Two-Dimensional Hardy Operators in Lebesgue Spaces Vladimir D. Stepanova , Elena P. Ushakovab and Sergey E. Zhukovskiyb a Computing Center of FEB RAS, 65 Kim Yu Chen street, Khabarovsk, 680000, Russia b V.A. Trapeznikov Institute of Control Sciences of RAS, 65 Profsoyuznaya street, Moscow, 117997, Russia Abstract Characterizations of linear and bilinear Lebesgue norm inequalities involving two- dimensional Hardy integral operators are obtained. Keywords 1 Hardy integral operator, weighted Lebesgue space, bilinear inequality. 1. Introduction Let Ϻ be the set of all Lebesgue measurable functions f on ℝ2+ ≔ (0, ∞)2 , and let Ϻ+ ⊂ Ϻ be the subset of all nonnegative f. If 𝑣 ∈ Ϻ+ and 0 < 𝑝 ≤ ∞ we define the weighted Lebesgue space 1 𝑝 p 𝐿𝑣 (ℝ2 ) = {𝑓 ∈ Ϻ: ‖𝑓‖p,v ≔ (∫|𝑓(𝑥)|𝑝 𝑣(𝑥)𝑑𝑥) < ∞} , 0 < 𝑝 < ∞, 𝐿∞ 2 𝑣 (ℝ ) = {𝑓 ∈ Ϻ: ‖𝑓‖∞,𝑣 ≔ ess sup𝑥∈ℝ2 𝑣(𝑥)|𝑓(𝑥)| < ∞}, 𝑝 = ∞. Let 𝑛 ∈ ℕ, 0 < 𝑞 ≤ ∞ and 1 ≤ 𝑝𝑖 ≤ ∞, 𝑤, 𝑣𝑖 ∈ Ϻ+ for all 𝑖 = 1, … 𝑛. Define the two-dimensional rectangular Hardy operator 𝑥 𝑦 𝐼2 𝑓(𝑥, 𝑦) ≔ ∫ ∫ 𝑓(𝑠, 𝑡)𝑑𝑠𝑑𝑡 , (𝑥, 𝑦) ∈ ℝ2+ , (1) 0 0 and consider the following multilinear inequality ‖(𝐼2 𝑓1 ) · … · (𝐼2 𝑓𝑛 )‖𝑞,𝑤 ≤ 𝐶‖𝑓1 ‖𝑝1 ,𝑣1 … ‖𝑓𝑛 ‖𝑝𝑛 ,𝑣𝑛 , 𝑓𝑖 ∈ Ϻ+ , (2) where a constant C>0 is independent of 𝑓𝑖 , 𝑖 = 1, … , 𝑛, and is supposed to be the least possible. The general problem is to characterize this inequality (2) by establishing a two-sided estimate 𝛼 𝐹(𝑣1 , … 𝑣𝑛 , 𝑤; 𝑝1 , … 𝑝𝑛 , 𝑞) ≤ 𝐶 ≤ 𝛽 𝐹(𝑣1 , … 𝑣𝑛 , 𝑤; 𝑝1 , … 𝑝𝑛 , 𝑞) with some irrelevant constants 𝛼 and β by a functional 𝐹(𝑣1 , … 𝑣𝑛 , 𝑤; 𝑝1 , … 𝑝𝑛 , 𝑞) of an explicit form depending on given weights 𝑣1 , … , 𝑣𝑛, 𝑤 and fixed parameters 𝑝1 , … , 𝑝𝑛 , 𝑞 only. An operator in the left-hand side of the inequality (2) is n-fold product of two-dimensional Hardy operators (1), it is acting on the product of n Lebesgue spaces. Multi(sub)linear maximal operators, which are related to (1), appeared in connection with multilinear Calderón-Zygmund theory. They were used for the study of multilinear singular integral operators of Calderón-Zygmund type and for building a theory of weights