# Parametric Marcinkiewicz Integral and Its Higher-Order Commutators on Variable Exponents Morrey-Herz Spaces

Authors: Omer Abdalrhman Omer; Khedoudj Saibi; Muhammad Zainul Abidin; Mawia Osman
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this article, we prove the boundedness of the parametric Marcinkiewicz integral and its higher-order commutators generated by BMO spaces on the variable Morrey-Herz space. All the results are new even whenα· is a constant.


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## Body

## 1. Introduction

Throughout the entirety of this article, we assume thatn≥2, ℝn is the n-dimensional Euclidean space, and Sn−1 is the unit sphere in ℝn equipped with the normalized Lebesgue measure dρ. The function Ω is assumed to be homogeneous of degree zero on ℝn with Ω∈L1Sn−1 and
(1)∫Sn−1Ωx′dρx′=0,where x′=x/x for any x∈ℝn\0. For ρ∈0,n, the parametric Marcinkiewicz integral MΩϱ of higher dimensions is defined as follows:
(2)MΩϱfx≔∫0∞t−ϱ∫x−y≤tΩx−yfyx−yn−ρdy2dtt1/2.LetB be a ball with a radius τ>0, and a center x∈ℝn. A locally integrable function Λ is said to be in the BMO space, if it satisfies
(3)Λ∗≔supB1B∫BΛz−ΛBdz<∞,where ΛB=B−1∫BΛtdt and E denotes the Lebesgue measure of the set E in ℝn. For Λ∈BMO,i∈ℕ, the i-order commutator for the parametric Marcinkiewicz integral MΩ,Λiϱ is defined as follows:
(4)MΩ,Λiϱfx≔∫0∞t−ϱ∫x−y≤tΛx−ΛyiΩx−yfyx−yn−ρdy2dtt1/2.Ifρ=1 in (2), then the operator MΩϱ is equivalent to the classical Marcinkiewicz function MΩ1, which was initially introduced by Stein [1] in 1958. When Ω∈LipβSn−1,β∈0,1, Stein [1] demonstrated that MΩ1 is bounded on Lp for p∈1,2. Subsequently, the authors of [2] established the Lp-boundedness of MΩ1 for every p∈1,∞ when Ω∈ℂ1Sn−1. On the other hand, Calderón [3] proved that the commutator the Hilbert transform H generated by Λ∈BMO, defined by Λ,Tf≔ΛTf−TΛf, is bounded on L2ℝn. Coifman et al. [4] arrived at the conclusion that the commutator, which was generated by the Calderón-Zygmund operator T and the Λ∈BMO, is bounded on Lp for p∈1,∞. Since then, the commutators of the Calderón-Zygmund operator have played an essential role in the study of the regularity of solutions to second-order elliptic, parabolic, and ultraparabolic partial differential equations, see for example [5–11]. Moreover, the boundedness of the commutators of various operators generated by a BMO function has been widely studied. Particularly, Torchinsky and Wang [12] studied the weighted Lp-boundedness of MΩ,Λi1, where MΩ,Λi1 is the i-order commutator of Marcinkiewicz integral. The authors of [13] studied the behaviour of the Hardy-Littlewood maximal operator and the action of commutators in generalized local Morrey spaces and generalized Morrey spaces. For further research works studying the commutators on different function spaces, we refer to [9, 14–21] and references therein.The parametric Marcinkiewicz integralMΩϱ was originally introduced by Hörmander in [22] where the author established the boundeness of MΩϱ on Lp for p∈1,∞ under the condition Ω∈LipβSn−1,β∈0,1 and ϱ>0. Shi and Jiang [23] investigated the weighted Lp-boundedness of MΩϱ and MΩ,Λiϱ. Since that time, the boundedness of the parametric Marcinkiewicz integral, as well as its related commutator, in several types of function spaces have attracted the attention of many researchers. Deringoz and Hasanov [24] considered the boundedness of the operator MΩϱ on generalized Orlicz-Morrey spaces. On generalized weighted Morrey spaces, Deringoz [25] investigated the boundedness of rough parametric Marcinkiewicz integral MΩϱ and its higher-order commutator MΩ,Λiϱ. For more applications and recent developments on the research of the parametric Marcinkiewicz function, see [26–31].In the last decades, the variable Lebesgue spaces have been intensively studied