Volume 2022, Issue 1 4845507

Research Article

Open Access

Boundedness of Multilinear Calderón-Zygmund Operators on Grand Variable Herz Spaces

Hammad Nafis,

Hammad Nafis

Department of Mathematics and Statistics, Riphah International University, I-14 Islamabad, Pakistan riphah.edu.pk

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Humberto Rafeiro,

Corresponding Author

Humberto Rafeiro

[email protected]

orcid.org/0000-0002-5304-8572

United Arab Emirates University, College of Sciences, Department of Mathematical Sciences, P.O. Box 15551, Al Ain, Abu Dhabi, UAE uaeu.ac.ae

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Muhammad Asad Zaighum,

Muhammad Asad Zaighum

Department of Mathematics and Statistics, Riphah International University, I-14 Islamabad, Pakistan riphah.edu.pk

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Hammad Nafis,

Hammad Nafis

Department of Mathematics and Statistics, Riphah International University, I-14 Islamabad, Pakistan riphah.edu.pk

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Humberto Rafeiro,

Corresponding Author

Humberto Rafeiro

[email protected]

orcid.org/0000-0002-5304-8572

United Arab Emirates University, College of Sciences, Department of Mathematical Sciences, P.O. Box 15551, Al Ain, Abu Dhabi, UAE uaeu.ac.ae

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Muhammad Asad Zaighum,

Muhammad Asad Zaighum

Department of Mathematics and Statistics, Riphah International University, I-14 Islamabad, Pakistan riphah.edu.pk

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First published: 28 March 2022

https://doi.org/10.1155/2022/4845507

Citations: 3

Academic Editor: Douadi Drihem

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Abstract

In this paper, we prove the boundedness of multilinear Calderón-Zygmund operators on product of grand variable Herz spaces. These results generalize the boundedness of multilinear Calderón-Zygmund operators on product of variable exponent Lebesgue spaces and variable Herz spaces.

1. Introduction

There has been increased interest in the study of multilinear singular integral operators in recent years. The class of multilinear singular integrals with standard Calderón-Zygmund kernels provides the foundation and starting point of research of the theory. Such a class of multilinear Calderón-Zygmund operators was introduced and first studied by Coifman and Meyer [1–3] and later by Grafakos and Torres [4]. For the boundedness and other properties of multilinear fractional integrals, we refer to, e.g., [5–8].

Variable Lebesgue spaces were introduced in [9], but stayed under the radar for a considerable amount of time. Apart from some previous sporadic episodes, the research boom on such spaces can be traced back to the foundational paper [10]. Since then, these spaces have attracted much attention of mathematicians, not only because of their connection with harmonic analysis but also due to their usefulness in application to a wide range of problems, see, e.g., [11]. The standard references to the general theory of variable Lebesgue spaces are [12, 13].

The classical definition of Herz spaces was introduced in [14]. Many studies can be found related to these spaces and its variations, which include variable Herz spaces, continual Herz spaces, and Herz spaces with variable smoothness and integrability. For details, see [15–21] and references therein.

Grand Lebesgue spaces on bounded sets, which proved to be useful in application to partial differential equations, were introduced in [22, 23]. In the last years, various operators of harmonic analysis have been intensively studied on grand spaces, see for instance [20, 24–32]. Grand Lebesgue sequence spaces were introduced recently in [33], where several operators of harmonic analysis were studied, e.g., maximal, convolutions, Hardy, Hilbert, and fractional operators.

In this paper, we prove the boundedness of multilinear Calderón-Zygmund operators on grand variable Herz spaces which were introduced in [34]. The present paper is organized in the following way. Apart from the introduction, in Section 2, we recall some definitions and results related to variable exponent spaces. Section 3 contains some details about multilinear Calderón-Zygmund kernels and the proof of the main result.

Notations.

(i)
ℕ is the set of natural numbers and ℕ₀≔ℕ ∪ {0}
(ii)
ℤ is the set of integers
(iii)
ℤ₋ is the set of negative integers
(iv)
for n, m ∈ ℤ and n < m
(v)
B(x, r) is the ball of radius r center at the point x
(vi)
B_k≔{x ∈ ℝⁿ : |x| ≤ 2^k} for all k ∈ ℤ
(vii)
R_t,τ≔B_τ\B_t = {x : t<∣x∣<τ} is a spherical layer
(viii)
R_k≔B_k\B_k−1
(ix)
(x)
f≲g means that f ≤ Cg and f≃g means that f≲g≲f
(xi)
constants (often different constant in the same chain of inequalities) will mainly be denoted by c or C

2. Function Spaces with Variable Exponent

In this section, we recall definitions and results related to variable exponent Lebesgue spaces, variable Herz spaces, and grand variable Herz spaces.