adapted to the multilinear setting [6, 3, 1]. Linear and multi-linear VI International Conference Information Technologies and High-Performance Computing (ITHPC-2021), September 14–16, 2021, Khabarovsk, Russia EMAIL elenau@inbox.ru (A. 2) ORCID: 0000-0002-3497-3762 (A. 2) ©️ 2020 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). CEUR Workshop Proceedings (CEUR-WS.org) inequalities with Hardy operators also play an important role in analysis and its applications [5]. The main purpose of this work is to survey the most recent characterizations of (2) by the authors in linear and bilinear cases. Starting in Section 2 from (quasi)linear case n = 1, we give the results for bilinear inequalities in Section 3. These findings can be similarly extended to any multilinear case. We use signs := and =: for determining new quantities. For positive functionals F and G we write 𝐹 ≪ 𝐺 if 𝐹 ≤ 𝛼 𝐺 with some constant 𝛼 > 0 depending, possibly, on irrelevant parameters only. Relations of the type 𝐹 ≈ 𝐺 mean F ≪ G ≪ 𝐹 or 𝐹 = 𝛼𝐺. 2. Two-dimensional Hardy inequality Weighted Hardy inequality ‖𝐼2 𝑓‖𝑞,𝑤 ≤ 𝐶‖𝑓‖𝑝,𝑣 , 𝑓 ∈ Ϻ+ , (3) with two-dimensional rectangular operator (1) was studied in [4, 8, 11, 12, 24]. In particular, the following criterion for the inequality (3) to hold was obtained by E. Sawyer in [12]. Theorem [12, Theorem 1A]. Let 1 < 𝑝 ≤ 𝑞 < ∞. Denote 𝑝′ ≔ 𝑝/(𝑝 − 1) and let (𝐼2∗ 𝑓)(𝑥, 𝑦) ≔ ∞ ∞ ∫𝑥 ∫𝑦 𝑓(𝑠, 𝑡)𝑑𝑠𝑑𝑡 be the adjoint to 𝐼2 operator. The inequality (3) holds if and only if 𝐴1 ≔ sup(𝑠,𝑡)∈ℝ2+ [𝐼2∗ 𝑤(𝑠, 𝑡)]1/𝑞 [𝐼2 𝑣 1−𝑝′ (𝑠, 𝑡)]1/𝑝′ < ∞, (4) 𝑠 𝑡 1/𝑞 1−𝑝′ )𝑞 [𝐼2 𝑣 1−𝑝′ (𝑠, 𝑡)]−1/𝑝 < ∞, (5) 𝐴2 ≔ sup(𝑠,𝑡)∈ℝ2+ (∫ ∫ (𝐼2 𝑣 𝑤) 0 0 ∞ ∞ 1/𝑝′ ∗ 𝑝′ (𝐼2 𝑤) 𝑣 1−𝑝′ [𝐼2∗ 𝑤(𝑠, 𝑡)]−1/𝑞′ < ∞. (6) 𝐴3 ≔ sup(𝑠,𝑡)∈ℝ2+ (∫ ∫ ) 𝑠 𝑡 Moreover, it holds for the least possible constant C>0 in (3) that 𝐶 ≈ 𝐴1 + 𝐴2 + 𝐴3 with equivalence constants depending of p and q only. The one-dimensional analog of the condition (4) is the boundedness of the Muckenhoupt constant [9]. Characteristics (5) and (6) are two-dimensional generalizations of the Tomaselli functional [23, definition (11)] in its direct and dual forms. In one-dimensional case all the conditions (4)-(6) are equivalent to each other (see e.g. [2]), that is 𝐴1 ≈ 𝐴2 ≈ 𝐴3 with equivalence constants depending of p and q. In two-dimensional case this generally is not true. Moreover, as it was shown in [12, § 4] for p=q=2 that no two of conditions (4)-(6) guarantee (3). But, it was discovered in the recent work [22] by the authors that the E. Sawyer’s theorem is actual for p=q only, while for p1 and generalized to all the types of boundedness constants in [11]. Theorem [11, Theorems 2.1, 2.2]. Let 1 < 𝑝 ≤ 𝑞 < ∞ and the weight v satisfy the condition (7). 𝑥 1−𝑝′ Denote 𝑉𝑖 (𝑥𝑖 ) ≔ ∫0 𝑖 𝑣𝑖 , 𝑖 = 1,2. Then the inequality (3) holds for all 𝑓 ≥ 0 if and only if 𝐴𝑀 ≔ sup(𝑠,𝑡)∈ℝ2+ [𝐼2∗ 𝑤(𝑠, 𝑡)]1/𝑞 [𝑉1 (𝑠)𝑉2 (𝑡)]1/𝑝′ < ∞, or if and only if 𝑠 𝑡 1/𝑞 𝐴 𝑇 ≔ sup(𝑠,𝑡)∈ℝ2+ (∫ ∫ [𝑉1 𝑉2 ]𝑞 𝑤 ) [𝑉1 (𝑠)𝑉2 (𝑡)]−1/𝑝 < ∞. 0 0 Besides, it holds for the least possible constant C>0 in (3) that 𝐶 ≈ 𝐴𝑀 ≈ 𝐴 𝑇 with equivalence constants depending of p and q only. Theorem [11, Theorems 2.4, 2.5]. Let 1 < 𝑝 ≤ 𝑞 < ∞ and the weight w satisfy the condition (8). ∞ Denote 𝑊𝑖 (𝑥𝑖 ) ≔ ∫𝑥 𝑤𝑖 , 𝑖 = 1,2. Then the inequality (3) holds for all 𝑓 ≥ 0 if and only if 𝑖 𝐴∗𝑀 ≔ sup(𝑠,𝑡)∈ℝ2+ [𝐼2 𝑣 1−𝑝′ (𝑠, 𝑡)]1/𝑝′ [𝑊1 (𝑠)𝑊2 (𝑡)]1/𝑞 < ∞, or if and only if ∞ ∞ 1/𝑝′ ∗ [𝑊 ]𝑝′ 1−𝑝′ [𝑊1 (𝑠)𝑊2 (𝑡)]−1/𝑞′ < ∞. 𝐴 𝑇 ≔ sup(𝑠,𝑡)∈ℝ2+ (∫ ∫ 1 𝑊2 𝑣 ) 𝑠 𝑡 Besides, 𝐶 ≈ 𝐴∗𝑀 ≈ 𝐴∗𝑇 with equivalence constants depending of p and q only. We complete the section by assertions similar to the last two above, but devoted to the case 1 < 𝑞 < 𝑝 < ∞. To state them we put 1/r=1/q-1/p and define two-dimensional analogs of Maz’ya-Rosin [7, § 1.3.2] and Persson-Stepanov [10, Theorem 3] functionals in their direct and dual forms: ′ ′ 1/𝑟 ∗ 𝑟/𝑞 [𝑉 (𝑠)𝑉 (𝑡)]𝑟/𝑞′ 1−𝑝 (𝑠) 1−𝑝 𝐵𝑀𝑅 ≔ (∫[𝐼2 𝑤(𝑠, 𝑡)] 1 2 𝑣1 𝑣2 (𝑡) 𝑑𝑠 𝑑𝑡) , 𝑟/𝑞 1/𝑟 𝑠 𝑡 ′ ′ 𝐵𝑃𝑆 ≔ (∫ (∫ ∫ [𝑉1 𝑉2 ]𝑞 𝑤) [𝑉1 (𝑠)𝑉2 (𝑡)]−𝑟/𝑞 𝑣11−𝑝 (𝑠) 𝑣21−𝑝 (𝑡) 𝑑𝑠 𝑑𝑡) , 0 0 1/𝑟 ∗ 𝐵𝑀𝑅 ≔ (∫[𝐼2 𝑣 1−𝑝′ (𝑠, 𝑡)]𝑟/𝑝′ [𝑊1 (𝑠)𝑊2 (𝑡)]𝑟/𝑝 𝑤1 (𝑠) 𝑤2 (𝑡) 𝑑𝑠 𝑑𝑡) , 𝑟/𝑝′ 1/𝑟 ∞ ∞ ∗ 𝑝′ 1−𝑝′ −𝑟/𝑝′ 𝐵𝑃𝑆 ≔ (∫ (∫ ∫ [𝑊1 𝑊2 ] 𝑣 ) [𝑊1 (𝑠)𝑊2 (𝑡)] 𝑤1 (𝑠) 𝑤2 (𝑡) 𝑑𝑠 𝑑𝑡) . 