since the pioneering work of [32] by Kovácık and Rákosnık. Additionally, different studies on variable function spaces, such as variable exponents Fourier-Besov-Morrey spaces [33–35], variable exponents Fourier-Besov spaces [36, 37], variable exponent Morrey spaces [38], variable Bessel potential spaces [39, 40], and variable exponent Hardy spaces [41, 42], were developed due to their applications in the modeling of electro-rheological fluids, PDEs with nonstandard growth, and image restoration. Recently, Izuki studied the Herz spaces K̇p·,qα in [43, 44]. As a generalization, Izuki [45] introduced the variable Morrey-Herz spaces MK̇p·,qα,σ. In fact, the author of [45] found that vector-valued sublinear operators which satisfy a certain size condition are bounded on the variable Morrey-Herz spaces. Furthermore, Almeida and Drihem [46] enhanced the variable case of the Herz spaces K̇p·,qα· and established the boundedness results for a class of sublinear operators. Lu and Zhu [47] generalized Izuki’s result for the MK̇p·,qα·,σ. For further information and applications, consult [48–54].Inspired by the research mentioned above, the main goal of this article is to prove the boundedness of the rough parametric Marcinkiewicz integral and its higher-order commutators on the variable exponents Morrey-Herz spaces.Henceforth, wherever the symbolC appears, it represents a positive constant whose value may vary but is independent of the basic variables. The expression f≲g denotes the existence of constant C such that f≤Cg, and f≍g means that f≲g≲f. If no further instructions are provided, the symbol for any space denoted by Xℝn is represented by X. For instance, Lpℝn is abbreviated as Lp.


## 2. Definitions and Preliminaries

In this section, we review some notations, definitions, and properties related to our work.A variable exponent is a measurable functionp·:ℝn⟶0,∞. For any variable exponent p·, we set p−≔essinfpx:x∈ℝn and p+≔esssuppx:x∈ℝn. Define the sets P by
(5)P≔pxismeasurablefunction:1<p−andp+<∞.Letp·∈P. The variable Lebesgue space Lp· consists of all measurable functions f on ℝn such that
(6)fLp·≔infϑ∈0,∞:℘p·fx≤1<∞,where
(7)℘p·f≔∫ℝnfxpxdx.It is obvious that the variable exponent Lebesgue norm has the following propertyfβLp·=fLβp·β,β≥1/p−.Define the setB by
(8)B≔px∈P:MHLisboundedonvariableLp,where MHL stands for the Hardy-Littlewood maximal function, which is defined as follows:
(9)MHLfx=supB∋x,B⊂ℝn1Bx,τ∫Bx,τfzdz,f∈Lloc1.Definition 1 (see [46]).
LetΘ· be real function on ℝn.
(i)
If there exists a constantClog>0 such that(10)Θx−Θ0≤Clogloge+1/x,forallx∈ℝn,then the function Θ· is said to be a log-Hölder continuous at the origin (or has a log decay at the origin).
(ii)
If there existΘ∞∈0,∞ and a constant Clog>0 such that(11)Θx−Θ∞≤Clogloge+x,forallx∈ℝn,then the function Θ· is said to be a log-Hölder continuous at the infinity (or has a log decay at the infinity).
Ifp·∈P, then the following expression of Hölder’s inequality is valid:
(12)∫ℝnfxgxdx≤1+1p−−1p+fLp·gLp′·.
See [55].
Here and hereafter,p′· denotes the conjugate exponent of p·, i.e., p′·≔p·/p·−1. It is well-known that if p· belongs to B, then p′·∈B (see [56]).
For anyℓ∈ℤ, let Bℓ≔x∈ℝn:x≤2ℓ,Rℓ≔Bℓ\Bℓ−1,anddenotebyχℓ≔χRℓ the characteristic function of Rℓ.Definition 2 (see [46]).
Letq∈0,∞,α·∈L∞ and let p·∈P. The homogeneous variable Herz space K̇p·,qα· is defined as the set of all functions f∈Llocp·ℝn\0 such that
(13)fK̇p·,qα·≔∑ℓ∈ℤ2ℓα·fχℓLp·q1/q<∞,for q<∞, and the usual modification should be made when q=∞.Definition 3 (see [47]).
Letσ∈0,∞,α·∈L∞,q∈0,∞ and let p·∈P. The homogeneous variable Morrey-Herz space MK̇q,p·α·,σ is defined as the set of all functions f∈Llocp·ℝn\0 such that
(14)fMK̇q,p·α·,σ≔supκ∈ℤ2−κσ∑ℓ=−∞κ2ℓα·fχℓLp·q1/q<∞,for q<∞, and the usual modification should be made when q=∞.Lemma 4 (see [44]).
