Home Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
Article Open Access

Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces

  • Aigerim Kalybay EMAIL logo
Published/Copyright: August 20, 2024
Download Article (PDF)
Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

In this study, we discuss new Hardy-type inequalities for operators involving iteration and provide explicit characterizations of these inequalities. As an application of our results, we consider the problem of the boundedness of the multidimensional Hardy operator from a Lebesgue space to a Morrey-type space.

Keywords: integral operator; Hardy-type inequality; weight function; Lebesgue space; Morrey-type space
MSC 2010: 26D10; 26D15; 47B38

1 Introduction

Let I = ( 0 , ∞ ) , 1 < p , q < ∞ , and 0 < θ < ∞ . Suppose u , v , and w are weight functions, i.e., positive measurable functions on I . Moreover, let ‖ f ‖ p , v = ∫ 0 ∞ v ( t ) ∣ f ( t ) ∣ p d t 1 p denote the norm of the Lebesgue space L p , v ( I ) .

In this paper, for f ≥ 0 , we consider the following Hardy-type inequalities:

(1) ‖ K + f ‖ q , u ≤ C ‖ f ‖ p , v ,

(2) ‖ K − f ‖ q , u ≤ C ‖ f ‖ p , v ,

where

K + f ( x ) = ∫ 0 x w ( t ) ∫ 0 t f ( s ) d s θ d t 1 θ ,

K − f ( x ) = ∫ x ∞ w ( t ) ∫ t ∞ f ( s ) d s θ d t 1 θ .

In 2013, in [1], it was proved that the boundedness of the multidimensional Hardy operator from a Lebesgue space to a Morrey-type space is equivalent to the validity of inequality (1). By the same method as in [1], one can prove that the boundedness of the conjugate multidimensional Hardy operator from a Lebesgue space to a complementary Morrey-type space is equivalent to the validity of (2). Motivated by this connection with Morrey-type spaces, inequalities (1) and (2) have been studied in many works over the last decade.

Hardy-type inequalities are usually studied separately for the cases p ≤ q and q < p . In [1], the study of inequality (1) is also separated, covering the case p ≤ q for any 0 < θ < ∞ , while the case q < p is covered only for 0 < θ ≤ q . In [2], both relations p ≤ q and q < p are characterized, but the obtained results are not explicit and depend on some auxiliary functions. In [3], all possible connections between p, q, and θ, including the case where 0 < q < 1, are investigated using the discretization method. In this study, using techniques different from those in [2] and [3], we find new explicit characterizations for the validity of inequalities (1) and (2) in the case where q < min { p , θ } , which was not covered in [1].

Note that there are a number of papers where the operators K ± are considered with certain kernels, but explicit characterizations for the case q < p are again given only for 0 < θ ≤ q (see the recent studies [4] and [5]).

The boundedness of the conjugate multidimensional Hardy operator from a Lebesgue space to a Morrey-type space can be reduced to the following inequality [6]:

(3) ‖ T + f ‖ q , u ≤ C ‖ f ‖ p , v ,

where

T + f ( x ) = ∫ 0 x w ( t ) ∫ t ∞ f ( s ) d s θ d t 1 θ ,

and the boundedness of the multidimensional Hardy operator from a Lebesgue space to a complementary Morrey-type space can be reduced to the following inequality:

(4) ‖ T − f ‖ q , u ≤ C ‖ f ‖ p , v ,

where

T − f ( x ) = ∫ x ∞ w ( t ) ∫ 0 t f ( s ) d s θ d t 1 θ .

These operators T + and T − have different types of integrals, while the operators K + and K − have the same type of integral. Therefore, the methods for characterizing inequalities (3) and (4) are not always applicable for characterizing inequalities (1) and (2). This difference has resulted in inequalities (3) and (4) being studied more comprehensively than inequalities (1) and (2), including cases where certain kernels are added next to f (e.g., [5,7–13] and references therein).

Note another motivation for studying inequalities (1)–(4). In [14], a bilinear version of the Hardy inequality was established. In [7], the result of [14] was re-proved using a simpler technique based on inequality (4). Thus, inequalities involving the operators K ± and T ± can be used to characterize bilinear Hardy inequalities (e.g., [5,13,15–17] and references therein).

