# Parameterθ-Type Marcinkiewicz Integral on Nonhomogeneous Weighted Generalized Morrey Spaces

Authors: Guanghui Lu
Journal: Journal of Function Spaces
 (2020)

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## Abstract

LetX,d,μ 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 Mθρ is bounded on the weighted generalized Morrey space Lp,ϕ,τω for p∈1,∞. Furthermore, the boudedness of Mθρ on weak weighted generalized Morrey space WLp,ϕ,τω is also obtained.


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## Body

## 1. Introduction

To unify the spaces of homogeneous type in the sense of Coifman and Weiss (see [1, 2]) and nondoubling measure spaces (see [3–8]), 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 [10–14]). On the other hand, the boundedness of singular integral operators on various of spaces is also obtained; the readers can see [15–20] and so on.In this paper, letX,d,μ 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 Mθρ. 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 [22–24] introduced generalized Morrey spaces Mp,φℝn and also obtained some boundedness of integral operators on Mp,φℝn. 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 Mp,ρω 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 onℝn 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 X,d,μ. Also, we obtain the boundedness of parameter θ-type Marcinkiewicz integral Mθρ 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 spaceX,d,μ is said to be upper doubling if μ is a Borel measure on X and there exist a dominating function λ:X×0,∞⟶0,∞ and a constant Cλ>0, depending on λ, for each x∈X, r⟶λx,r is nondecreasing and, for all x∈X and r∈0,∞(1)μBx,r≤λx,r≤Cλλx,r2.Moreover, Hytönen et al. [12] have showed that there exists another dominating function λ~ such that λ~≤λ, Cλ~≤Cλ, and for all x,y∈X with dx,y≤r(2)λ~x,r≤Cλ~λ~y,r.If there is no special explanation in this paper, we always assume thatλ satisfies (2).Definition 2 [9].
A metric spaceX,d is said to be geometrically doubling if there exist some N0∈ℕ such that, for any ball Bx,r⊂X with x∈X and r∈0,∞, there exists a finite ball covering Bxi,r/2i of Bx,r such that the cardinality of this covering is at most N0.Remark 3.
LetX,d be a metric measure space. Hytönen in [9] pointed out that the geometrically doubling X,d is equivalent to the following statement: for any ε∈0,1 and any ball Bx,r⊂X with x∈X and r∈0,∞, there exists a finite ball covering Bxi,εri of Bx,r such that the cardinality of this covering is at most N0ε−n0, where n0≔log2N0.Although the measure doubling condition is not assumed uniformly for all balls in the nonhomogeneous metric measure spaceX,d,μ, Hytönen in [9] showed that there exist many balls which have the α,β-doubling properties. That is, for all α,β∈1,∞, a ball B⊂X is said to be α,β-doubling if μαB≤βμB. To be precise, Hytönen [9] pointed out that, if a metric measure space X,d,μ is upper doubling and α,β∈1,∞ with β>Cλlog2α≕αν, then there exists some j∈ℤ+ such that αjB is α,β-doubling. Moreover, let X,d be a geometrically doubling, β>αn0 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 α,β-doubling balls with centers at x. Furthermore, the radii of these balls may be chosen to be of the form α−jr for j∈ℕ and any preassigned number r∈0,∞. Throughout this paper, for any α∈1,∞ and ball B, the smallest α,βα-doubling ball of the form αjB with j∈ℤ+ is simply denoted by B~α, where
(3)βα≔α3maxn0,μ+max5α,30n0+max3α,30ν.Here and in what follows, we always assumeα=6 and denote by B~ the smallest 6,β6-doubling ball of the form 6jB with j∈ℤ+.The following discrete coefficientK~B,Sκ introduced by Bui and Duong [33] is very similar to the quantity KB,S which is introduced by Tolsa in [7].Definition 4 [33]).
For anyκ∈1,∞ and two balls B,S∈X satisfying B⊂S, define
(4)K~B,Sκ=1+∑l=−logκ2NB,SκμκlBλcB,κlrB,where NB,Sκ represents the smallest integer satisfying κNB,SκrB≥rS.Remark 5.
(i)
By the definition ofNB,Sκ and the fact rB≤2rS, it is not difficult to get NB,Sκ≥−logκ2=−logκ2, which guarantees the definition of the K~B,Sκ to make sense. Furthermore, Lin et al. [16] showed that, via a change of variables and (4), it is obvious to see that
