛼1,𝛼2β€–β‰ˆmax{𝑃1,𝑃2}.(ii)Let 𝑀1βˆ’π‘ξ…žβˆˆDC(π‘₯). Then the operator ℛ𝛼1,𝛼2 is bounded from 𝐿𝑝(𝑀,ℝ2+) to πΏπ‘ž(𝑣,ℝ+) if and only if 𝑃1∢=supπ‘Ž,𝑏>0βŽ›βŽœβŽœβŽξ€œπ‘Ž0ξ€œπ‘0𝑀1βˆ’π‘β€²ξ€·π‘₯1,π‘₯2ξ€Έξ€·π‘βˆ’π‘₯2ξ€Έ(1βˆ’π›Ό2)𝑝′𝑑π‘₯1𝑑π‘₯2⎞⎟⎟⎠1/π‘β€²ξƒ©ξ€œβˆžπ‘Žξ€œβˆžπ‘π‘£ξ€·π‘₯1,π‘₯2ξ€Έπ‘₯(1βˆ’π›Ό11)π‘žπ‘‘π‘₯1𝑑π‘₯2ξƒͺ1/π‘žξ‚π‘ƒ<∞,2∢=supπ‘Ž,𝑏>0ξ‚΅ξ€œπ‘Ž0ξ€œπ‘0𝑀1βˆ’π‘β€²ξ€·π‘₯1,π‘₯2𝑑π‘₯1𝑑π‘₯2ξ‚Ά1/π‘β€²ξƒ©ξ€œβˆžπ‘Žξ€œβˆžπ‘π‘£ξ€·π‘₯1,π‘₯2ξ€Έξ€·π‘₯2ξ€Έβˆ’π‘(1βˆ’π›Ό2)π‘žπ‘₯(1βˆ’π›Ό11)π‘žπ‘‘π‘₯1𝑑π‘₯2ξƒͺ1/π‘ž<∞.(4.5) Moreover, ‖ℛ𝛼1,𝛼2ξ‚π‘ƒβ€–β‰ˆmax{1,𝑃2}.

Let us introduce the following multiple integral operators: (β„‹β„›)𝛼1,𝛼2𝑓π‘₯1,π‘₯2ξ€Έ=π‘₯𝛼11βˆ’1ξ€œπ‘₯10ξ€œπ‘₯20𝑓𝑑1,𝑑2𝑑𝑑1𝑑𝑑2ξ€·π‘₯2βˆ’π‘‘2ξ€Έ1βˆ’π›Ό2,(β„›β„‹)𝛼1,𝛼2𝑓π‘₯1,π‘₯2ξ€Έ=π‘₯𝛼22βˆ’1ξ€œπ‘₯10ξ€œπ‘₯20𝑓𝑑1,𝑑2𝑑𝑑1𝑑𝑑2ξ€·π‘₯1βˆ’π‘‘1ξ€Έ1βˆ’π›Ό1,(ℋ𝒲)𝛼1,𝛼2𝑓π‘₯1,π‘₯2ξ€Έ=π‘₯𝛼11βˆ’1ξ€œπ‘₯10ξ€œβˆžπ‘₯2𝑓𝑑1,𝑑2𝑑𝑑1𝑑𝑑2𝑑2βˆ’π‘₯2ξ€Έ1βˆ’π›Ό2,(𝒲ℋ)𝛼1,𝛼2𝑓π‘₯1,π‘₯2ξ€Έ=π‘₯𝛼22βˆ’1ξ€œβˆžπ‘₯1ξ€œπ‘₯20𝑓𝑑1,𝑑2𝑑𝑑1𝑑𝑑2𝑑1βˆ’π‘₯1ξ€Έ1βˆ’π›Ό1,ξ€·β„‹ξ…žβ„›ξ€Έπ›Ό1,𝛼2𝑓π‘₯1,π‘₯2ξ€Έ=ξ€œβˆžπ‘₯1ξ€œπ‘₯20𝑓𝑑1,𝑑2𝑑𝑑1𝑑𝑑2𝑑1βˆ’π›Ό11ξ€·π‘₯2βˆ’π‘‘2ξ€Έ1βˆ’π›Ό2,ξ€·β„›β„‹ξ…žξ€Έπ›Ό1,𝛼2𝑓π‘₯1,π‘₯2ξ€Έ=ξ€œπ‘₯10ξ€œβˆžπ‘₯2𝑓𝑑1,𝑑2𝑑𝑑1𝑑𝑑2ξ€·π‘₯1βˆ’π‘‘1ξ€Έ1βˆ’π›Ό1𝑑1βˆ’π›Ό22,ξ€·β„‹ξ…žπ’²ξ€Έπ›Ό1,𝛼2𝑓π‘₯1,π‘₯2ξ€Έ=ξ€œβˆžπ‘₯1ξ€œβˆžπ‘₯2𝑓𝑑1,𝑑2𝑑𝑑1𝑑𝑑2𝑑1βˆ’π›Ό11𝑑2βˆ’π‘₯2ξ€Έ1βˆ’π›Ό2,ξ€·π’²β„‹ξ…žξ€Έπ›Ό1,𝛼2𝑓π‘₯1,π‘₯2ξ€Έ=ξ€œβˆžπ‘₯1ξ€œβˆžπ‘₯2𝑓𝑑1,𝑑2𝑑𝑑1𝑑𝑑2𝑑1βˆ’π‘₯1ξ€Έ1βˆ’π›Ό1𝑑1βˆ’π›Ό22.(4.6)

Now we prove some auxiliary statements.