The study is organized as follows: Section 2 contains the notations and a lemma required to prove the main results. In Section 3, we state and prove our main results concerning Hardy-type inequalities (1) and (2). In Section 4, we discuss an application of the main results to the theory of Morrey-type spaces.

2 Preliminaries

Let 1 p + 1 p ′ = 1 . Throughout the study, the symbol A ≪ B means that A ≤ c B for some constant c > 0 . The symbol A ≈ B means that A ≪ B and B ≪ A . Moreover, χ ( α , β ) ( ⋅ ) stands for the characteristic function of the interval ( α , β ) ⊂ R .

In order to prove the main result, we need the following lemma and remark from [18]:

Lemma 2.1

Let λ be a σ -finite measure on [ a , b ] , where a , b ∈ R ∪ { − ∞ , + ∞ } . For γ > 0 , the relations

(5) ∫ [ a , b ] f ( x ) d λ ( x ) γ + 1 ≪ ∫ [ a , b ] f ( t ) ∫ [ a , t ] f ( x ) d λ ( x ) γ d λ ( t ) ≪ ∫ [ a , b ] f ( x ) d λ ( x ) γ + 1 , f ≥ 0 ,

(6) ∫ [ a , b ] f ( x ) d λ ( x ) γ + 1 ≪ ∫ [ a , b ] f ( t ) ∫ [ t , b ] f ( x ) d λ ( x ) γ d λ ( t ) ≪ ∫ [ a , b ] f ( x ) d λ ( x ) γ + 1 , f ≥ 0 ,

hold. For γ ∈ ( − 1 , 0 ) and ∫ [ a , b ] f ( x ) d λ ( x ) < ∞ , relations (5) and (6) also hold.

Remark 2.1

If γ ∈ ( − 1 , 0 ) and ∫ [ a , b ] f ( x ) d λ ( x ) = ∞ , then the second inequalities in (5) and (6) obviously hold.

3 Main results

The main result concerning inequality (1) is as follows:

Theorem 3.1

Let 1 < q < min { p , θ } < ∞ . Then, inequality (1) holds if and only if

B + = ∫ 0 ∞ ∫ z ∞ u ( x ) ∫ z x w ( t ) d t q θ d x p p − q ∫ 0 z v 1 − p ′ ( s ) d s p ( q − 1 ) p − q v 1 − p ′ ( z ) d z p − q p q < ∞ .

Moreover, C ≈ B + , where C is the best constant in (1).

The main result concerning inequality (2) reads:

Theorem 3.2

Let 1 < q < min { p , θ } < ∞ . Then, inequality (2) holds if and only if

B − = ∫ 0 ∞ ∫ 0 z u ( x ) ∫ x z w ( t ) d t q θ d x p p − q ∫ z ∞ v 1 − p ′ ( s ) d s p ( q − 1 ) p − q v 1 − p ′ ( z ) d z p − q p q < ∞ .

Moreover, C ≈ B − , where C is the best constant in (2).

We will present the proof of Theorem 3.2 and omit the proof of Theorem 3.1, as they are similar.

Proof of Theorem 3.2

Sufficiency. Let B − < ∞ . Let 0 ≤ f be such that ∫ 0 ∞ f ( s ) d s < ∞ . Without loss of generality, we assume that ∫ 0 ∞ f ( s ) d s = 1 . For any k ∈ N , we define

t k = inf t ∈ I : ∫ t ∞ f ( s ) d s ≤ 2 − ( k − 1 ) .

It is obvious that 0 = t 1 < t 2 < … and I = ∑ k = 1 ∞ ( t k , t k + 1 ] . Moreover,

(7) ∫ t k ∞ f ( s ) d s = 2 − ( k − 1 ) , k = 1 , 2 , …

Therefore,

2 − ( k − 1 ) = ∫ t k t k + 1 f ( s ) d s + ∫ t k + 1 ∞ f ( s ) d s = ∫ t k t k + 1 f ( s ) d s + 2 − k ,

which implies that

(8) ∫ t k t k + 1 f ( s ) d s = 2 − ( k − 1 ) − 2 − k = 2 − k .

Replacing k with k + 1 in (8), we obtain

2 − ( k + 1 ) = ∫ t k + 1 t k + 2 f ( s ) d s .

