Parameter θ-Type Marcinkiewicz Integral on Nonhomogeneous Weighted Generalized Morrey Spaces

Let ðX , d, μÞ be a nonhomogeneous metric measure space satisfying the upper doubling and geometrically doubling conditions in the sense of Hytönen. In this setting, the author proves that parameter θ-type Marcinkiewicz integral Mρθ is bounded on the weighted generalized Morrey space Lp,φ,τðωÞ for p ∈ ð1,∞Þ. Furthermore, the boudedness of Mρθ on weak weighted generalized Morrey space WLp,φ,τðωÞ is also obtained.


Introduction
To unify the spaces of homogeneous type in the sense of Coifman and Weiss (see [1,2]) and nondoubling measure spaces (see [3][4][5][6][7][8]), in 2010, Hytönen [9] first introduced a new class of metric measure space satisfying the so-called upper doubling and geometrically doubling conditions. For the sake of convenience, the new space is now called a nonhomogeneous metric measure space. Since then, the research on the space has been widely focused, for example, some authors established the properties of function spaces on the nonhomogeneous metric measure space (see [10][11][12][13][14]). On the other hand, the boundedness of singular integral operators on various of spaces is also obtained; the readers can see [15][16][17][18][19][20] and so on.
In this paper, let ðX, d, μÞ be a nonhomogeneous metric measure space in the sense of Hytönen [9]. In this setting, we will give out the definition of weighted (weak) generalized Morrey space and then obtain the boundedness of parameter θ-type Marcinkiewicz integral M ρ θ . In 1938, Morrey [21] first introduced the definition of Morrey space when regularity of the solution of elliptic differential equations in terms of the solutions themselves and their derivatives is considered. Later, many researchers studied Morrey spaces from various point of view. After studying Morrey spaces in detail, some researchers passed to generalized Morrey spaces, weighted Morrey spaces, and generalized Morrey spaces, for example, the Guliyev, Mizuhara, and Nakai in [22][23][24] introduced generalized Morrey spaces M p,φ ðℝ n Þ and also obtained some boundedness of integral operators on M p,φ ðℝ n Þ. In addition, we can see [25,26] to study the research and development about generalized Morrey space and weak generalized Morrey space. In 2009, Komori and Shirai [27] defined the weighted Morrey space and studied the boundedness of some classical operators such as the Hardy-Littlewood maximal operator and Calderón-Zygmund operator on these spaces. Based on this, Nakamura and Sawano established the boundedness of singular integral operator and its commutator on weighted Morrey space (see [28]). In 2012, Guliyev [29] first introduced the generalized weighted Morrey spaces M p,ρ ðωÞ and studied the boundedness of the sublinear operators and their higher order commutators which is generated by Calderón-Zygmund operators and Riese potentials on these spaces (see also [30,31]). In 2016, Nakamura defined another definition of generalized weighted Morrey space and established the boundedness of classical operators on this space (see [32]).
Recently, the Morrey space, weighted Morrey space, and generalized Morrey space on ℝ n have been extended to nonhomogeneous metric measure space, for example, we can see [10,13,14]. Motivated by these, in this paper, we first give out the definition of weighted generalized Morrey space and weighted weak generalized Morrey space on ðX, d, μÞ. Also, we obtain the boundedness of parameter θ-type Marcinkiewicz integral M ρ θ on the weighted generalized Morrey space and weak generalized Morrey space.
Before stating the main results of this paper, we first recall some necessary notions. The following definitions of upper doubling condition and geometrically doubling condition are from [9]. Definition 1 [9]. A metric measure space ðX, d, μÞ is said to be upper doubling if μ is a Borel measure on X and there exist a dominating function λ : X × ð0, ∞Þ ⟶ ð0, ∞Þ and a constant C ðλÞ > 0, depending on λ, for each x ∈ X, r ⟶ λ ðx, rÞ is nondecreasing and, for all x ∈ X and r ∈ ð0, ∞Þ Moreover, Hytönen et al. [12] have showed that there exists another dominating functionλ such thatλ ≤ λ, C ðλÞ ≤ C ðλÞ , and for all x, y ∈ X with dðx, yÞ ≤ r If there is no special explanation in this paper, we always assume that λ satisfies (2). Definition 2 [9]. A metric space ðX, dÞ is said to be geometrically doubling if there exist some N 0 ∈ ℕ such that, for any ball Bðx, rÞ ⊂ X with x ∈ X and r ∈ ð0, ∞Þ, there exists a finite ball covering fBðx i , r/2Þg i of Bðx, rÞ such that the cardinality of this covering is at most N 0 .
Remark 3. Let ðX, dÞ be a metric measure space. Hytönen in [9] pointed out that the geometrically doubling ðX, dÞ is equivalent to the following statement: for any ε ∈ ð0, 1Þ and any ball Bðx, rÞ ⊂ X with x ∈ X and r ∈ ð0, ∞Þ, there exists a finite ball covering fBðx i , εrÞg i of Bðx, rÞ such that the cardinality of this covering is at most N 0 ε −n 0 , where n 0 ≔ log 2 N 0 .
Although the measure doubling condition is not assumed uniformly for all balls in the nonhomogeneous metric measure space ðX, d, μÞ, Hytönen in [9] showed that there exist many balls which have the ðα, βÞ-doubling properties. That is, for all α, β ∈ ð1, ∞Þ, a ball B ⊂ X is said to be ðα, βÞ-doubling if μðαBÞ ≤ βμðBÞ. To be precise, Hytönen [9] pointed out that, if a metric measure space ðX, d, μÞ is upper doubling and α, β ∈ ð1, ∞Þ with β > ½C ðλÞ log 2 α ≕ α ν , then there exists some j ∈ ℤ + such that α j B is ðα, βÞ-doubling. Moreover, let ðX, dÞ be a geometrically doubling, β > α n 0 and μ be a Borel measure on X being finite on bounded sets. Hytönen also showed that, for μ-a.e. x ∈ X, there exist arbitrary small ðα, βÞ-doubling balls with centers at x. Furthermore, the radii of these balls may be chosen to be of the form α −j r for j ∈ ℕ and any preassigned number r ∈ ð0, ∞Þ. Throughout this paper, for any α ∈ ð1, ∞Þ and ball B, the smallest Here and in what follows, we always assume α = 6 and denote byB the smallest ð6, β 6 Þ-doubling ball of the form The following discrete coefficientK ðκÞ B,S introduced by Bui and Duong [33] is very similar to the quantity K B,S which is introduced by Tolsa in [7].
Definition 4 [33]). For any κ ∈ ð1, ∞Þ and two balls B, S ∈ X satisfying B ⊂ S, definẽ where N   [16] showed that, via a change of variables and (4), it is obvious to see that holds, where the implicit equivalent positive constants do not rely on the balls B ⊂ S ⊂ X but depend on the choice of κ with κ ∈ ð1, ∞Þ (ii) Hytönen in [9] introduced a continuous version K B,S (also see [12]). That is, for any two balls B, S ∈ X satisfying B ⊂ S, set Via the simple computation, it is not difficult to see that  Journal of Function Spaces Definition 6. Let θ be a nonnegative, nondecreasing function on ð0, ∞Þ satisfying the following condition: Suppose that Kð·, · Þ is a locally integrable function defined on X × X \ fðx, yÞ: x = yg. Then, there exists a positive constant C such that, for all x, y ∈ X with x ≠ y and, for all x, x ′ , y with satisfying dðx, yÞ ≥ 2dðx, Remark 7. Especially, if we take θðtÞ = t ε with ε ∈ ð0, 1 as in (9), then the above kernel is just the standard kernel given in [20].
The parameter θ-type Marcinkiewicz integral M ρ θ associated with the above kernel K satisfying (8) and (9) is defined by, for all x ∈ X and ρ > 0 Remark 8.
Next, we recall the definition of A ρ p ðωÞ weight given in [14].
And a weight ω is called an A ρ 1 ðμÞ weight if there exists a positive constant C such that, for all balls B ⊂ X As in the classical setting, let A ρ ∞ ðμÞ ≔ The weighted generalized Morrey space L p,ϕ,τ ðωÞ is defined as follows.
Definition 10. Let τ > 1 and p ∈ ð1, ∞Þ and ω be a weight. Suppose that ϕ : ð0, ∞Þ ⟶ ð0, ∞Þ is an increasing function. Then a weighted generalized Morrey space L p,ϕ,τ ðωÞ is defined by where We also denoted WL p,ϕ,τ ðωÞ by the weighted weak generalized Morrey space of all locally integrable functions satisfying