𝑠 𝑡 Theorem [11, Theorems 3.1, 3.2]. Let 1 < 𝑞 < 𝑝 < ∞ . Suppose that the weight v in (3) satisfies the condition (7) and 𝑉1 (∞) = 𝑉2 (∞) = ∞. Then the inequality (3) is valid for all 𝑓 ∈ Ϻ+ if and only if 𝐵𝑀𝑅 < ∞, or if and only if 𝐵𝑃𝑆 < ∞. Moreover, 𝐶 ≈ 𝐵𝑀𝑅 ≈ 𝐵𝑃𝑆 . Theorem [11, Theorems 3.3, 3.4]. Let 1 < 𝑞 < 𝑝 < ∞ . Assume that the weight function w in (3) satisfies the condition (8) and 𝑊1 (0) = 𝑊2 (0) = ∞. Then the inequality (3) is valid for all ∗ ∗ ∗ ∗ 𝑓 ∈ Ϻ+ if and only if 𝐵𝑀𝑅 < ∞, or if and only if 𝐵𝑃𝑆 < ∞. Moreover, 𝐶 ≈ 𝐵𝑀𝑅 ≈ 𝐵𝑃𝑆 . 163 3. Bilinear two-dimensional Hardy inequality In this section we demonstrate some of the new characteristics from [20] obtained for the inequality ‖(𝐼2 𝑓)(𝐼2 𝑔)‖𝑞,𝑤 ≤ 𝐶‖𝑓‖𝑝,𝑣 ‖𝑔‖𝑠,𝑢 , 𝑓, 𝑔 ∈ Ϻ+ . (9) These results are based on statements for the linear two-dimensional Hardy inequality from Section 2. Distinguish the following zones for the relations between integration parameters 1 < 𝑝, 𝑠, 𝑞 < ∞: (I) 1 < max{𝑝, 𝑠} ≤ 𝑞 < ∞, (II) 1 < min{𝑝, 𝑠} ≤ 𝑞 < max{𝑝, 𝑠} < ∞, (III) 1 < 𝑞 < min{𝑝, 𝑠}. The required characteristics for (I), (II) and (III) are given in the assertions below. Theorem [20, Theorem 4]. Let 𝑝, 𝑠, 𝑞 ∈ (𝐼) . Assume that the weight v in (9) is of product type, that is v satisfies the condition (7). Then the best constant C in the inequality (9) is estimated as 𝐶 ≈ 𝐷𝐼 ≔ sup(𝑥,𝑦)∈ℝ2+ (𝐷1 (𝑥, 𝑦) + 𝐷2 (𝑥, 𝑦) + 𝐷3 (𝑥, 𝑦))[𝑉1 (𝑥)𝑉2 (𝑦)]1/𝑝′ , (10) 𝑥 1−𝑝′ where 𝑉𝑖 (𝑥𝑖 ) ≔ ∫0 𝑖 𝑣𝑖 , 𝑖 = 1,2, as before and 1/𝑞 𝐷1 (𝑥, 𝑦) ≔ sup(𝜚,𝜏)∈ℝ2+ [𝐼2∗ (𝑤𝜒(𝑥,∞)×(𝑦,∞) )(𝜚, 𝜏)] [𝐼2 𝑢1−𝑠′ (𝜚, 𝜏)]1/𝑠′ , 𝜚 𝜏 1/𝑞 1−𝑠′ )𝑞 [𝐼2 𝑢1−𝑠′ (𝜚, 𝜏)]−1/𝑠 , 𝐷2 (𝑥, 𝑦) ≔ sup(𝜚,𝜏)∈ℝ2+ (∫ ∫ (𝐼2 𝑢 𝑤𝜒(𝑥,∞)×(𝑦,∞) ) 0 0 ∞ ∞ 1/𝑠′ 𝑠′ −1/𝑞′ 𝐷3 (𝑥, 𝑦) ≔ sup(𝜚,𝜏)∈ℝ2+ (∫ ∫ (𝐼2∗ (𝑤𝜒(𝑥,∞)×(𝑦,∞) )) 𝑢 1−𝑠′ ) [𝐼2∗ (𝑤𝜒(𝑥,∞)×(𝑦,∞) )(𝜚, 𝜏)] . 