Two-Dimensional Hardy Operators in Lebesgue Spaces

Abstract

Characterizations of linear and bilinear Lebesgue norm inequalities involving twodimensional Hardy integral operators are obtained. Hardy integral operator, weighted Lebesgue space, bilinear inequality. the subset of all nonnegative f. If ∈ Ϻ+ and 0 < ≤ ∞ we define the weighted Lebesgue space Let Ϻ be the set of all Lebesgue measurable functions f on ℝ2+ ≔ (0, ∞)2, and let Ϻ+ ⊂ Ϻ be VI International Conference Information Technologies and High-Performance Computing (ITHPC-2021),

1. Introduction

0 < < ∞, ∞(ℝ2) = { ∈ Ϻ: ‖ ‖∞, ≔ ess sup ∈ℝ2 ( )| ( )| < ∞}, = ∞. rectangular Hardy operator Let ∈ ℕ, 0 < ≤ ∞ and 1 ≤ ≤ ∞, , ∈ Ϻ+ for all = 1, … . Define the two-dimensional and consider the following multilinear inequality 2 ( , ) ≔ ∫ ∫ ( , ) ‖( 2 1)· … · ( 2 )‖ , ≤ ‖ 1‖ 1, 1 … ‖ ‖ , , where a constant C>0 is independent of , = 1, … , , and is supposed to be the least possible. The general problem is to characterize this inequality (2) by establishing a two-sided estimate ( 1, … , ; 1, … , ) ≤ ≤ ( 1, … , ; 1, … , ) depending on given weights 1, … , , and fixed parameters 1, … , , only. with some irrelevant constants and β by a functional ( 1, … , ; 1, … , )of an explicit form 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 ( , ) ∈ ℝ2+, ∈ Ϻ+, (1) (2)

2020 Copyright for this paper by its authors. 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 ≪ if ≤ with some constant

> 0 depending, possibly, on irrelevant parameters only.

Relations of the type ≈ mean F ≪ G ≪ or = .

2. Two-dimensional Hardy inequality

Weighted Hardy inequality ‖ 2 ‖ , ≤ ‖ ‖ , , ∈ 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 < ≤ < ∞. Denote ′ ≔ /( − 1) and let ( 2∗ )( , ) ≔ be the adjoint to 2 operator. The inequality ( 3 ) holds if and only if 0 0 ∞

∞ ∫ ( 2∗ 1 ≔ sup( , )∈ℝ2+[ 2∗

( , )]1/ [ 2 1− ′( , )]1/ ′ < ∞, 2 ≔ sup( , )∈ℝ2+ (∫ ∫ ( 2 1− ′) )

[ 2 1− ′( , )]−1/ < ∞, 3 ≔ sup( , )∈ℝ2+ (∫ ) ′ 1− ′) [ 2∗

( , )]−1/ ′ < ∞. 1/

Moreover, it holds for the least possible constant C>0 in ( 3 ) that ≈ 1 + 2 + 3 with equivalence 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 1 ≈ 2 ≈ 3 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 p<q the inequality ( 3 ) is characterized by only one Muckenhoupt functional ≔ 1 of the form ( 4 ). 24 3 Theorem [22, Theorem 2]. Let 1 < < < ∞. Denote ≔ ( , ) ≔ ℂ , ′ ≔ 33 [ max{ , 2 ( ′) / ′} (2 −1 − 1 ) + 31/ +1/ ′( ′)1/ ′].

The inequality ( 3 ) holds if and only if < ∞. Besides, ≤ ≤ ℂ , ′ .

The results of [12, Theorems 1A] and [22, Theorem 2] are valid for any type of weights v and w. It was established in [24] that if one of the two weights v or w is factorizable, that is if 2 −1 /

− 2( −1), ′ ≔ ( ′, ′) and or ( 1, 2) = 1( 1) 2( 2) ( 1, 2) = 1( 1) 2( 2), then it is possible to characterize ( 3 ) by only one functional for 1 < ≤ < ∞. This result was extended to all p,q>1 and generalized to all the types of boundedness constants in [11]. Denote ( )≔ ∫0 1− ′ , = 1,2. Then the inequality ( 3 ) holds for all ≥ 0 if and only if Theorem [11, Theorems 2.1, 2.2]. Let 1 < ≤ < ∞ and the weight v satisfy the condition ( 7 ). or if and only if ≔ sup( , )∈ℝ2+[ 2∗

( , )]1/ [ 1( ) 2( )]1/ ′ < ∞,

1/ ≔ sup( , )∈ℝ2+ (∫0 ∫0 [ 1 2 ] ) [ 1( ) 2( )]−1/ < ∞.

Denote ( )≔ ∫

∞ or if and only if constants depending of p and q only.

Besides, it holds for the least possible constant C>0 in ( 3 ) that ≈ ≈ with equivalence Theorem [11, Theorems 2.4, 2.5]. Let 1 < ≤ < ∞ and the weight w satisfy the condition ( 8 ).

, = 1,2. Then the inequality ( 3 ) holds for all ≥ 0 if and only if ∗ ≔ sup( , )∈ℝ2+[ 2 1− ′( , )]1/ ′[ 1( ) 2( )]1/ < ∞, ∗ ≔ sup( , )∈ℝ2+ (∫ ∞ ∞

Besides, ≈ ∗ ≈ ∗ 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 < < < ∞. 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: ≔ (∫[ 2∗ ( , )] / [ 1( ) 2( )] / ′ 11− ′( ) 21− ′ ( ) ) , < ∞, or if and only if < ∞. Moreover,

( 3 ) ≈ is

valid ≈ .

for all ∈ Ϻ+ if and only if ∗ < ∞, or if and only if ∗ < ∞. Moreover, ≈ ∗ ≈ ∗ .