Then, for the left-hand side J of (2), we obtain

J q = ∫ 0 ∞ u ( x ) ∫ x ∞ w ( t ) ∫ t ∞ f ( s ) d s θ d t q θ d x = ∑ k = 1 ∞ ∫ t k t k + 1 u ( x ) ∫ x ∞ w ( t ) ∫ t ∞ f ( s ) d s θ d t q θ d x .

Since x ≥ t k , for t m = t k , we have that x < t m + 1 , so J q can be rewritten as follows:

J q = ∑ k = 1 ∞ ∫ t k t k + 1 u ( x ) ∫ x t k + 1 w ( t ) ∫ t ∞ f ( s ) d s θ d t + ∑ m = k ∞ ∫ t m + 1 t m + 2 w ( t ) ∫ t ∞ f ( s ) d s θ d t q θ d x = ∑ k = 1 ∞ ∫ t k t k + 1 u ( x ) ∑ m = k ∞ ∫ max { x , t m } t m + 1 w ( t ) ∫ t ∞ f ( s ) d s θ d t q θ d x .

Since q θ < 1 and t ≥ t m , we find

(9) J q ≤ ∑ k = 1 ∞ ∫ t k t k + 1 u ( x ) ∑ m = k ∞ ∫ max { x , t m } t m + 1 w ( t ) d t q θ d x ∫ t m ∞ f ( s ) d s q = J q .

Changing the order of sums and taking into account (7), the latter gives that

(10) J q = ∑ m = 1 ∞ ∫ t m ∞ f ( s ) d s q ∑ k = 1 m ∫ t k t k + 1 u ( x ) ∫ max { x , t m } t m + 1 w ( t ) d t q θ d x = ∑ m = 1 ∞ 2 − q ( m − 1 ) ∑ k = 1 m − 1 ∫ t k t k + 1 u ( x ) ∫ max { x , t m } t m + 1 w ( t ) d t q θ d x + ∫ t m t m + 1 u ( x ) ∫ max { x , t m } t m + 1 w ( t ) d t q θ d x .

For the first term of (10), we have t k ≤ x ≤ t k + 1 ≤ t m for 1 ≤ k < m , while for the second term of (10), we have x ≥ t m . Hence, we deduce

(11) J q = ∑ m = 1 ∞ 2 − q ( m − 1 ) ∑ k = 1 m − 1 ∫ t k t k + 1 u ( x ) ∫ t m t m + 1 w ( t ) d t q θ d x + ∫ t m t m + 1 u ( x ) ∫ x t m + 1 w ( t ) d t q θ d x .

Thus, from (9) and (11), we obtain

(12) J q ≤ ∑ m = 1 ∞ 2 − q ( m − 1 ) ∑ k = 1 m ∫ t k t k + 1 u ( x ) ∫ x t m + 1 w ( t ) d t q θ d x = ∑ m = 1 ∞ 2 − q ( m − 1 ) ∫ 0 t m + 1 u ( x ) ∫ x t m + 1 w ( t ) d t q θ d x .

Taking into account that

2 − ( m − 1 ) = 2 − ( m + 1 ) ⋅ 4 = 4 ∫ t m + 1 t m + 2 f ( s ) d s

and using Hölder’s inequality twice, (12) implies that

(13) J q ≪ ∑ m = 1 ∞ ∫ t m + 1 t m + 2 f ( s ) d s q ∫ 0 t m + 1 u ( x ) ∫ x t m + 1 w ( t ) d t q θ d x ≤ ∑ m = 1 ∞ ∫ t m + 1 t m + 2 v ( s ) ∣ f ( s ) ∣ p d s q p ∫ t m + 1 t m + 2 v ( s ) 1 − p ′ d s q p ′ ∫ 0 t m + 1 u ( x ) ∫ x t m + 1 w ( t ) d t q θ d x ≤ ∑ m = 1 ∞ ∫ t m + 1 t m + 2 v ( s ) 1 − p ′ d s q ( p − 1 ) p − q ∫ 0 t m + 1 u ( x ) ∫ x t m + 1 w ( t ) d t q θ d x p p − q p − q p × ∑ m = 1 ∞ ∫ t m + 1 t m + 2 v ( s ) ∣ f ( s ) ∣ p d s q p ≪ ℬ ‖ f ‖ p , v q .