𝜚 𝜏 Remark [20, Remark 3]. If the weight u in (9) is also of product type, that is if 𝑢(𝑥1 , 𝑥2 ) = 𝑢1 (𝑥1 )𝑢(𝑥2 ), (11) then the expression for the functional 𝐷𝐼 in (10) simplifies as follows: 𝐷𝐼 ≔ sup(𝑥,𝑦)∈ℝ2+ [𝐼2∗ 𝑤(𝑥, 𝑦)]1/𝑞 [𝑉1 (𝑥)𝑉2 (𝑦)]1/𝑝′ [𝑈1 (𝑥)𝑈2 (𝑦)]1/𝑝′ < ∞, 𝑥 1−𝑝′ 𝑥 where 𝑉𝑖 (𝑥𝑖 ) ≔ ∫0 𝑖 𝑣𝑖 and 𝑈𝑖 (𝑥𝑖 ) ≔ ∫0 𝑖 𝑢1−𝑠′ 𝑖 , 𝑖 = 1,2. Theorem [20, Theorem 5]. Let 𝑝, 𝑠, 𝑞 ∈ (𝐼𝐼) . Assume that the weights v and u in (9) are of product type, that is v and u satisfy the conditions (7) and (11), respectively. Then 𝐶 ≈ 𝐷𝐼𝐼 , where for 1 < 𝑝 ≤ 𝑞 < 𝑠 < ∞, under the condition 𝑈𝑖 (∞) = ∞, 𝑖 = 1,2, ∞ ∞ 1/𝑡 ∗ 𝑡/𝑞 𝑡/𝑞′ 1−𝑠′ 1−𝑠′ 𝐷𝐼𝐼 ≔ sup(𝑥,𝑦)∈ℝ2+ (∫ ∫ [𝐼2 𝑤] [𝑈1 𝑈2 ] 𝑢1 𝑢2 ) [𝑉1 (𝑥)𝑉2 (𝑦)]1/𝑝′ , 𝑥 𝑦 and for 1 < 𝑠 ≤ 𝑞 < 𝑝 < ∞, under the condition 𝑉𝑖 (∞) = ∞, 𝑖 = 1,2, ∞ ∞ 1/𝑟 ∗ 𝑟/𝑞 𝑟/𝑞′ 1−𝑝′ 1−𝑝′ 𝐷𝐼𝐼 ≔ sup(𝑥,𝑦)∈ℝ2+ (∫ ∫ [𝐼2 𝑤] [𝑉1 𝑉2 ] 𝑣1 𝑣2 ) [𝑈1 (𝑥)𝑈2 (𝑦)]1/𝑠′ , 𝑥 𝑦 where 1/r:=1/q-1/p and 1/t=1/q-1/s. 164 Theorem [20, Theorem 6]. Let 𝑝, 𝑠, 𝑞 ∈ (𝐼𝐼𝐼) . Assume that all the weights in (9) are of product type, that is v,u and w satisfy the conditions (7), (11) and (8), respectively. Then, under the conditions 𝑉𝑖 (∞) = ∞, 𝑖 = 1,2, and 𝑈𝑖 (∞) = ∞, 𝑖 = 1,2, it holds 𝐶 ≈ ∑4𝑖=1 𝐷𝐼𝐼 (𝑖), where for 1/𝑞 ≤ 1/𝑝 + 1/𝑠 ∞ ∞ 𝑡 𝑡 1/𝑡 𝐷𝐼𝐼 (1) ≔ sup(𝑥,𝑦)∈ℝ2+ (∫ ∫ [𝑊1 𝑊2 ]𝑞 [𝑈1 𝑈2 ]𝑞′ 𝑑𝑈1 𝑑𝑈2 ) [𝑉1 (𝑥)𝑉2 (𝑦)]1/𝑝′ , 𝑥 𝑦 ∞ ∞ 𝑟 𝑟 1/𝑡 𝐷𝐼𝐼 (2) ≔ sup(𝑥,𝑦)∈ℝ2+ (∫ ∫ [𝑊1 𝑊2 ] [𝑉1 𝑉2 ] 𝑑𝑉1 𝑑𝑉2 ) 𝑞 𝑞′ [𝑈1 (𝑥)𝑈2 (𝑦)]1/𝑠′ , 𝑥 𝑦 ∞ 𝑡 𝑡 1/𝑡 ∞ 𝑟 𝑟 1/𝑟 𝐷𝐼𝐼 (3) ≔ sup(𝑥,𝑦)∈ℝ2+ (∫ [𝑊1 ]𝑞 [𝑈1 ]𝑞′ 𝑑𝑈1 ) (∫ [𝑊2 ]𝑞 [𝑉2 ]𝑞′ 𝑑𝑉2 ) [𝑉1 (𝑥)]1/𝑝′ [𝑈2 (𝑦)]1/𝑠′ , 𝑥 𝑦 ∞ 𝑟 𝑟 1/𝑟 ∞ 𝑡 𝑡 1/𝑡 1 1 𝐷𝐼𝐼 (4) ≔ sup(𝑥,𝑦)∈ℝ2+ (∫ [𝑊1 ] [𝑉1 ] 𝑑𝑉1 ) 𝑞 𝑞′ (∫ [𝑊2 ] [𝑈2 ] 𝑑𝑈2 ) 𝑞 𝑞′ [𝑈1 (𝑥)]𝑠′ [𝑉2 (𝑦)]𝑝′ ; 𝑥 𝑦 and for 1/𝑞 > 1/𝑝 + 1/𝑠 with 1/𝜅 ≔ 1/𝑞 − 1/𝑝 − 1/𝑠 𝜅/𝑡 1/𝜅 ∞ ∞ ∞ ∞ 𝑡 𝑡 𝐷𝐼𝐼 (1) ≔ (∫ ∫ (∫ ∫ [𝑊1 𝑊2 ] [𝑈1 𝑈2 ] 𝑑𝑈1 𝑑𝑈2 ) 𝑞 𝑞′ [𝑉1 (𝑥)𝑉2 (𝑦)]𝜅/𝑡′ 𝑑𝑉1 (𝑥) 𝑑𝑉2 (𝑦)) , 0 0 𝑥 𝑦 𝜅/𝑟 1/𝜅 ∞ ∞ ∞ ∞ 𝑟 𝑟 𝑞′ 𝜅/𝑟′ 𝐷𝐼𝐼 (2) ≔ (∫ ∫ (∫ ∫ [𝑊1 𝑊2 ] [𝑉1 𝑉2 ] 𝑑𝑉1 𝑑𝑉2 ) 𝑞 [𝑈1 (𝑥)𝑈2 (𝑦)] 𝑑𝑈1 (𝑥) 𝑑𝑈2 (𝑦)) , 0 0 𝑥 𝑦 ∞ ∞ ∞ 𝑡 𝑡 𝜅/𝑡 ∞ 𝑟 𝑟 𝜅/𝑠 𝜅 [𝐷𝐼𝐼 (3)] ≔ ∫ ∫ (∫ [𝑊1 ] [𝑈1 ] 𝑞′ 𝑞′ [𝑉1 (𝑥)]𝜅/𝑡′ 𝑞 𝑑𝑈1 ) (∫ [𝑊2 ] [𝑉2 ] 𝑑𝑉2 ) 𝑞 0 0 𝑥 𝑦 𝑟 𝑟 × [𝑈2 (𝑦)]𝜅/𝑠′ [𝑊2 (𝑦)]𝑞 [𝑉2 (𝑦)]𝑞′ 𝑑𝑉1 (𝑥) 𝑑𝑉2 (𝑦), ∞ ∞ ∞ 𝑟 𝑟 𝜅/𝑠 ∞ 𝑡 𝑡 𝜅/𝑡 [𝐷𝐼𝐼 (4)]𝜅 ≔ ∫ ∫ (∫ [𝑊1 ]𝑞 [𝑉1 ]𝑞′ 𝑑𝑉1 ) (∫ [𝑊2 ]𝑞 [𝑈2 ]𝑞′ 𝑑𝑈2 ) [𝑈1 (𝑥)]𝜅/𝑠′ 0 0 𝑥 𝑦 𝑟 𝑟 × [𝑉2 (𝑦)]𝜅/𝑡′ [𝑊1 (𝑥)] [𝑉1 (𝑥)] 𝑑𝑉1 (𝑥) 𝑑𝑉2 (𝑦), 𝑞 𝑞′ 1−𝑝′ 𝑥 𝑥 ′ ∞ where 1/r:=1/q-1/p, 1/t:=1/q-1/s, 𝑉𝑖 (𝑥𝑖 ) ≔ ∫0 𝑖 𝑣𝑖 , 𝑈𝑖 (𝑥𝑖 ) ≔ ∫0 𝑖 𝑢1−𝑠 𝑖 , 𝑊𝑖 (𝑥𝑖 ) ≔ ∫𝑥 𝑤𝑖 , 𝑖 = 1,2. 