Proposition 4.3. Let 1<π‘β‰€π‘ž<∞, and let 𝛼1,𝛼2β‰₯1. Suppose that either 𝑀(π‘₯1,π‘₯2)=𝑀1(π‘₯1)𝑀2(π‘₯2) or 𝑣(π‘₯1,π‘₯2)=𝑣1(π‘₯1)𝑣2(π‘₯2) for some one-dimensional weights 𝑀1, 𝑀2, 𝑣1, and 𝑣2. (i)The operator (β„›β„‹)𝛼1,𝛼2 is bounded from 𝐿𝑝(𝑀,ℝ2+) to πΏπ‘ž(𝑣,ℝ+) if and only if 𝐼1∢=supπ‘Ž,𝑏>0βŽ›βŽœβŽœβŽξ€œπ‘Ž0ξ€œπ‘0𝑀1βˆ’π‘β€²ξ€·π‘₯1,π‘₯2ξ€Έξ€·π‘Žβˆ’π‘₯1ξ€Έ(1βˆ’π›Ό1)𝑝′𝑑π‘₯1𝑑π‘₯2⎞⎟⎟⎠1/π‘β€²ξƒ©ξ€œβˆžπ‘Žξ€œβˆžπ‘π‘£ξ€·π‘₯1,π‘₯2ξ€Έπ‘₯(1βˆ’π›Ό22)π‘žπ‘‘π‘₯1𝑑π‘₯2ξƒͺ1/π‘žξ‚πΌ<∞,2∢=supπ‘Ž,𝑏>0ξ‚΅ξ€œπ‘Ž0ξ€œπ‘0𝑀1βˆ’π‘β€²ξ€·π‘₯1,π‘₯2𝑑π‘₯1𝑑π‘₯2ξ‚Ά1/π‘β€²ξƒ©ξ€œβˆžπ‘Žξ€œβˆžπ‘π‘£ξ€·π‘₯1,π‘₯2ξ€Έξ€·π‘₯1ξ€Έβˆ’π‘Ž(1βˆ’π›Ό1)π‘žπ‘₯(1βˆ’π›Ό22)π‘žπ‘‘π‘₯1𝑑π‘₯2ξƒͺ1/π‘ž<∞.(4.7) Moreover, β€–(β„›β„‹)𝛼1,𝛼2ξ‚πΌβ€–β‰ˆmax{1,𝐼2}.(ii)The operator (𝒲ℋ)𝛼1,𝛼2 is bounded from 𝐿𝑝(𝑀,ℝ2+) to πΏπ‘ž(𝑣ℝ+) if and only if 𝐽1∢=supπ‘Ž,𝑏>0ξƒ©ξ€œπ‘Ž0ξ€œβˆžπ‘π‘£ξ€·π‘₯1,π‘₯2ξ€Έξ€·π‘Žβˆ’π‘₯1ξ€Έ(1βˆ’π›Ό1)π‘žπ‘₯(1βˆ’π›Ό22)π‘žπ‘‘π‘₯1𝑑π‘₯2ξƒͺ1/π‘žξ‚΅ξ€œβˆžπ‘Žξ€œπ‘0𝑀1βˆ’π‘β€²ξ€·π‘₯1,π‘₯2𝑑π‘₯1𝑑π‘₯2ξ‚Ά1/π‘žξ‚π½<∞,2∢=supπ‘Ž,𝑏>0ξ‚΅ξ€œπ‘Ž0ξ€œβˆžπ‘π‘£ξ€·π‘₯1,π‘₯2ξ€Έπ‘₯(1βˆ’π›Ό22)π‘žπ‘‘π‘₯1𝑑π‘₯2ξ‚Ά1/π‘žβŽ›βŽœβŽœβŽξ€œβˆžπ‘Žξ€œπ‘0𝑀1βˆ’π‘β€²ξ€·π‘₯1,π‘₯2ξ€Έξ€·π‘₯1ξ€Έβˆ’π‘Ž(1βˆ’π›Ό1)𝑝′𝑑π‘₯1𝑑π‘₯2⎞⎟⎟⎠1/𝑝′<∞.