Using Lemma 2.1, we estimate ℬ as follows:

ℬ p p − q = ∑ m = 1 ∞ ∫ t m + 1 t m + 2 v ( s ) 1 − p ′ d s q ( p − 1 ) p − q ∫ 0 t m + 1 u ( x ) ∫ x t m + 1 w ( t ) d t q θ d x p p − q ≪ ∑ m = 1 ∞ ∫ t m + 1 t m + 2 v ( z ) 1 − p ′ ∫ z ∞ v ( s ) 1 − p ′ d s p ( q − 1 ) p − q d z ∫ 0 t m + 1 u ( x ) ∫ x t m + 1 w ( t ) d t q θ d x p p − q ≪ ∫ 0 ∞ ∫ 0 z u ( x ) ∫ x z w ( t ) d t q θ d x p p − q ∫ z ∞ v 1 − p ′ ( s ) d s p ( q − 1 ) p − q v 1 − p ′ ( z ) d z = ( B − ) p q p − q .

Hence, from (13) we finally obtain

J ≪ B − ‖ f ‖ p , v

and

(14) C ≪ B − ,

where C is the best constant in (2).

Necessity. Suppose (2) holds with the best constant C > 0 . Consider the test function:

f y , z ( t ) = χ ( y , z ) v 1 − p ′ ( t ) ∫ t z v 1 − p ′ ( x ) d x q − 1 p − q ∫ y t u ( s ) ∫ s t w ( τ ) d τ q θ d s 1 p − q .

Substituting the function f y , z ( t ) into the right-hand side of (2), we obtain

(15) ‖ f y , z ‖ p , v = ∫ 0 ∞ v ( t ) f y , z p ( t ) d t 1 p = ∫ y z v 1 − p ′ ( t ) ∫ t z v 1 − p ′ ( x ) d x p ( q − 1 ) p − q ∫ y t u ( s ) ∫ s t w ( τ ) d τ q θ d s p p − q d t 1 p = B < ∞ .

Next substituting the function f y , z ( t ) into the left-hand side of (2), we obtain

J q = ∫ 0 ∞ u ( x ) ∫ x ∞ w ( t ) ∫ t ∞ f y , z ( s ) d s θ d t q θ d x ≥ ∫ y z u ( x ) ∫ x z w ( t ) ∫ t z f y , z ( s ) d s θ d t q θ d x .

Using Lemma 2.1, the latter gives us

(16) J q ≫ ∫ y z u ( x ) ∫ x z w ( t ) ∫ t z f y , z ( τ ) ∫ τ z f y , z ( s ) d s θ − 1 d τ d t q θ d x .

Changing the order of integration, from (16) we find that

(17) J q ≫ ∫ y z u ( x ) ∫ x z f y , z ( τ ) ∫ τ z f y , z ( s ) d s θ − 1 ∫ x τ w ( t ) d t d τ q θ d x .

Using Lemma 2.1, changing the order of integration, and using Lemma 2.1 again, for μ ≤ τ from (17) we have

(18) J q ≫ ∫ y z u ( x ) ∫ x z f y , z ( μ ) ∫ μ z f y , z ( s ) d s θ − 1 ∫ x μ w ( t ) d t ∫ μ z f y , z ( τ ) ∫ τ z f y , z ( s ) d s θ − 1 ∫ x τ w ( t ) d t d τ q − θ θ d μ d x ≥ ∫ y z f y , z ( μ ) ∫ μ z f y , z ( s ) d s θ − 1 ∫ μ z f y , z ( τ ) ∫ τ z f y , z ( s ) d s θ − 1 d τ q − θ θ ∫ y μ u ( x ) ∫ x μ w ( t ) d t q θ d x d μ

(19) ≫ ∫ y z f y , z ( μ ) ∫ μ z f y , z ( s ) d s q − 1 ∫ y μ u ( x ) ∫ x μ w ( t ) d t q θ d x d μ .