(5)K~B,Sκ~1+∑l=1NB,Sκ+logκ2+1μκlBλcB,κlrBholds, where the implicit equivalent positive constants do not rely on the balls B⊂S⊂X but depend on the choice of κ with κ∈1,∞(ii)
Hytönen in [9] introduced a continuous version KB,S (also see [12]). That is, for any two balls B,S∈X satisfying B⊂S, set(6)KB,S=1+∫2S\B1λcB,dx,cBdμx.Via the simple computation, it is not difficult to see thatKB,S≤CK~B,Sκ. Unfortunately, in general, KB,S and K~B,Sκ are not equivalent, but, under the case of nondoubling measure, KB,S~K~B,Sκ.We now give out the definition of parameterθ-type Marcinkiewicz integral Mθρ.Definition 6.
Letθ be a nonnegative, nondecreasing function on 0,∞ satisfying the following condition:
(7)∫01θttlog1tdt<∞.Suppose thatK·,· is a locally integrable function defined on X×X\x,y:x=y. Then, there exists a positive constant C such that, for all x,y∈X with x≠y(8)Kx,y≤Cdx,yλx,dx,y,and, for all x,x′,y with satisfying dx,y≥2dx,x′(9)Kx,y−Kx′,y+Ky,x−Ky,x′≤Cθdx,x′dx,ydx,yλx,dx,y.Remark 7.
Especially, if we takeθt=tε with ε∈0,1 as in (9), then the above kernel is just the standard kernel given in [20].The parameterθ-type Marcinkiewicz integral Mθρ associated with the above kernel K satisfying (8) and (9) is defined by, for all x∈X and ρ>0(10)Mθρfx=∫0∞1tρ∫dx,y≤tKx,ydx,y1−ρfydμy2dtt12.Remark 8.
(1)
If we takeθt=tε with ε∈0,1 as in (9), and ρ=1 in (10), then the parameter θ-type Marcinkiewicz integral Mθρ is just the Marcinkiewicz integral M on X,d,μ given in [17](2)
If we takeX,d,μ=ℝn,·,μ, θt=tε, and ρ≡1, then the parameter θ-type Marcinkiewicz integral Mθρ is just the Marcinkiewicz integral M under nondoubling measure (see [34])(3)
If we takeX,d,μ=ℝn,·,dx, θt=tε, ρ≡1, and Kx,y=Ωx−y/x−yn−1, then the parameter θ-type Marcinkiewicz integral Mθρ is just the classical Marcinkiewicz integral MΩ introduced by Stein in [35] and its form as follows:
(11)MΩfx=∫0∞∫dx,y≤tΩx−yx−yn−1fydy2dtt31/2,x∈ℝn(4)
If we takeX,d,μ=ℝn,·,dx, θt=tε, and Kx,y=Ωx−y/x−yn−1, then Mθρ defined in (10) is just the classical parameter Marcinkiewicz integral MΩρ introduced by Hörmander in [36], that is
(12)MΩρfx=∫0∞1tρ∫dx,y≤tΩx−yx−yn−ρfydy2dtt1/2,where x∈ℝn and ρ>0.Next, we recall the definition ofApρω weight given in [14].Definition 9.
[14] Let ρ∈1,∞ and p∈1,∞. A nonnegative μ-measurable function ω is called an Apρμ weight if there exists a positive constant C such that, for all balls B⊂X(13)1μρB∫Bωxdμx1μρB∫Bωx1−p′dμxp−1≤C.
And a weightω is called an A1ρμ weight if there exists a positive constant C such that, for all balls B⊂X(14)1μρB∫Bωxdμx≤Cinfx∈Bωx.
As in the classical setting, letA∞ρμ≔⋃p=1∞Apρμ.The weighted generalized Morrey spaceLp,ϕ,τω is defined as follows.Definition 10.
Letτ>1 and p∈1,∞ and ω be a weight. Suppose that ϕ:0,∞⟶0,∞ is an increasing function. Then a weighted generalized Morrey space Lp,ϕ,τω is defined by
(15)Lp,ϕ,τω=f∈Llocpω:fLp,ϕ,τω<∞,where
(16)fLp,ϕ,τω=supB1ϕωτB∫Bfxpωxdμx1/p.We also denotedWLp,ϕ,τω by the weighted weak generalized Morrey space of all locally integrable functions satisfying
(17)fWLp,ϕ,τω=supBsupt>01ϕωτB1/ptωx∈B:fx>t1/p<∞.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 ·Lp,ϕ,τω is independent of the choice of the parameter τ∈1,∞.The main results of this paper are stated as follows.Theorem 12.
Letτ,p∈1,∞, ρ∈1,∞, ω∈Apρμ, and ϕ:0,∞⟶0,∞ be an increasing function satisfying the following condition:
(18)∫m∞ϕzzdzz≤Cϕmm,for1<m<∞.Suppose thatMθρ defined by (10) associated with K satisfying (8) and (9) is bounded on L2μ, and the mapping t↦ϕt/t is almost decreasing and there exists a positive constant C such that
(19)ϕtt≤Cϕss,for s≥t. Then, Mθρ is bounded from Lp,ϕ,τω∩L2μ into itself.Theorem 13.
LetK satisfy (8) and (9), ω∈Apρμ, and ϕ:0,∞⟶0,∞ be an increasing function satisfying (18) and (19). Suppose that Mθρ defined in (10) is bounded on L2μ. Then, Mθρ is bounded from Lp,ϕ,τω into WLp,ϕ,τω for τ,p∈1,∞.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, χE denotes its characteristic function. For any p∈1,∞, we denote by p′ its conjugate index, that is 1/p+1/p′=1. For any ball B, cB and rB represent the center and radius of ball B, respectively. Furthermore, mBf denotes the mean value of the function f over ball B, i.e., mBf=1/μB∫Bfydμy.