(4.8) Moreover, β€–(𝒲ℋ)𝛼1,𝛼2ξ‚π½β€–β‰ˆmax{1,𝐽2}.(iii)The operator (β„›β„‹β€²)𝛼1,𝛼2 is bounded from 𝐿𝑝(𝑀,ℝ2+) to πΏπ‘ž(𝑣,ℝ+) if and only if ξ‚π½ξ…ž1∢=supπ‘Ž,𝑏>0ξ‚΅ξ€œβˆžπ‘Žξ€œπ‘0𝑣π‘₯1,π‘₯2𝑑π‘₯1𝑑π‘₯2ξ‚Ά1/π‘žβŽ›βŽœβŽœβŽξ€œπ‘Ž0ξ€œβˆžπ‘π‘€1βˆ’π‘β€²ξ€·π‘₯1,π‘₯2ξ€Έπ‘₯(1βˆ’π›Ό2)𝑝′2ξ€·π‘Žβˆ’π‘₯1ξ€Έ(1βˆ’π›Ό1)𝑝′𝑑π‘₯1𝑑π‘₯2⎞⎟⎟⎠1/𝑝′𝐽<∞,ξ…ž2∢=supπ‘Ž,𝑏>0ξƒ©ξ€œβˆžπ‘Žξ€œπ‘0𝑣π‘₯1,π‘₯2ξ€Έξ€·π‘₯1ξ€Έβˆ’π‘Ž(1βˆ’π›Ό1)π‘žπ‘‘π‘₯1𝑑π‘₯2ξƒͺ1/π‘žξƒ©ξ€œπ‘Ž0ξ€œβˆžπ‘π‘€1βˆ’π‘β€²ξ€·π‘₯1,π‘₯2ξ€Έπ‘₯(1βˆ’π›Ό2)𝑝′2𝑑π‘₯1𝑑π‘₯2ξƒͺ1/𝑝′<∞.(4.9) Moreover, β€–(β„›β„‹β€²)𝛼1,𝛼2ξ‚π½β€–β‰ˆmax{ξ…ž1,ξ‚π½ξ…ž2}.(iv)The operator (𝒲ℋ′)𝛼1,𝛼2 is bounded from 𝐿𝑝(𝑀,ℝ2+) to πΏπ‘ž(𝑣,ℝ+) if and only if ξ‚πΌξ…ž1∢=supπ‘Ž,𝑏>0ξƒ©ξ€œπ‘Ž0ξ€œπ‘0𝑣π‘₯1,π‘₯2ξ€Έξ€·π‘Žβˆ’π‘₯1ξ€Έ(1βˆ’π›Ό1)π‘žπ‘‘π‘₯1𝑑π‘₯2ξƒͺ1/π‘žξƒ©ξ€œβˆžπ‘Žξ€œβˆžπ‘π‘€1βˆ’π‘β€²ξ€·π‘₯1,π‘₯2ξ€Έπ‘₯(1βˆ’π›Ό2)𝑝′2𝑑π‘₯1𝑑π‘₯2ξƒͺ1/𝑝′𝐼<∞,ξ…ž2∢=supπ‘Ž,𝑏>0ξ‚΅ξ€œπ‘Ž0ξ€œπ‘0𝑣π‘₯1,π‘₯2𝑑π‘₯1𝑑π‘₯2ξ‚Ά1/π‘žβŽ›βŽœβŽœβŽξ€œβˆžπ‘Žξ€œβˆžπ‘π‘€1βˆ’π‘β€²ξ€·π‘₯1,π‘₯2ξ€Έπ‘₯(1βˆ’π›Ό2)𝑝′2ξ€·π‘₯1ξ€Έβˆ’π‘Ž(1βˆ’π›Ό1)𝑝′𝑑π‘₯1𝑑π‘₯2⎞⎟⎟⎠1/𝑝′<∞.(4.10) Moreover, β€–(𝒲ℋ′)𝛼1,𝛼2ξ‚πΌβ€–β‰ˆmax{ξ…ž1,ξ‚πΌξ…ž2}.