𝑖 For some other types of bilinear inequalities with Hardy type operators one can consult [13-19, 21]. 4. Acknowledgements The research work of the first author and the second author was partially funded by the Russian Foundation for Basic Research (project No. 19-01-00223). The studies were carried out using the resources of the Center for Shared Use of Scientific Equipment "Center for Processing and Storage of Scientific Data of the Far Eastern Branch of the Russian Academy of Sciences", funded by the Russian Federation represented by the Ministry of Science and Higher Education of the Russian Federation under project No. 075-15-2021-663. 5. References [1] W. Damián, A. Lerner, C. Pérez. "Sharp weighted bounds for multilinear maximal functions and Carderón-Zygmund operators." J. Fourier Anal. Appl. 21.1 (2015): 161-181. [2] A. Gogatishvili, A. Kufner, L.-E. Persson, A. Wedestig. "An equivalence theorem for integral conditions related to Hardy's inequality." Real Anal. Exchange 29.2 (2003/04): 867-880. 165 [3] V. Kokalishvili, M. Mastył, A. Meskhi. "The multisublinear maximal type operators in Banach function lattices." J. Math. Anal. Appl. 421.1 (2015): 656-668. [4] V. Kokilashvili, A. Meskhi, L.-E. Persson, Weighted norm inequalities for integral transforms with product kernels, Nova Science Publishers, NY, 2009. [5] A. Kufner, L.-E. Persson, N. Samko, Weighted inequalities of Hardy type, Second Edition, World Scientific Publishing Co. Pte. Ltd., NJ, 2017. [6] A. Lerner, S. Ombrosi, C. Pérez, R.H. Torres, R. Trujillo-González. "New multiple weights for the multilinear Calderón-Zygmund theory. " , Adv. Math. 220.4 (2009): 1222-1264. [7] V. G. Maz’ya, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. [8] A. Meskhi. "A note on two-weight inequalities for multiple Hardy-type operators." J. Funct. Spaces Appl. 3.3 (2005): 223–237. [9] B. Muckenhoupt. "Hardy inequalities with weights." Studia Math. 44.3 (1972): 31-38. [10] L.