## 2. Proof of Main Theorems

In this section, we will give out the proofs of Theorems16 and 17. First, we need do recall the following lemmas.We now recall the following properties ofApρμ weights from [15].Lemma 14.
[30] Let ρ,p∈1,∞, ω∈Apρω and η∈5ρ,∞. Then, there exist positive constants C1,C2∈1,∞ such that
(i)
for any ballB and μ-measurable set E⊂B(20)ωEωB≥C2−1μEμηBp(ii)
for any6,β6-doubling ball B and μ-measurable set E⊂B(21)ωEωB≤1−C1−11−μEμB1p.Finally, we recall the following lemma ensuring the integrability of functions [14].Lemma 15.
[27] Let ψ:0,∞⟶0,∞ be a function satisfying
(22)∫m∞ψsdss≤Cψm,forallm>0.
Then there existsε>0 such that ∫m∞ψssεds/s≤Cψmmε for all m>0. In particular, for every η≤1, there exists a positive constant C such that ∫m∞ψssηds/s≤Cψmmη for all m>0.The proofs of Theorems16 and 17 are stated as follows.Proof of Theorem 16.
From Remark11, we may assume that τ=6 in (16). For a fixed doubling ball B, decompose fx=f1x+f2x, where f1x=fxχ6Bx. Write
(23)MfLp,ϕ,τω≤Mf1Lp,ϕ,τω+Mf2Lp,ϕ,τω≕I+II.
In order to estimateI, we first consider Mf1. For any x∈B, by applying
(24)Mθρf1x≤∫XKx,ydx,y1−ρf1y∫dx,y∞dtt1+2ρ1/2dμy≤C∫6Bfyλx,dx,yωy1/pωy−1/pdμy≤C∫6Bfypωyλx,dx,ypdμy1/pμ12Bω12B1/pμ12B1/p′×1μ12B∫6Bωy−p′/pdμy1μ12B∫6Bωydμy1/p≤C∫6Bfypωyλx,dx,ypdμy1/pμ12Bω12B1/p≤C∑j=0∞∫6−j+1B\6−jBfypωyλx,dx,ypdμy1/pμ12Bω12B1/p≤CfLp,ϕ,τω∑j=0∞ϕω6−j+2B1/pλcB,6−jrBμ6−jBμ6−jBμ12Bω12B1/p≤CfLp,ϕ,τω∑j=0∞ϕω6−j+2B1/pω12B1/pμ12Bμ6−jB≤CfLp,ϕ,τωϕω6Bω12B1/p,and further, by applying (19) and Lemma 14, we can obtain that
(25)I=supB1ϕω6B∫BMf1xpωxdμx1/p≤CfLp,ϕ,τωsupBϕω6Bω12B1/pωBϕω6B1/p≤CfLp,ϕ,τω.
Now we estimateII. For any x∈B, by applying (8), the Hölder inequality, and (13) and (16), we can get
(26)Mθρf2x≤C∑j=1∞1λcB,6jrB∫6j+1Bfydμy≤C∑j=1∞1λcB,6jrB∫6j+1Bfyωy1/pωy−1/pdμy≤C∑j=1∞1λcB,6jrB∫6j+1Bfypωydμy1/p×μ6j+2Bω6j+1B1/p1μ6j+1B∫6j+1Bωydμy1/p×1μ6j+1B∫6j+1Bωy−p′/pdμy1/p′≤C∑j=1∞1λcB,6jrB∫6j+1Bfypωydμy1/pμ6j+2Bω6j+1B1/p≤CfLp,ϕ,τω∑j=1∞ϕω6j+2Bω6j+2B1/p,and by the assumption (19) and Lemma 15, we have
(27)∑j=1∞ϕ6j+1ω6B6j+1ω6B1/p≤Cϕω6Bω6B1/p.
Thus, we haveII≤C∥f∥Lp,ϕ,τω. So the proof of Theorem 16 is completed.Proof of Theorem 17.
By applying the definition ofWLp,ϕ,τω and an argument similar to that used in the estimate of Theorem 12, it is not difficult to obtain that Theorem 13 holds. So, here we omit the detail.



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*Source: 4532312-2020-12-16.xml*