Proof. Let 𝑀(π‘₯1,π‘₯2)=𝑀1(π‘₯1)𝑀2(π‘₯2). The proof of the case 𝑣(π‘₯1,π‘₯2)=𝑣1(π‘₯1)𝑣2(π‘₯2) 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 βˆ«π‘†βˆΆ=∞0𝑀21βˆ’π‘ξ…ž(π‘₯2)𝑑π‘₯2=∞. Let {π‘Žπ‘˜}+βˆžπ‘˜=βˆ’βˆž be a sequence of positive numbers for which the equality 2π‘˜=ξ€œπ‘Žπ‘˜0𝑀1βˆ’π‘β€²2ξ€·π‘₯2𝑑π‘₯2(4.11) holds for all π‘˜βˆˆβ„€. It is clear that {π‘Žπ‘˜} is increasing and ℝ+=βˆͺπ‘˜βˆˆβ„€[π‘Žπ‘˜,π‘Žπ‘˜+1). Moreover, it is easy to verify that 2π‘˜=ξ€œπ‘Žπ‘˜+1π‘Žπ‘˜π‘€1βˆ’π‘β€²2ξ€·π‘₯2𝑑π‘₯2.(4.12) Let 𝑓β‰₯0. We have that β€–β€–(β„›β„‹)𝛼1,𝛼2π‘“β€–β€–π‘žπΏπ‘žξ€·π‘£,ℝ2+ξ€Έ=ξ€œβ„2+𝑣π‘₯1,π‘₯2ξ€Έξ€·(β„›β„‹)𝛼1,𝛼2π‘“ξ€Έπ‘žξ€·π‘₯1,π‘₯2𝑑π‘₯1𝑑π‘₯2β‰€ξ“π‘˜βˆˆβ„€ξ€œβˆž0ξ€œπ‘Žπ‘˜+1π‘Žπ‘˜π‘£ξ€·π‘₯1,π‘₯2ξ€Έπ‘₯(1βˆ’π›Ό22)π‘žξƒ©ξ€œπ‘₯10ξ€œπ‘₯20𝑓𝑑1,𝑑2ξ€Έξ€·π‘₯1βˆ’π‘‘1ξ€Έ1βˆ’π›Ό1𝑑𝑑1𝑑𝑑2ξƒͺπ‘žπ‘‘π‘₯1𝑑π‘₯2β‰€ξ“π‘˜βˆˆβ„€ξ€œβˆž0ξƒ©ξ€œπ‘Žπ‘˜+1π‘Žπ‘˜π‘£ξ€·π‘₯1,π‘₯2ξ€Έπ‘₯(1βˆ’π›Ό22)π‘žπ‘‘π‘₯2ξƒͺξ‚΅ξ€œπ‘₯10ξ€·π‘₯1βˆ’π‘‘1𝛼1βˆ’1ξ‚΅ξ€œπ‘Žπ‘˜+10𝑓𝑑1,𝑑2𝑑𝑑2𝑑𝑑1ξ‚Άπ‘žπ‘‘π‘₯1=ξ“π‘˜βˆˆβ„€ξ€œβˆž0π‘‰π‘˜ξ€·π‘₯1ξ€Έξ‚΅ξ€œπ‘₯10ξ€·π‘₯1βˆ’π‘‘1ξ€Έ(𝛼1βˆ’1)πΉπ‘˜ξ€·π‘‘1𝑑𝑑1ξ‚Άπ‘žπ‘‘π‘₯1,(4.13) where π‘‰π‘˜ξ€·π‘₯1ξ€Έξ€œβˆΆ=π‘Žπ‘˜+1π‘Žπ‘˜π‘£ξ€·π‘₯1,π‘₯2ξ€Έπ‘₯(1βˆ’π›Ό22)π‘žπ‘‘π‘₯2,πΉπ‘˜ξ€·π‘‘1ξ€Έξ€œβˆΆ=π‘Žπ‘˜+10𝑓𝑑1,𝑑2𝑑𝑑2.(4.14)