By Lemma 2.1, it follows that

∫ μ z f y , z ( s ) d s = ∫ μ z v 1 − p ′ ( s ) ∫ s z v 1 − p ′ ( x ) d x q − 1 p − q ∫ y s u ( t ) ∫ t s w ( τ ) d τ q θ d t 1 p − q d s ≫ ∫ μ z v 1 − p ′ ( x ) d x p − 1 p − q ∫ y μ u ( t ) ∫ t μ w ( τ ) d τ q θ d t 1 p − q .

From (18), we obtain

(20) J q ≫ ∫ y z v 1 − p ′ ( μ ) ∫ μ z v 1 − p ′ ( x ) d x p ( q − 1 ) p − q ∫ y μ u ( t ) ∫ t μ w ( τ ) d τ q θ d t p p − q d μ = B p .

Taking into account the validity of (2), and using (15) and (20), we deduce that B p − q q ≪ C . Proceeding to limits y → 0 and z → ∞ , from the last estimate, we obtain

B − ≪ C ,

which, together with (14), implies that

B − ≈ C .

The proof of Theorem 3.2 is complete.□

4 Applications

Let B r be the open ball in R n centered at the origin with radius r , and C B r be the complement of the ball B r in R n . We consider the local Morrey-type space L M θ q , u ≡ L M θ q , u ( R n ) of all functions g ∈ L θ loc ( R n ) , for which

‖ g ‖ L M θ q , u = ∫ B r ∣ g ( x ) ∣ θ d x 1 θ L q , u ( I ) < ∞ .

We also consider the complementary local Morrey-type space L M θ q , u C ≡ L M θ q , u C ( R n ) of all functions g ∈ L θ loc ( R n ) such that

‖ g ‖ L M θ q , u C = ∫ C B r ∣ g ( x ) ∣ θ d x 1 θ L q , u ( I ) < ∞ .

The multidimensional Hardy operator is defined as follows:

H n , φ g ( x ) = φ ( ∣ x ∣ ) ∫ B ∣ x ∣ g ( y ) d y , x ∈ R n ,

where φ is a positive measurable function on I . The conjugate multidimensional Hardy operator is defined as follows:

H ˜ n , φ g ( x ) = φ ( ∣ x ∣ ) ∫ C B ∣ x ∣ g ( y ) d y , x ∈ R n .

The problem of the boundedness of the multidimensional Hardy operator from a Lebesgue space L p , V ≡ L p , V ( R n ) to a local Morrey-type space L M θ q , u ( R n ) is equivalent to the validity of the following inequality:

(21) ‖ H n , φ g ‖ L M θ q , u ≤ C ˜ ‖ g ‖ L p , V ,

where V is a positive measurable function on R n . Moreover, the problem of the boundedness of the conjugate multidimensional Hardy operator from a Lebesgue space L p , V ( R n ) to a complementary local Morrey-type space L M θ q , u C ( R n ) is equivalent to the validity of the following inequality:

(22) ‖ H ˜ n , φ g ‖ L M θ q , u C ≤ C ˜ ‖ g ‖ L p , V .

In [1], assuming that V ( x ) ≡ v ˜ ( ∣ x ∣ ) , where v ˜ is a positive measurable function on I , it was proved that the validity of inequalities (21) and (22) for all non-negative functions g ∈ L p , V ( R n ) is equivalent to the validity of inequalities (1) and (2) for all non-negative functions f ∈ L p , v ( I ) , respectively. Here, w ( t ) = φ θ ( t ) t n − 1 , v ( t ) = v ˜ ( t ) t ( n − 1 ) ( 1 − p ) , and C = C ˜ σ n − ( 1 p ′ + 1 q ) , where σ n is the surface area of the unit sphere S n − 1 in R n .

Therefore, replacing w ( t ) by φ θ ( t ) t n − 1 and v 1 − p ′ ( t ) by v ˜ 1 − p ′ ( t ) t n − 1 in the expression B + , from Theorem 3.1 we deduce:

Theorem 4.1

Let 1 < q < min { p , θ } < ∞ . Then, the operator H n , φ is bounded from L p , V to L M θ q , u if and only if B + < ∞ . Moreover, ‖ H n , φ ‖ L p , V → L M θ q , u ≈ B + .