2.1. Lebesgue Space with Variable Exponent

For the current section, we refer to [10–13, 35] unless and until stated otherwise. Let q be a real-valued measurable function on ℝⁿ with values in [1, ∞). For X ⊂ ℝⁿ, we suppose that

(1)

where q⁻(X)≔essinf_x∈Xq(x) and q⁺(X)≔esssup_x∈Xq(x). By L^q(·)(ℝⁿ), we denote the space of measurable function f on ℝⁿ such that

(2)

It is a Banach space, see [13, 35], endowed with norm:

(3)

By q^′, we denote the conjugate exponent of q, defined by q^′(x) = q(x)/(q(x) − 1). In the sequel, we use

log condition:

(4)

where A = A(q) > 0 does not depend on x, y;

decay condition at 0:

(5)

holds for some q(0) ∈ (1, ∞); and

decay condition at ∞: there exists a number q(∞) ∈ (1, ∞), such that

(6)

where A = A(q) does not depend on x.

Given a function

, the Hardy-Littlewood maximal operator M is defined by

(7)

We adopt the following notations:

(i)
(ii)
consists of all measurable functions q satisfying q⁻ > 1 and q⁺ < ∞
(iii)
and denote the classes of which satisfy (5) and (6), respectively
(iv)
is the set of all for which M is bounded on L^q(·)(ℝⁿ)

For the following lemma, we refer to, e.g., [12].

Lemma 1 (Generalized Hölder’s Inequality). Given , define by 1/r(x) = 1/p(x) + 1/q(x). Then, there exists a constant c such that for all f ∈ L^p(·)(ℝⁿ) and g ∈ L^q(·)(ℝⁿ), fg ∈ L^r(·)(ℝⁿ) and

(8)

The following lemma appears in [21].

Lemma 2. Let D > 1 and Then

(9)

(10)

where c₀ ≥ 1 and c_∞ ≥ 1 depend on D, but do not depend on r.

2.2. Herz Spaces with Variable Exponent

The classical Herz spaces were first introduced in [14]. We recall the definition of variable exponent Herz spaces.

Definition 3. Let 1 < p < ∞, α ∈ ℝ, and . The homogeneous Herz space is defined by

(11)

where

(12)

The nonhomogeneous Herz space is defined by

(13)

where

(14)

For the boundedness of integral operators on Herz type spaces, we refer to, e.g., [18, 19, 36].

2.3. Grand Lebesgue Sequence Space

In this section, we recall the definition of grand Lebesgue sequence space. For the following definition and statements, see [33]. In what follows, stands for one of the sets ℤⁿ, ℤ, ℕ, and ℕ₀.

Definition 4. Let 1 ≤ p < ∞ and θ > 0. The grand Lebesgue sequence space l^p),θ is defined by the norm

(15)

2.4. Grand Variable Herz Space

Following [34], we now introduce the grand variable Herz spaces.

Definition 5. Let α ∈ L^∞(ℝ), 1 ≤ p < ∞, q : ℝⁿ⟶[1, ∞), and θ > 0. The homogeneous grand variable Herz space is defined by

(16)

where

(17)

The following lemma, see [15], is helpful to estimate the norm of characteristics functions.

Lemma 6. Let and let R = B(0, r)\B(0, (r/2)). If |R| ≤ 2⁻ⁿ, then

(18)

with the implicit constants independent of r and x ∈ R.

The left-hand side equivalence remains true for every |R| > 0 if we assume, additionally, that

3. Boundedness of Multilinear Calderón-Zygmund Operators

Consider the multilinear operator T of the form

(19)

where

and

, the space of compactly supported functions. Let K(x, y₁, ⋯, y_m) be a locally integrable function defined away from the diagonal

, which satisfies the size estimate

(20)

for some c > 0 and all

with x ≠ y_j for some j. For smoothness, assume that for some ε > 0,

(21)

provided that |x − x^′| ≤ 1/2max{|x − y₁|, ⋯, |x − y_m|} and

(22)

whenever

for all

Such kernels are called Calderón-Zygmund kernels and the class of all functions satisfying (20), (21), and (22) with parameters m, c, and ε will be denoted by m-CZK(c, ε), compare [4]. We say that T be as in (19) is an m-linear Calderón-Zygmund operator, if

(i)
The related kernel belongs to m-CZK(c, ε) class
(ii)
T is bounded from to L^q for some 1 < q₁, q₂, ⋯, q_m < ∞, and 1/q = 1/q₁ + ⋯+1/q_m

Grafakos and Torres [4] proved the boundedness of T from to L^q,∞ for some 1 < q₁, q₂, ⋯, q_m < ∞, and 1/q = 1/q₁ + ⋯+1/q_m, and from L¹ × ⋯×L¹ to L^1/m,∞.