Letp·∈B. Then, for any ball B in ℝn,
(15)χBLp·χBLp′·≤CB.Lemma 5 (see [44]).
Letp♭·∈B,♭=1,2. Then, there are positive constants δ♭1,δ♭2∈0,1, such that for any ball B in ℝn and any measurable subset S⊂B,
(16)χSLp♭′·χBLp♭′·≲SBδ♭2,andχSLp♭·χBLp♭·≲SBδ♭1.Proposition 6 (see [47]).
Letα·∈L∞,q∈0,∞,σ∈0,∞, and let p·∈P. If the function α· is log-Hölder continuous function both at origin and at infinity, then the following inequalities hold:
(17)fMK̇q,p·α·,σq≍maxsupκ<0κ∈ℤ2−κσq∑ℓ=−∞κ2ℓqα0fχℓLp·q,supκ⩾0κ∈ℤ2−κσq∑ℓ=−∞−12ℓqα0fχℓLp·q+supκ⩾0κ∈ℤ2−κσq∑ℓ=0κ2ℓqα∞fχℓLp·q.Lemma 7 (see [46]).
Letαx∈L∞ and τ1>0. If the function αx is log-Hölder continuous both at origin and infinity, then the following inequality holds:
(18)τ1αx≲τ2αy×τ1τ2α+,0<τ2⩽τ12,1,τ12<τ2⩽2τ1,τ1τ2α−,τ2>2τ1,for every x∈B0,τ1\B0,τ1/2 and y∈B0,τ2\B0,τ2/2.Lemma 8 (see [56]).
Letp·,q∗x∈P, and let q∈0,∞ such that 1/px=1/q+1/q∗x. Then, for any measurable functions f∈Lp· and g∈Lq∗·,
(19)fgLp·≲fLq∗·gLq.Lemma 9 (see [57]).
Leti be a positive integer, Λ∈BMO and let p·∈B. Then, there exists a positive C such that for all ℓ,j∈ℤℓ>j,
(20)supB⊂ℝn1χBLp·Λ−ΛBiχBLp·≍Λ∗i,and
(21)Λ−ΛBjiχBℓLp·≲ℓ−jiΛ∗iχBℓLp·.
The main results of this article are as follows.Theorem 10.
Suppose thatp·∈B and Ω∈LsSn−1 with s>p′+ satisfying (1). Let ϱ∈0,n,q∈0,∞,σ>0, and α· be log-Hölder continuous both at the origin and at infinity, such that
(22)−nδ1+σ<α−≤α+<−n−1s+nδ2,where δ1,δ2∈0,1 are the constants mentioned in Lemma 5. Then, the operator MΩρ is bounded on MK̇q,p·α·,σ.Theorem 11.
Suppose thatp·∈B and Ω∈LsSn−1 with s>p′+ satisfies (1). Let ϱ∈0,n,q∈0,∞,b∈BMO,σ>0, i∈ℕandα· be log-Hölder continuous both at the origin and at infinity, such that
(23)−nδ1+σ<α−≤α+<−n−1s+nδ2,where δ1,δ2∈0,1 are the constants mentioned in Lemma 5. Then, the operator MΩ,Λiϱ is bounded on MK̇q,p·α·,σ.It is worth noting that ifσ=0, then the variable Morrey-Herz space MK̇q,p·α·,0 dates back to the variable Herz space K̇q,p·α·. Thus, by letting σ=0 in Theorems 10 and 11, we will get the following results on the variable exponents Herz spaces.Corollary 12.
Suppose thatp·∈B and Ω∈LsSn−1 with s>p′+ satisfies (1). Let ρ∈0,n,q∈0,∞, and α· be log-Hölder continuous both at the origin and at infinity, such that
(24)−nδ1<α−≤α+<−n−1s+nδ2,where δ1,δ2∈0,1 are the constants mentioned in Lemma 5. Then, the operator MΩϱ is bounded on K̇q,p·α·.Corollary 13.
Suppose thatp·∈B and Ω∈LsSn−1 with s>p′+ satisfies (1). Let ϱ∈0,n,q∈0,∞,b∈BMO and i∈ℕ and α· be log-Hölder continuous both at the origin and at infinity, such that
(25)−nδ1<α−≤α+<−n−1s+nδ2,where δ1,δ2∈0,1 are the constants mentioned in Lemma 5. Then, the operator MΩ,Λiϱ is bounded on K̇q,p·α·.Remark 14.