–E. Persson, V.D. Stepanov. "Weighted integral inequalities with the geometric mean operator." J. Inequal. Appl. 7 (2002): 727-746. [11] L.–E. Persson, E.P. Ushakova. "Some multi--dimensional Hardy type integral inequalities." J. Math. Inequal. 1.3 (2007): 301-319. [12] E. Sawyer. "Weighted inequalities for two-dimensional Hardy operator". Studia Math. 82.1 (1985): 1-16. [13] V.D. Stepanov, G.E. Shambilova. "Multidimensional bilinear Hardy inequalities." Doklady Math. 100.1 (2019): 374–376. [14] V.D. Stepanov, G.E. Shambilova. "On bilinear weighted inequalities with Volterra integral operators." Doklady Math. 99.3 (2019): 290–294. [15] V.D. Stepanov, G.E. Shambilova. "Discrete Bilinear Hardy Inequalities." Doklady Math. 100.3 (2019): 554–557. [16] V.D. Stepanov, G.E. Shambilova. "On iterated and bilinear integral Hardy-type operators." Math. Inequal. Appl. 22.4 (2019): 1505–1533. [17] V.D. Stepanov, G.E. Shambilova. "On bilinear weighted inequalities with Volterra integral operators." Doklady Math. 100.3 (2019): 554–557. [18] V.D. Stepanov, G.E. Shambilova. "Multidimensional bilinear Hardy inequalities." Sib. Math. J. 61.4 (2020): 725-742. [19] V.D. Stepanov, G.E. Shambilova. "Bilinear weighted inequalities with two-dimensional operators." Doklady Math. 102.2 (2020): 406-408. [20] V.D. Stepanov, G.E. Shambilova. "On two-dimensional bilinear inequalities with rectangular Hardy operators in weighted Lebesgue spaces." Proc. Stekl. Inst. Math. 312 (2021): 241-248. [21] V.D. Stepanov, E.P. Ushakova. "Bilinear Hardy-type inequalities in weighted Lebesgue spaces." Nonlinear Stud . 26.4 (2019): 939–953. [22] V.D. Stepanov, E.P. Ushakova. "On weighted Hardy inequality with two-dimensional rectangular operator - extension of the E. Sawyer theorem." Math. Ineq. & Appl. 24.3 (2021): 617-634. [23] G. Tomaselli. "A class of inequalities." Boll. Unione Mat. Ital. 2 (1969): 622-631. [24] A. Wedestig. "Weighted inequalities for the Sawyer two-dimensional Hardy operator and its limiting geometric mean operator. " J. Inequal. Appl. 4 (2005): 387-394. 166