Similarly, replacing w ( t ) by φ θ ( t ) t n − 1 and v 1 − p ′ ( t ) by v ˜ 1 − p ′ ( t ) t n − 1 in the expression B − , from Theorem 3.2, we obtain

Theorem 4.2

Let 1 < q < min { p , θ } < ∞ . Then, the operator H ˜ n , φ is bounded from L p , V to L M θ q , u C if and only if B − < ∞ . Moreover, ‖ H ˜ n , φ ‖ L p , V → L M θ q , u C ≈ B − .

Acknowledgements

The author would like to thank the anonymous referees for their valuable comments and suggestions, which have helped improve the manuscript.

  1. Funding information: This work was supported by the Ministry of Science and Higher Education of the Republic of Kazakhstan, Grant no. AP19677836.

  2. Author contribution: The author confirms the sole responsibility for the conception of the study, presented results, and manuscript preparation.

  3. Conflict of interest: The author states no conflict of interest.

References

[1] V. I. Burenkov and R. Oinarov, Necessary and sufficient conditions for boundedness of the Hardy-type operator from a weighted Lebesgue space to a Morrey-type space, Math. Inequal. Appl. 16 (2013), no. 1, 1–19, DOI: https://dx.doi.org/10.7153/mia-16-01. Search in Google Scholar

[2] D. V. Prokhorov and V. D. Stepanov, On weighted Hardy inequalities in mixed norms, Proc. Steklov Inst. Math. 283 (2013), 149–164, DOI: https://doi.org/10.1134/S0081543813080117. Search in Google Scholar

[3] A. Gogatishvili and T. Ünver, New characterization of weighted inequalities involving superposition of Hardy integral operators, Math. Nachr. (2024), DOI: https://doi.org/10.1002/mana.202400007. Search in Google Scholar

[4] A. Kalybay and R. Oinarov, On weighted inequalities for a class of quasilinear integral operators, Banach J. Math. Anal. 17 (2023), no. 1, 3, DOI: https://doi.org/10.1007/s43037-022-00226-1. Search in Google Scholar

[5] V. D. Stepanov and G. E. Shambilova, On iterated and bilinear integral Hardy-type operators, Math. Inequal. Appl. 22 (2019), no. 4, 1505–1533, DOI: https://doi.org/10.7153/mia-2019-22-105. Search in Google Scholar

[6] R. Oinarov and A. Kalybay, Weighted inequalities for a class of semiadditive operators, Ann. Funct. Anal. 6 (2015), no. 4, 155–171, DOI: https://doi.org/10.15352/afa/06-4-155. Search in Google Scholar

[7] A. L. Bernardis and P. Ortega Salvador, Some new iterated Hardy-type inequalities and applications, J. Math. Inequal. 11 (2017), no. 2, 577–594, DOI: https://doi.org/10.7153/jmi-11-47. Search in Google Scholar

[8] A. Gogatishvili, Z. Mihula, L. Pick, H. Turčinová, and T. Ünver, Weighted inequalities for a superposition of the Copson operator and the Hardy operator, J. Fourier Anal. Appl. 28 (2022), no. 2, 24, DOI: https://doi.org/10.1007/s00041-022-09918-6. Search in Google Scholar

[9] A. Gogatishvili and R. Mustafayev, Weighted iterated Hardy-type inequalities, Math. Inequal. Appl. 20 (2017), no. 3, 683–728, DOI: https://doi.org/10.7153/mia-20-45. Search in Google Scholar

[10] A. Gogatishvili, R. Mustafayev, and L.-E. Persson, Some new iterated Hardy-type inequalities, J. Funct. Spaces Appl. 2012 (2012), 734194, DOI: http://doi.org/10.1155/2012/734194. Search in Google Scholar

[11] A. Gogatishvili, R. Mustafayev, and L.-E. Persson, Some new iterated Hardy-type inequalities: The case θ=1, J. Inequal. Appl. 2013 (2013), no. 1, 515, DOI: https://doi.org/10.1186/1029-242X-2013-515. Search in Google Scholar

[12] A. Kalybay, Weighted estimates for a class of quasilinear integral operators, Sib. Math. J. 60 (2019), 291–303, DOI: https://doi.org/10.1134/S0037446619020095. Search in Google Scholar

[13] D. V. Prokhorov, On a class of weighted inequalities containing quasilinear operators, Proc. Steklov Inst. Math. 293 (2016), 272–287, DOI: https://doi.org/10.1134/S0081543816040192. Search in Google Scholar