Ifα· is a constant function, i.e., α·=α, then the results of Corollaries 12 and 13 can be founded in [30].


## 3. Proofs of Theorems10 and 11

Proof of Theorem 10.
Letf∈MK̇q,p·α·,σ. For any j∈ℤ, let fj≔fχj, then we have
(26)fx=∑j=−∞∞fχjx≔∑j=ℓ−1ℓ+1fjx+∑j=−∞ℓ−2fjx+∑j=ℓ+2∞fjx.Using the definition ofMK̇q,p·α·,σ, we have
(27)MΩϱfMK̇q,p·α·,σq=supκ∈ℤ2−κσq∑ℓ=−∞κ2ℓα·MΩϱfχℓLp·q≲supκ∈ℤ2−κσq∑ℓ=−∞κ2ℓα·∑j=ℓ−1ℓ+1MΩϱfjχℓLp·q+supκ∈ℤ2−κσq∑ℓ=−∞κ2ℓα·∑j=−∞ℓ−2MΩϱfjχℓLp·q+supκ∈ℤ2−κσq∑ℓ=−∞κ2ℓα·∑j=ℓ+2∞MΩϱfjχℓLp·q≕T1+T2+T3.First, we estimateT1. By Proposition 6, we obtain
(28)T1≍maxsupκ<0κ∈ℤ2−κσq∑ℓ=−∞κ2ℓα0q∑j=ℓ−1ℓ+1MΩϱfjχℓLp·q,supκ≥0κ∈ℤ2−κσq∑ℓ=−∞−12ℓα0q∑j=ℓ−1ℓ+1MΩϱfjχℓLp·q+supκ≥0κ∈ℤ2−κσq∑ℓ=0κ2ℓα∞q∑j=ℓ−1ℓ+1MΩϱfjχℓLp·q.Using the fact that2α·≍2α0, and the boundedness of MΩρ on Lp·(see [30]), we deduce
(29)T1≲maxsupκ<0κ∈ℤ2−κσq∑ℓ=−∞κ2ℓα0fχℓLp·q,supκ≥0κ∈ℤ2−κσq∑ℓ=−∞−12ℓα0fχℓLp·q+supκ≥0κ∈ℤ2−κσq∑ℓ=0κ2ℓα∞fχℓLp·q≲fMK̇q,p·α·,σq.For the termT2, we firstly estimate 2ℓαxMΩϱfjx.
(30)2ℓαxMΩϱfjx·χℓx≲2ℓαx∫0xt−ϱ∫x−y≤tΩx−yfjyx−yn−ϱdy2dtt1/2·χℓx+2ℓαx∫x∞t−ϱ∫x−y≤tΩx−yfjyx−yn−ρdy2dtt1/2·χℓx.It is clear that ifx∈Rk,j+2≤ℓ,andy∈Rj, then x−y≍x≍2ℓ. Thus, for ϱ∈0,n, by the mean value theorem, we have
(31)1x−y2ϱ−1x2ϱ≤yx−y2ρ+1.By (31) and the Minkowski’s inequality, it follows that
(32)2ℓαxMΩϱfjx·χℓx≲2ℓαx∫ℝnΩx−yfjyx−yn−ϱ·∫x−yxdtt2ρ+11/2dy·χℓx+2ℓαx∫ℝnΩx−yfjyx−yn−ϱ·∫x∞dtt2ρ+11/2dy·χℓx≲2ℓαx∫RjΩx−yfjyx−yn−ϱ·1x−y2ϱ−1x2ϱ1/2dy·χℓx+2ℓαx∫RjΩx−yfjyx−yn−ϱ·1xϱdy·χℓx≲2ℓαx∫RjΩx−yfjyx−yn−ϱ·y1/2x−yρ+1/2dy·χℓx+2ℓαx∫RjΩx−yfjyx−yndy·χℓx≲2ℓαxxn∫RjfyΩx−ydy·χℓx.By Lemma7, we deduce
(33)2ℓαxMΩϱfjx·χℓx≲1xn∫Rj2ℓαxfyΩx−ydy·χℓx≲1xn∫Rj2ℓαx2j−jαyfyΩx−ydy·χℓx≲2ℓ−jα+xn∫Rj2jαyfyΩx−ydy·χℓx.It follows from Hölder’s inequality (12) that
(34)2ℓαxMΩϱfjx·χℓx≲2ℓ−jα+x−nΩx−·χjLp′·2jα·fLp·χℓx.Sinces>p′+, we can find a variable exponent p∗·>1 such that 1/p′·=1/s+1/p∗·, then by Lemma 8, it follows that
(35)Ωx−·χjLp′·≲Ωx−·χjLsχjLp∗·≲ΩLsSn−12ℓ−jn−1/sχBjLp′·,where the last inequality (35) is based on the fact that χjLp∗·≍Bj−1/sχjLp′·; see [18]. From (34) and (35), Lemmas 4 and 5, we can deduce
(36)2ℓα·MΩϱfjχℓLp·≲12ℓn≲2ℓ−jα+−n−1/sχBjLp′·2jα·fjLp·χBℓLp·≲2ℓ−jα+−n−1/sχBjLp′·2jα·fjLp·2ℓnBℓχBℓLp′·≲2j−ℓnδ2+n−1/s−α+2jα·fjLp·.To estimateT2, we need to consider two cases: 0<q≤1 and 1<q<∞.Case 1.
