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

There has been increased interest in the study of multilinear singular integral operators in recent years. The class of multilinear singular integrals with standard CalderónZygmund 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.


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][2][3] and later by Grafakos and Torres [4]. For the boundedness and other properties of multilinear fractional integrals, we refer to, e.g., [5][6][7][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][16][17][18][19][20][21] and references therein.
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 ℕ 0 ≔ ℕ ∪ f0g (ii) ℤ is the set of integers (iii) ℤ − is the set of negative integers (iv) n, m ≔ fn, n + 1, n + 2, ⋯, m − 1, mg for n, m ∈ ℤ and n < m (v) Bðx, rÞ is the ball of radius r center at the point x (xi) constants (often different constant in the same chain of inequalities) will mainly be denoted by c or C

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.

Lebesgue Space with Variable Exponent.
For the current section, we refer to [10][11][12][13]35] unless and until stated otherwise. Let q be a real-valued measurable function on ℝ n with values in ½1, ∞Þ. For X ⊂ ℝ n , we suppose that where q − ðXÞ ≔ ess inf x∈X qðxÞ and q + ðXÞ ≔ ess sup x∈X qðxÞ. By L qð·Þ ðℝ n Þ, we denote the space of measurable function f on ℝ n such that It is a Banach space, see [13,35], endowed with norm: By q′, we denote the conjugate exponent of q, defined by q ′ ðxÞ = qðxÞ/ðqðxÞ − 1Þ. In the sequel, we use log condition: where A = AðqÞ > 0 does not depend on x, y; decay condition at 0: holds for some qð0Þ ∈ ð1,∞Þ; and decay condition at ∞: there exists a number qð∞Þ ∈ ð1,∞Þ, such that where A = AðqÞ does not depend on x.
Given a function f ∈ L 1 loc ðℝ n Þ, the Hardy-Littlewood maximal operator M is defined by We adopt the following notations: (i) L qð·Þ loc ðℝ n Þ ≔ f f : f ∈ L qð·Þ ðKÞ for all compact subsets K ⊂ ℝ n g (ii) P ðℝ n Þ consists of all measurable functions q satisfying q − > 1 and q + < ∞ (iii) P log 0 ðℝ n Þ and P log ∞ ðℝ n Þ denote the classes of q ∈ P ðℝ n Þ which satisfy (5) and (6), respectively (iv) Bðℝ n Þ is the set of all q ∈ P ðℝ n Þ for which M is bounded on L qð·Þ ðℝ n Þ For the following lemma, we refer to, e.g., [12].

Lemma 2.
Let D > 1 and q ∈ P log 0 ðℝ n Þ ∩ P log ∞ ðℝ n Þ: Then where c 0 ≥ 1 and c ∞ ≥ 1 depend on D, but do not depend on r.

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

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, X stands for one of the sets ℤ n , ℤ, ℕ, and ℕ 0 .
2.4. Grand Variable Herz Space. Following [34], we now introduce the grand variable Herz spaces.

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

Lemma 7.
Let q i ∈ Bðℝ n Þ, i ∈ 1, m, q ∈ P log 0 ðℝ n Þ with 1/qðxÞ = 1/q 1 ðxÞ + ⋯ + 1/q m ðxÞ, and ðq/sÞ ′ ∈ Bðℝ n Þ for some 0 < s < q − . Then, the m-linear Calderón-Zygmund operator T is bounded on the product of variable exponent Lebesgue spaces. Moreover, with the constant C independent of f We now state and prove the boundedness of multilinear Calderón-Zygmund operator on grand variable Herz spaces.

Journal of Function Spaces
Similar estimate, with the corresponding changes, is obtained for A 2 , from which we obtain I 12 ≲ k f 1 k _ K α 1 ð·Þ,p 1 Þ,θ q 1 ð·Þ ðℝ n Þ k f 2 k _ K α 2 ð·Þ,p 2 Þ,θ q 2 ð·Þ ðℝ n Þ : Hence Estimation for I 2 : as in the case of I 1 , we obtain the following estimate Notice that, for x ∈ R k , y 1 ∈ R φ , y 2 ∈ R σ , φ ∈ L k , and σ ∈ M k , we have from which, taking Lemma 2 into consideration and elementary computations, we obtain From the estimate for ν k ðL k , M k Þ and Hölder's inequality, we get Notice that I 211 = I 111 . For the estimate I 212 , we reason as follows The term I 22 is estimated by To estimate I 221 , by Hölder's inequality, Lemma 2, and the inequality q 1 ð0Þ ⩽ q 1 ð∞Þ, we have Thus using (51) and by Hölder's inequality, we have The term B 1 is equal to A 1 and for B 2 we use similar arguments as for I 212 , replacing α 2 ð0Þ with α 2 ð∞Þ.