[14] M. I. Aguilar Cannnestro, P. Ortega Salvador, and C. Ramírez Torreblanca, Weighted bilinear Hardy inequalities, J. Math. Anal. Appl. 387 (2012), no. 1, 320–334, DOI: https://doi.org/10.1016/j.jmaa.2011.08.078. Search in Google Scholar

[15] V. García García and P. Ortega Salvador, Weighted weak-type iterated and bilinear Hardy inequalities, J. Math. Anal. Appl. 525 (2023), no. 2, 127284, DOI: https://doi.org/10.1016/j.jmaa.2023.127284. Search in Google Scholar

[16] A. Gogatishvili, P. Jain, and S. Kanjilal, On bilinear Hardy inequality and corresponding geometric mean inequality, Ricerche Mat. 71 (2022), no. 2, 581–608, DOI: https://doi.org/10.1007/s11587-020-00536-2. Search in Google Scholar

[17] P. Jain, S. Kanjilal, V. D. Stepanov, and E. P. Ushakova, Bilinear Hardy-Steklov operators, Math. Notes 104 (2018), no. 6, 823–832, DOI: https://doi.org/10.1134/S0001434618110275. Search in Google Scholar

[18] D. V. Prokhorov, Hardy’s inequality with three measures, Proc. Steklov Inst. Math. 255 (2006), 221–233, DOI: https://doi.org/10.1134/S0081543806040183. Search in Google Scholar
Received: 2023-12-21
Revised: 2024-07-16
Accepted: 2024-07-24
Published Online: 2024-08-20

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Special Issue on Contemporary Developments in Graph Topological Indices
  2. On the maximum atom-bond sum-connectivity index of graphs
  3. Upper bounds for the global cyclicity index
  4. Zagreb connection indices on polyomino chains and random polyomino chains
  5. On the multiplicative sum Zagreb index of molecular graphs
  6. The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two
  7. Special Issue on Convex Analysis and Applications - Part I
  8. Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function
  9. Scattering threshold for the focusing energy-critical generalized Hartree equation
  10. (p, q)-Compactness in spaces of holomorphic mappings
  11. Characterizations of minimal elements of upper support with applications in minimizing DC functions
  12. Some new Hermite-Hadamard-type inequalities for strongly h-convex functions on co-ordinates
  13. Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void
  14. Extension of Fejér's inequality to the class of sub-biharmonic functions
  15. On sup- and inf-attaining functionals
  16. Regularization method and a posteriori error estimates for the two membranes problem
  17. Rapid Communication
  18. Note on quasivarieties generated by finite pointed abelian groups
  19. Review Articles
  20. Amitsur's theorem, semicentral idempotents, and additively idempotent semirings
  21. A comprehensive review of the recent numerical methods for solving FPDEs
  22. On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating
  23. Pullback and uniform exponential attractors for non-autonomous Oregonator systems
  24. Regular Articles
  25. On certain functional equation related to derivations
  26. The product of a quartic and a sextic number cannot be octic
  27. Combined system of additive functional equations in Banach algebras
  28. Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices
  29. Local and global solvability for the Boussinesq system in Besov spaces
  30. Construction of 4 x 4 symmetric stochastic matrices with given spectra
  31. A conjecture of Mallows and Sloane with the universal denominator of Hilbert series
  32. The uniqueness of expression for generalized quadratic matrices
  33. On the generalized exponential sums and their fourth power mean
  34. Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions
  35. Computing the determinant of a signed graph
  36. Two results on the value distribution of meromorphic functions
  37. Zariski topology on the secondary-like spectrum of a module
  38. On deferred f-statistical convergence for double sequences
  39. About j-Noetherian rings
  40. Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model
  41. On the distribution of powered numbers
  42. Almost periodic dynamics for a delayed differential neoclassical growth model with discontinuous control strategy
  43. A new distributionally robust reward-risk model for portfolio optimization
  44. Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results
  45. Silting modules over a class of Morita rings
  46. Non-oscillation of linear differential equations with coefficients containing powers of natural logarithm
  47. Mutually unbiased bases via complex projective trigonometry
  48. Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three
  49. On second-order linear Stieltjes differential equations with non-constant coefficients
  50. Complex dynamics of a nonlinear discrete predator-prey system with Allee effect
  51. The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains
  52. On discrete inequalities for some classes of sequences
  53. Boundary value problems for integro-differential and singular higher-order differential equations
  54. Existence and properties of soliton solution for the quasilinear Schrödinger system
  55. Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension
  56. Endpoint boundedness of toroidal pseudo-differential operators
  57. Matrix stretching
  58. A singular perturbation result for a class of periodic-parabolic BVPs
  59. On Laguerre-Sobolev matrix orthogonal polynomials
  60. Pullback attractors for fractional lattice systems with delays in weighted space
  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
https://doi.org/10.1515/math-2024-0046
Keywords for this article
integral operator; Hardy-type inequality; weight function; Lebesgue space; Morrey-type space
Creative Commons
BY 4.0