1<q<∞, by the fact that nδ2−α++n−1/s>0, the Hölder’s inequality, and the inequality (36), it follows that
(37)T2≲supκ∈ℤ2−κσq∑ℓ=−∞κ∑j=−∞ℓ−22ℓα·MΩϱfjχℓLp·q≲supκ∈ℤ2−κσq∑ℓ=−∞κ∑j=−∞ℓ−22q/2j−ℓnδ2−α++n−1/s2jα·fjLp·q·∑j=−∞ℓ−22q′/2j−ℓnδ2−α++n−1/sq/q′≲supκ∈ℤ2−κσq∑j=−∞κ−22jα·fjLp·q∑ℓ=j+2κ21/2j−ℓqnδ2−α++n−1/s≲supκ∈ℤ2−κσq∑j=−∞κ−22jα·fjLp·q≲fMK̇q,p·α·,σq.Case 2.
0<q≤1, we use the inequality
(38)∑j=1∞ℵjq≤∑j=1∞ℵjq,ℵ1,ℵ2,⋯>0,and obtain
(39)T2≲supκ∈ℤ2−κσq∑ℓ=−∞κ∑j=−∞ℓ−22ℓα·MΩϱfjχℓLp·q≲supκ∈ℤ2−κσq∑ℓ=−∞κ∑j=−∞ℓ−22qj−ℓnδ2−α++n−1/s2jα·fjLp·q≲supκ∈ℤ2−κσq∑j=−∞κ−22jα·fjLp·q∑ℓ=j+2κ2j−ℓqnδ2−α++n−1/s≲fMK̇q,p·α·,σq.For the termT3, by applying Proposition 6, we can get
(40)T3≲maxsupκ<0,κ∈ℤ2−κσq∑ℓ=−∞κ2ℓα0∑j=ℓ+2∞MΩϱfjχℓLp·q,supκ≥0,κ∈ℤ2−κσq∑ℓ=−∞−12ℓα0∑j=ℓ+2∞MΩϱfjχℓLp·q+supκ≥0,κ∈ℤ2−κσq∑ℓ=0κ2ℓα∞∑j=ℓ+2∞MΩϱfjχℓLp·q≕maxT31,T32.ForT3, it is clear that if x∈Rℓ,j−2≥ℓ,andy∈Rj, then x−y≍y. By (31), (12), and the Minkowski’s inequality, we deduce that
(41)MΩϱfjx·χℓx≲∫0yt−ϱ∫x−y≤tΩx−yx−yn−ϱfjydy2dtt1/2·χℓx+∫y∞t−ϱ∫x−y≤tΩx−yx−yn−ϱfjydy2dtt1/2·χℓx≲∫RjfjyΩx−yx−yn−ϱ·x1/2x−yϱ+1/2dy·χℓx+∫RjfjyΩx−yx−yndy·χℓx≲1yn∫RjfjyΩx−ydy·χℓx≲2−njΩx−·χjLp′·fjLp·χℓx.Similar to (35), we conclude
(42)Ωx−·χjLp′·≲Ωx−·χjLsχjLp∗·≲ΩLsSn−1χBjLp′·.From (42), Lemmas 4 and 5, it follows that
(43)∑j=ℓ+2∞MΩϱfjχℓLp·≲∑j=ℓ+2∞MΩϱfjχℓLp·≲∑j=ℓ+2∞12jnχBjLp′·fjLp·χBℓLp·≲∑j=ℓ+2∞fjLp·χBℓLp·2−jnBjχBjLp·≲∑j=ℓ+2∞fjLp·2ℓ−jnδ1.To estimateT31, we need to consider two cases below: 0<q≤1 and 1<q<∞.Case 3.