It is obvious that ξ‚πΌπ‘ž1β‰₯supπ‘Ž>0π‘—βˆˆβ„€ξƒ©ξ€œβˆžπ‘Žξ€œπ‘Žπ‘—+1π‘Žπ‘—π‘£ξ€·π‘₯1,π‘₯2ξ€Έξ€·π‘₯1ξ€Έβˆ’π‘Ž(1βˆ’π›Ό1)π‘žπ‘₯(1βˆ’π›Ό22)π‘žπ‘‘π‘₯1𝑑π‘₯2ξƒͺβŽ›βŽœβŽœβŽξ€œπ‘Ž0ξ€œπ‘Žπ‘—0𝑀1βˆ’π‘β€²ξ€·π‘₯1,π‘₯2ξ€Έξ€·π‘Žβˆ’π‘₯1ξ€Έ(1βˆ’π›Ό1)𝑝′𝑑π‘₯1𝑑π‘₯2βŽžβŽŸβŽŸβŽ π‘ž/𝑝′,ξ‚πΌπ‘ž2β‰₯supπ‘Ž>0π‘—βˆˆβ„€ξƒ©ξ€œβˆžπ‘Žξ€œπ‘Žπ‘—+1π‘Žπ‘—π‘£ξ€·π‘₯1,π‘₯2ξ€Έπ‘₯(1βˆ’π›Ό22)π‘žπ‘‘π‘₯1𝑑π‘₯2ξƒͺβŽ›βŽœβŽœβŽξ€œπ‘Ž0ξ€œπ‘Žπ‘—0𝑀1βˆ’π‘β€²ξ€·π‘₯1,π‘₯2ξ€Έξ€·π‘Žβˆ’π‘₯1ξ€Έ(1βˆ’π›Ό1)𝑝′𝑑π‘₯1𝑑π‘₯2βŽžβŽŸβŽŸβŽ π‘ž/𝑝′.(4.15) Hence, by using the two-weight criteria for the one-dimensional Riemann-Liouville operator without singularity (see [24]), we find that β€–β€–(β„›β„‹)𝛼1,𝛼2π‘“β€–β€–π‘žπΏπ‘žξ€·π‘£,ℝ2+ξ€Έξ‚πΌβ‰€π‘π‘žξ“π‘—βˆˆβ„€ξƒ¬ξ€œβˆž0𝑀1ξ€·π‘₯1ξ€Έξ‚΅ξ€œπ‘Žπ‘—0𝑀1βˆ’π‘β€²2ξ€·π‘₯2𝑑π‘₯2ξ‚Ά1βˆ’π‘ξ€·πΉπ‘—ξ€·π‘₯1𝑝𝑑π‘₯1ξƒ­π‘ž/π‘ξ‚πΌβ‰€π‘π‘žξƒ¬ξ€œβˆž0𝑀1ξ€·π‘₯1ξ€Έξ“π‘—βˆˆπ‘ξ‚΅ξ€œπ‘Žπ‘—0𝑀1βˆ’π‘β€²2ξ€·π‘₯2𝑑π‘₯2ξ‚Ά1βˆ’π‘ξƒ©π‘—ξ“π‘˜=βˆ’βˆžξ€œπ‘Žπ‘˜+1π‘Žπ‘˜π‘“ξ€·π‘₯1,𝑑2𝑑𝑑2ξƒͺ𝑝𝑑π‘₯1ξƒ­π‘ž/𝑝,(4.16) where 𝐼𝐼=max{1,𝐼2}.