Articles in the same Issue

  1. Special Issue on Contemporary Developments in Graph Topological Indices
  2. On the maximum atom-bond sum-connectivity index of graphs
  3. Upper bounds for the global cyclicity index
  4. Zagreb connection indices on polyomino chains and random polyomino chains
  5. On the multiplicative sum Zagreb index of molecular graphs
  6. The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two
  7. Special Issue on Convex Analysis and Applications - Part I
  8. Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function
  9. Scattering threshold for the focusing energy-critical generalized Hartree equation
  10. (p, q)-Compactness in spaces of holomorphic mappings
  11. Characterizations of minimal elements of upper support with applications in minimizing DC functions
  12. Some new Hermite-Hadamard-type inequalities for strongly h-convex functions on co-ordinates
  13. Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void
  14. Extension of Fejér's inequality to the class of sub-biharmonic functions
  15. On sup- and inf-attaining functionals
  16. Regularization method and a posteriori error estimates for the two membranes problem
  17. Rapid Communication
  18. Note on quasivarieties generated by finite pointed abelian groups
  19. Review Articles
  20. Amitsur's theorem, semicentral idempotents, and additively idempotent semirings
  21. A comprehensive review of the recent numerical methods for solving FPDEs
  22. On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating
  23. Pullback and uniform exponential attractors for non-autonomous Oregonator systems
  24. Regular Articles
  25. On certain functional equation related to derivations
  26. The product of a quartic and a sextic number cannot be octic
  27. Combined system of additive functional equations in Banach algebras
  28. Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices
  29. Local and global solvability for the Boussinesq system in Besov spaces
  30. Construction of 4 x 4 symmetric stochastic matrices with given spectra
  31. A conjecture of Mallows and Sloane with the universal denominator of Hilbert series
  32. The uniqueness of expression for generalized quadratic matrices
  33. On the generalized exponential sums and their fourth power mean
  34. Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions
  35. Computing the determinant of a signed graph
  36. Two results on the value distribution of meromorphic functions
  37. Zariski topology on the secondary-like spectrum of a module
  38. On deferred f-statistical convergence for double sequences
  39. About j-Noetherian rings
  40. Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model
  41. On the distribution of powered numbers
  42. Almost periodic dynamics for a delayed differential neoclassical growth model with discontinuous control strategy
  43. A new distributionally robust reward-risk model for portfolio optimization
  44. Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results
  45. Silting modules over a class of Morita rings
  46. Non-oscillation of linear differential equations with coefficients containing powers of natural logarithm
  47. Mutually unbiased bases via complex projective trigonometry
  48. Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three
  49. On second-order linear Stieltjes differential equations with non-constant coefficients
  50. Complex dynamics of a nonlinear discrete predator-prey system with Allee effect
  51. The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains
  52. On discrete inequalities for some classes of sequences
  53. Boundary value problems for integro-differential and singular higher-order differential equations
  54. Existence and properties of soliton solution for the quasilinear Schrödinger system
  55. Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension
  56. Endpoint boundedness of toroidal pseudo-differential operators
  57. Matrix stretching
  58. A singular perturbation result for a class of periodic-parabolic BVPs
  59. On Laguerre-Sobolev matrix orthogonal polynomials
  60. Pullback attractors for fractional lattice systems with delays in weighted space
  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 8.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/math-2024-0046/html
Scroll to top button