The boundedness of the multilinear Calderón-Zygmund operator T on variable exponent Lebesgue spaces was proved in [37], as stated below.

Lemma 7. Let , , with 1/q(x) = 1/q₁(x) + ⋯+1/q_m(x), and for some 0 < s < q⁻. Then, the m-linear Calderón-Zygmund operator T is bounded on the product of variable exponent Lebesgue spaces. Moreover,

(23)

with the constant C independent of

We now state and prove the boundedness of multilinear Calderón-Zygmund operator on grand variable Herz spaces.

Theorem 8. Let , , such that q_i(0) ≤ q_i(∞), 1/q(x) = 1/q₁(x) + ⋯+1/q_m(x), , and for some 0 < s < q⁻. Let θ > 0, 1 < p_i < ∞, and α_i ∈ L^∞(ℝ) be log-Hölder continuous both at the origin and at infinity for with

(24)

Suppose that and Then, the m-linear Calderón-Zygmund operator T is bounded on the product of grand variable Herz spaces. Moreover,

(25)

with the constant C > 0 independent of

Proof. We restrict ourselves to m = 2, the general case following in a similar manner. Defining and , we decompose the component functions of as

(26)

For future usage, we divide ℤ into the following sets

(27)

and, for X and Y arbitrary subsets of ℤ, we define

(28)

From Definition 5 and (26), we have

(29)

where

(30)

(31)

It is necessary to estimate I₁, I₂, I₃, I₅, I₆, and I₉, since I₄, I₇, and I₈ can be obtained in a similar manner as I₂, I₃, and I₆, respectively. Estimation for I₁: splitting ℤ = ℤ₋ ∪ ℕ₀ and by the asymptotic 2^kα(x)≃2^kα(0) (x ∈ R_k and k < 0) and 2^kα(x)≃2^kα(∞) (x ∈ R_k and k ≥ 0), we get

(32)

For φ, σ ∈ L_k, x ∈ R_k, y₁ ∈ R_φ, and y₂ ∈ R_σ, we have

(33)

(34)

yielding

(35)

From estimate (35), Hölder’s inequality, 1/q(x) = 1/q₁(x) + 1/q₂(x), and Lemma 2, we obtain

(36)

Taking into account the previous estimate for ν_k(L_k, L_k), the equality 1p = 1p₁ + 1p₂, and Hölder’s inequality, we have

(37)

By Hölder’s inequality, Fubini’s theorem for series, , and defining b₁≔n/q₁(0) − n + α₁(0) < 0, we obtain

(38)

The estimate is obtained, mutatis mutandis, via the estimation for I₁₁₁. With this estimates at hand, we obtain

(39)

To estimate I₁₂, we split as follow

(40)

The estimate I₁₂₂ follows in similar manner as in I₁₁ with simply replaced α_i(0) by α_i(∞) and used the fact q_i(0) ≤ q_i(∞).

For estimate I₁₂₁, by Hölder’s inequality, Lemma 2, and the inequality q_i(0) ≤ q_i(∞), we obtain

(41)

From the estimate of ν_k(ℤ₋, ℤ₋), the equality 1/p = 1/p₁ + 1/p₂, and Hölder’s inequality, we have

(42)

Invoking the Hölder inequality and defining ξ₁≔n/q₁(0) − n + α₁(∞) < 0, we have

(43)

Similar estimate, with the corresponding changes, is obtained for A₂, from which we obtain Hence

(44)

Estimation for I₂: as in the case of I₁, we obtain the following estimate

(45)

Notice that, for x ∈ R_k, y₁ ∈ R_φ, y₂ ∈ R_σ, φ ∈ L_k, and σ ∈ M_k, we have

(46)

from which, taking Lemma 2 into consideration and elementary computations, we obtain

(47)

From the estimate for ν_k(L_k, M_k) and Hölder’s inequality, we get

(48)

Notice that I₂₁₁ = I₁₁₁. For the estimate I₂₁₂, we reason as follows

(49)

The term I₂₂ is estimated by

(50)

To estimate I₂₂₁, by Hölder’s inequality, Lemma 2, and the inequality q₁(0) ⩽ q₁(∞), we have

(51)

Thus using (51) and by Hölder’s inequality, we have

(52)

The term B₁ is equal to A₁ and for B₂ we use similar arguments as for I₂₁₂, replacing α₂(0) with α₂(∞).

For the term I₂₂₂, by Hölder’s inequality and Lemma 2, we have

(53)

Taking into consideration (53) and applying Hölder’s inequality, we get

(54)

Noting that and estimating in a similar fashion as I₂₁₂, we obtain , from which we get

(55)

Estimation for I₃: we have

(56)

For φ ∈ L_k, σ ∈ N_k, x ∈ R_k, y₁ ∈ R_φ, and y₂ ∈ R_σ, we get

(57)

from which, taking into account the elementary inequality

, it follows

(58)

By Hölder’s inequality, Lemma 2, q₂(0)≲q₂(∞), and k ∈ ℤ₋, we obtain

(59)

To estimate the term I₃₁, using (59) and the Hölder inequality, we obtain

(60)

We have , since I₃₁₁ = I₁₁₁.