On the other hand, (4.11) yields +βˆžξ“π‘˜=π‘›ξ‚΅ξ€œπ‘Žπ‘˜0𝑀1βˆ’π‘β€²2ξ€·π‘₯2𝑑π‘₯2ξ‚Ά1βˆ’π‘ξƒ©π‘›ξ“π‘˜=βˆ’βˆžξ€œπ‘Žπ‘˜+1π‘Žπ‘˜π‘€1βˆ’π‘β€²2ξ€·π‘₯2𝑑π‘₯2ξƒͺπ‘βˆ’1=+βˆžξ“π‘˜=π‘›ξ‚΅ξ€œπ‘Žπ‘˜0𝑀1βˆ’π‘β€²2ξ€·π‘₯2𝑑π‘₯2ξ‚Ά1βˆ’π‘ξ‚΅ξ€œπ‘Žπ‘›+10𝑀1βˆ’π‘β€²2ξ€·π‘₯2𝑑π‘₯2ξ‚Άπ‘βˆ’1=+βˆžξ“π‘˜=𝑛2π‘˜(1βˆ’π‘)ξƒͺ2(𝑛+1)(π‘βˆ’1)≀𝑐(4.17) for all π‘›βˆˆβ„€. Hence by Hardy’s inequality in discrete case (see, for example, [25, 26]) and HΓΆlder’s inequality we have that β€–β€–(β„›β„‹)𝛼1,𝛼2π‘“β€–β€–π‘žπΏπ‘žξ€·π‘£,ℝ2+ξ€Έξ‚πΌβ‰€π‘π‘žβŽ‘βŽ’βŽ’βŽ£ξ€œβˆž0𝑀1ξ€·π‘₯1ξ€Έξ“π‘—βˆˆβ„€ξƒ©ξ€œπ‘Žπ‘—+1π‘Žπ‘—π‘€1βˆ’π‘β€²2ξ€·π‘₯2𝑑π‘₯2ξƒͺ1βˆ’π‘ξƒ©ξ€œπ‘Žπ‘—+1π‘Žπ‘—π‘“ξ€·π‘₯1,𝑑2𝑑𝑑2ξƒͺ𝑝𝑑π‘₯1⎀βŽ₯βŽ₯βŽ¦π‘ž/π‘ξ‚πΌβ‰€π‘π‘žξƒ¬ξ€œβˆž0𝑀1ξ€·π‘₯1ξ€Έξ“π‘—βˆˆβ„€ξƒ©ξ€œπ‘Žπ‘—+1π‘Žπ‘—π‘€2𝑑2𝑓𝑝π‘₯1,𝑑2𝑑𝑑2ξƒͺ𝑑π‘₯1ξƒ­π‘ž/𝑝𝐼=π‘π‘žβ€–π‘“β€–π‘žπΏπ‘ξ€·π‘€,ℝ2+ξ€Έ.(4.18)
If 𝑆<∞, then without loss of generality we can assume that 𝑆=1. In this case we choose the sequence {π‘Žπ‘˜}0π‘˜=βˆ’βˆž for which (4.11) holds for all π‘˜βˆˆβ„€βˆ’. 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 {π‘₯π‘˜} so that ∫∞π‘₯π‘˜π‘€21βˆ’π‘ξ…ž(π‘₯)𝑑π‘₯=2π‘˜ (notice that π‘₯π‘˜ is decreasing) and argue as in the proof of (i).

Proposition 4.4. Let 1<π‘β‰€π‘ž<∞, and let 𝛼1,𝛼2β‰₯1. Suppose that either 𝑀(π‘₯1,π‘₯2)=𝑀1(π‘₯1)𝑀2(π‘₯2) or 𝑣(π‘₯1,π‘₯2)=𝑣1(π‘₯1)𝑣2(π‘₯2) for some one-dimensional weights: 𝑀1, 𝑀2, 𝑣1, and 𝑣2.