Theorem [11, Theorems 3.3, 3.4]. Let 1 < < < ∞ . Assume that the weight function w in ( 3 ) satisfies the condition ( 8 ) and 1(0)= 2(0)= ∞. Then the inequality ( 3 ) is valid for all

/ ′ ≔ (∫ (∫ ∫ [ 1 2 ] ) [ 1( ) 2( )]− / 11− ′( ) 21− ′ ( ) ) , ∗ ≔ (∫[ 2 1− ′( , )] / ′[ 1( ) 2( )] / 1( ) 2( ) ) , ∗ ≔ (∫ (∫ ] ′ 1− ′)

[ 1( ) 2( )]− / ′ 1( ) 2( ) ) . 0 0 ∞ ∞ 1/ 1/ 1/ 1/

3. Bilinear two-dimensional Hardy inequality

In this section we demonstrate some of the new characteristics from [20] obtained for the

The required characteristics for (I), (II) and (III) are given in the assertions below.

Theorem [20, Theorem 4]. Let , , ∈ ( ). 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 ≈ ≔ sup( , )∈ℝ2+( 1( , )+ 2( , )+ 3( , ))[ 1( ) 2( )]1/ ′, ( 10 ) where ( )≔ ∫0 1− ′, = 1,2, as before and 3( , )≔ sup( , )∈ℝ2+ (∫ 0 0 1( , )≔ sup( , )∈ℝ2+[ 2∗( ( ,∞)×( ,∞))( , )] [ 2 1− ′( , )]1/ ′, 2( , )≔ sup( , )∈ℝ2+ (∫ ∫ ( 2 1− ′) ( ,∞)×( ,∞)) [ 2 1− ′( , )]−1/ , 1/ 1/ ′

1/ ∞ ∞

′ ∫ ( 2∗( ( ,∞)×( ,∞))) 1− ′) [ 2∗( ( ,∞)×( ,∞))( , )] Remark [20, Remark 3]. If the weight u in ( 9 ) is also of product type, that is if

( 1, 2)= 1( 1) ( 2), then the expression for the functional in ( 10 ) simplifies as follows:

≔ sup( , )∈ℝ2+[ 2∗ ( , )]1/ [ 1( ) 2( )]1/ ′[ 1( ) 2( )]1/ ′ < ∞, where ( )≔ ∫0 1− ′ and ( )≔ ∫0

1− ′, = 1,2. ≤ < < ∞, under the condition (∞)= ∞, = 1,2, Theorem [20, Theorem 5]. Let , ,

∈ ( ). 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 ≈ , where for 1 < 1 < max{ , } ≤ < ∞, 1 < min{ , } ≤ < max{ , } < ∞, 1 < < min{ , }.

−1/ ′.

( 11 ) ≔ sup( , )∈ℝ2+ (∫ ≔ sup( , )∈ℝ2+ (∫ ∞ ∞

∫ [ 2∗ ∞

∞ where 1/r:=1/q-1/p and 1/t=1/q-1/s. and for 1 < ≤ < < ∞, under the condition (∞)= ∞, = 1,2, 1/ 1/ ∫ [ 2∗ ] / [ 1 2 ] / ′ 11− ′ 21− ′)

[ 1( ) 2( )]1/ ′, ] / [ 1 2 ] / ′ 11− ′ 21− ′) Theorem [20, Theorem 6]. Let , , ∈ (

). 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 (∞)= ∞, = 1,2, and (∞)= ∞, = 1,2, it holds ≈ ∑4=1 ( ), where for 1/ ≤ 1/ + 1/ (1)≔ sup( , )∈ℝ2+ (∫ (2)≔ sup( , )∈ℝ2+ (∫ ( 3 )≔ sup( , )∈ℝ2+ (∫ [ 1 ( 4 )≔ sup( , )∈ℝ2+ (∫ [ 1 ∞ ∞ and for 1/ > 1/ + 1/ with 1/ ≔ 1/ − 1/ − 1/ [ 1( ) 2( )] / ′ 1( ) 2( )) , (2)≔ (∫ ∫ (∫ ∫ [ 1 2 ] [ 1 2 ] ′ 1 2) [ 1( ) 2( )] / ′ 1( ) 2( )) , [ ( 3 )] ≔ ∫ ∫ (∫ [ 1 ] [ 1 ] ′ 1) (∫ [ 2 ] [ 2 ] ′ 2) [ 1( )] / ′ × [ 2( )] / ′[ 2( )] [ 2( )] ′ 1( ) 2( ), [ ( 4 )] ≔ ∫ ∫ (∫ [ 1 ] [ 1 ] ′ 1) (∫ [ 2 ] [ 2 ] ′ 2) [ 1( )] / ′ × [ 2( )] / ′[ 1( )] [ 1( )] ′ 1( ) 2( ), where 1/r:=1/q-1/p, 1/t:=1/q-1/s, ( )≔ ∫0 1− ′, ( )≔ ∫0 1− ′, ( )≔ ∫ ∞ , = 1,2.

For some other types of bilinear inequalities with Hardy type operators one can consult [13-19, 1/ 1/

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.

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