Journal of Function Spaces
Remark 11. With an argument similar to that used in the proof of Lemma 2.3 and 2.4 in [14], it is not difficult to show that the norm of the weighted generalized Morrey space k·k L p,ϕ,τ ðωÞ is independent of the choice of the parameter τ ∈ ð1, ∞Þ.
The main results of this paper are stated as follows.
Finally, we make some conventions on notation. Throughout the paper, C represents a positive constant which is independent of the main parameters but may be different from line to line. For a μ-measurable set E ⊂ X, χ E denotes its characteristic function. For any p ∈ ½1, ∞, we denote by p′ its conjugate index, that is ð1/pÞ + ð1/p′Þ = 1.

Proof of Main Theorems
In this section, we will give out the proofs of Theorems 16 and 17. First, we need do recall the following lemmas. We now recall the following properties of A ρ p ðμÞ weights from [15]. Lemma 14. [30] Let ρ, p ∈ ½1, ∞Þ, ω ∈ A ρ p ðωÞ and η ∈ ½5ρ, ∞Þ. Then, there exist positive constants C 1 , C 2 ∈ ½1, ∞Þ such that (i) for any ball B and μ-measurable (ii) for any ð6, β 6 Þ-doubling ball B and μ-measurable set Finally, we recall the following lemma ensuring the integrability of functions [14].