(i)The operator (β„‹β„›)𝛼1,𝛼2 is bounded from 𝐿𝑝(𝑀,ℝ2+) to πΏπ‘ž(𝑣,ℝ2+) if and only if 𝐼1∢=supπ‘Ž,𝑏>0βŽ›βŽœβŽœβŽξ€œπ‘Ž0ξ€œπ‘0𝑀1βˆ’π‘β€²ξ€·π‘₯1,π‘₯2ξ€Έξ€·π‘βˆ’π‘₯2ξ€Έ(1βˆ’π›Ό2)𝑝′𝑑π‘₯1𝑑π‘₯2⎞⎟⎟⎠1/π‘β€²ξƒ©ξ€œβˆžπ‘Žξ€œβˆžπ‘π‘£ξ€·π‘₯1,π‘₯2ξ€Έπ‘₯(1βˆ’π›Ό11)π‘žπ‘‘π‘₯1𝑑π‘₯2ξƒͺ1/π‘žπΌ<∞,2∢=supπ‘Ž,𝑏>0ξ‚΅ξ€œπ‘Ž0ξ€œπ‘0𝑀1βˆ’π‘β€²ξ€·π‘₯1,π‘₯2𝑑π‘₯1𝑑π‘₯2ξ‚Ά1/π‘β€²ξƒ©ξ€œβˆžπ‘Žξ€œβˆžπ‘π‘£ξ€·π‘₯1,π‘₯2ξ€Έπ‘₯(1βˆ’π›Ό11)π‘žξ€·π‘₯2ξ€Έβˆ’π‘(1βˆ’π›Ό2)π‘žπ‘‘π‘₯1𝑑π‘₯2ξƒͺ1/π‘ž<∞.(4.19) Moreover, β€–(β„‹β„›)𝛼1,𝛼2β€–β‰ˆmax{𝐼1,𝐼2}.(ii)The operator (ℋ𝒲)𝛼1,𝛼2 is bounded from 𝐿𝑝(𝑀,ℝ2+) to πΏπ‘ž(𝑣,ℝ+) if and only if 𝐽1∢=supπ‘Ž,𝑏>0ξƒ©ξ€œβˆžπ‘Žξ€œπ‘0𝑣π‘₯1,π‘₯2ξ€Έπ‘₯(1βˆ’π›Ό11)π‘žξ€·π‘βˆ’π‘₯2ξ€Έ(1βˆ’π›Ό2)π‘žπ‘‘π‘₯1𝑑π‘₯2ξƒͺ1/π‘žξ‚΅ξ€œπ‘Ž0ξ€œβˆžπ‘π‘€1βˆ’π‘β€²ξ€·π‘₯1,π‘₯2𝑑π‘₯1𝑑π‘₯2ξ‚Ά1/𝑝′𝐽<∞,2∢=supπ‘Ž,𝑏>0ξ‚΅ξ€œβˆžπ‘Žξ€œπ‘0𝑣π‘₯1,π‘₯2ξ€Έπ‘₯(𝛼11βˆ’1)π‘žπ‘‘π‘₯1𝑑π‘₯2ξ‚Ά1/π‘žβŽ›βŽœβŽœβŽξ€œπ‘Ž0ξ€œβˆžπ‘π‘€1βˆ’π‘β€²ξ€·π‘₯1,π‘₯2ξ€Έξ€·π‘₯2ξ€Έβˆ’π‘(1βˆ’π›Ό2)𝑝′𝑑π‘₯1𝑑π‘₯2⎞⎟⎟⎠1/𝑝′<∞.(4.20) Moreover, β€–(ℋ𝒲)𝛼1,𝛼2β€–β‰ˆmax{𝐽1,𝐽2}.(iii)The operator (β„‹β€²β„›)𝛼1,𝛼2 is bounded from 𝐿𝑝(𝑀,ℝ2+) to πΏπ‘ž(𝑣,ℝ+) if and only if π½ξ…ž1∢=supπ‘Ž,𝑏>0ξ‚΅ξ€œπ‘Ž0ξ€œβˆžπ‘π‘£ξ€·π‘₯1,π‘₯2𝑑π‘₯1𝑑π‘₯2ξ‚Ά1/π‘žβŽ›βŽœβŽœβŽξ€œβˆžπ‘Žξ€œπ‘0𝑀1βˆ’π‘β€²ξ€·π‘₯1,π‘₯2ξ€Έπ‘₯(1βˆ’π›Ό1)𝑝′1ξ€·π‘βˆ’π‘₯2ξ€Έ(1βˆ’π›Ό2)𝑝′𝑑π‘₯1𝑑π‘₯2⎞⎟⎟⎠1/𝑝′𝐽<∞,ξ…ž2∢=supπ‘Ž,𝑏>0ξƒ©ξ€œπ‘Ž0ξ€œβˆžπ‘π‘£ξ€·π‘₯1,π‘₯2ξ€Έξ€·π‘₯2ξ€Έβˆ’π‘(1βˆ’π›Ό2)π‘žπ‘‘π‘₯1𝑑π‘₯2ξƒͺ1/π‘žξƒ©ξ€œβˆžπ‘Žξ€œπ‘0𝑀1βˆ’π‘β€²ξ€·π‘₯1,π‘₯2ξ€Έπ‘₯(1βˆ’π›Ό1)𝑝′1𝑑π‘₯1𝑑π‘₯2ξƒͺ1/𝑝′<∞.(4.21) Moreover, β€–(β„‹β€²β„›)𝛼1,𝛼2β€–β‰ˆmax{π½ξ…ž1,π½ξ…ž2}.(iv)The operator (ℋ′𝒲)𝛼1,𝛼2 is bounded from 𝐿𝑝(𝑀,ℝ2+) to πΏπ‘ž(𝑣,ℝ+) if and only if