As for I₃₁₂, we split as follow

(61)

By Hölder’s inequality, Fubini’s theorem for series, the inequality n/q₂(∞) + α₂(0) > 0 and , we obtain

(62)

For D₂, using the same argument as above and the inequality α₂(0) ⩽ α₂(∞), we obtain

(63)

The term I₃₂ can be estimated as

(64)

For k ∈ ℕ₀, φ ∈ ℤ₋, σ ∈ N_k, q₁(0)≲q₁(∞), and Lemma 2, we obtain

(65)

By (65) and Hölder’s inequality, we get

(66)

Note that The estimate for can be obtained in a similar way as for I₃₁₂, by replacing α₂(0) with α₂(∞).

For estimate I₃₂₂, by Hölder’s inequality and Lemma 2, we have

(67)

Using (67) and Hölder’s inequality, we obtain

(68)

We have and the estimate for the term is similar to Taking all the estimates into account yields

(69)

Estimation for I₅: We have

(70)

By the L^q(·)-boundedness of T, see Lemma 7, we obtain that

(71)

By Hölder’s inequality, we have

(72)

Similarly, we can obtain similar estimate for I₅₂, replacing α_i(0) by α_i(∞) with k ∈ ℕ₀. Therefore

(73)

Estimation for I₆: we have

(74)

For k ∈ ℤ₋, φ ∈ M_k, σ ∈ N_k, and x ∈ R_k, we have

(75)

yielding

(76)

By Hölder’s inequality, Lemma 2, k ∈ ℤ₋, and inequality q₂(0) ⩽ q₂(∞), we obtain

(77)

By (77) and invoking Hölder’s inequality, we get

(78)

Note that the estimate I₆₁₁ is equal to that of I₅₁ and I₆₁₂ is obtained by similar argument used in I₃₁₂.

For k ∈ ℕ₀, we have

(79)

Using (79) and Hölder’s inequality, we get

(80)

The estimate I₆₂₁ is similar to I₅₂ and Hence

(81)

Estimation for I₉: we estimate I₉ as usual

(82)

For φ ∈ N_k, y₁ ∈ R_φ, and x ∈ R_k, we have ∣x − y₁ | 2^φ, whereas for σ ∈ N_k, y₂ ∈ R_σ, and x ∈ R_k, we have ∣x − y₂ | 2^σ. Under the assumptions for the obtained inequalities, we have

(83)

When k ∈ ℤ₋, by Hölder’s inequality, Lemma 2, and q_j(0) ⩽ q_j(∞), we obtain

(84)

Thus

(85)

The estimates for the terms I₉₁₁ and I₉₁₂ are obtained in the same way as for I₃₁₂. Therefore

(86)

Finally, for the term I₉₂, we have

(87)

The terms I₉₂₁ and I₉₂₂ can be estimated in a similar manner as for that I₃₂₂. Thus

(88)

Taking into account the estimates (44), (55), (69), (73), (81), and (88), we get

(89)

which completes the proof.

Conflicts of Interest

The authors declare that there is no conflict of interest.

Authors’ Contributions

All the authors have contributed equally in preparation and finalization of the manuscript.

Acknowledgments

We gratefully acknowledge the constructive comments of the anonymous referees.

Open Research

Data Availability

No data were used to support this study.

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Boundedness of Multilinear Calderón-Zygmund Operators on Grand Variable Herz Spaces

Abstract

1. Introduction

2. Function Spaces with Variable Exponent

2.1. Lebesgue Space with Variable Exponent

2.2. Herz Spaces with Variable Exponent

2.3. Grand Lebesgue Sequence Space

2.4. Grand Variable Herz Space

3. Boundedness of Multilinear Calderón-Zygmund Operators

Conflicts of Interest

Authors’ Contributions

Acknowledgments

Open Research

Data Availability

References

Citing Literature

References

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Boundedness of Multilinear Calderón-Zygmund Operators on Grand Variable Herz Spaces

Abstract

1. Introduction

2. Function Spaces with Variable Exponent

2.1. Lebesgue Space with Variable Exponent

2.2. Herz Spaces with Variable Exponent

2.3. Grand Lebesgue Sequence Space

2.4. Grand Variable Herz Space

3. Boundedness of Multilinear Calderón-Zygmund Operators

Conflicts of Interest

Authors’ Contributions

Acknowledgments

Open Research

Data Availability

References

Citing Literature

References

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