0<q≤1, combining the above inequalities and using (38), we can obtain
(44)T31≲supκ<0κ∈ℤ2−κσq∑ℓ=−∞κ2ℓα0q∑j=ℓ+2κ−12qℓ−jnδ1fjLp·q+supκ<0κ∈ℤ2−κσq∑ℓ=−∞κ2ℓα0q∑j=κ∞2qℓ−jnδ1fjLp·q≕T31′+T31′′.ForT31′, in view of α0+nδ1>α−+nδ1>0, we get
(45)T31′≲supκ<0κ∈ℤ2−κσq∑j=−∞κ−12qjα0fjLp·q∑ℓ=−∞j−22qℓ−jα0+nδ1≲supκ<0κ∈ℤ2−κσq∑j=−∞κ−12qjα0fjLp·q≲fMK̇q,p·α·,σq.Now, let us deal withT31′′, noting that σ−nδ1−α0<σ−nδ1−α−<0, it follows that
(46)T31′′≲supκ<0κ∈ℤ2−κσq∑ℓ=−∞κ2qℓα0∑j=κ∞2qℓ−jnδ12−jqα02jσq2−jσq∑I=−∞j2Iα0qfILp·q≲supκ<0κ∈ℤ2−κσq∑ℓ=−∞κ2qkα0+nδ1∑j=κ∞2jqσ−nδ1−α0fMK̇q,p·α·,σq≲fMK̇q,p·α·,σ.Case 4.
1<q<∞, we have
(47)T31≲supκ<0κ∈ℤ2−κσq∑ℓ=∞κ2ℓqα0∑j=ℓ+2∞2ℓ−jnδ1fjLp·q≲supκ<0κ∈ℤ2−κσq∑ℓ=−∞κ2ℓqα0∑j=ℓ+2κ2ℓ−jnδ1fjLp·q+supκ<0κ∈ℤ2−κσq∑ℓ=−∞κ2ℓqα0∑j=κ+1∞2ℓ−jnδ1fjLp·q≕T31⋄+T31⋄⋄.ForT31⋄, by the fact that α0+nδ1>0 and using Hölder inequality, we infer that
(48)T31⋄≲supκ<0κ∈ℤ2−κσq∑ℓ=∞κ∑j=l+2κ2qjα0fjLp·q2q/2ℓ−jα0+nδ1·∑j=l+2κ2q′/2ℓ−jα0+nδ1q/q′≲supκ<0κ∈ℤ2−κσq∑j=−∞κ2jqα0fjLp·q∑ℓ=−∞j−22q/2ℓ−jα0+nδ1≲fMK̇q,p·α·,σ.ForT31⋄⋄, from the inequality (36) and using the method as for T31′′, we obtain
(49)T31⋄⋄≲fMK̇q,p·α·,σq.By combiningT31′, T31′′, T31⋄, and T31⋄⋄ estimates, we arrive at
(50)T31≲fMK̇q,p·α·,σq.By the similar method used in the estimate forT31, it is not difficult to show that
(51)T32≲fMK̇q,p·α·,σq.Thus, we have(52)MΩϱfMK̇q,p·α·,σ≲T1+T2+T3≲fMK̇q,p·α·,σ.The proof for Theorem10 is finished.Proof of Theorem 11.
LetΛ∈BMO,f∈MK̇q,p·α·,σ. For any j∈ℤ, let fj≔fχj, then we have
(53)fx=∑j∈ℤfχjx≔∑j=ℓ−1ℓ+1fjx+∑j=−∞ℓ−2fjx+∑j=ℓ+2∞fjx.Using the definition ofMK̇q,p·α·,σ, we have
(54)MΩ,ΛiϱfMK̇q,p·α·,σq=supκ∈ℤ2−κσq∑ℓ=−∞κ2ℓα·MΩ,ΛiϱfχℓLp·q≲supκ∈ℤ2−κσq∑ℓ=−∞κ2ℓα·∑j=ℓ−1ℓ+1MΩ,ΛiϱfjχℓLp·q+supκ∈ℤ2−κσq∑ℓ=−∞κ2ℓα·∑j=−∞ℓ−2MΩ,ΛiϱfjχℓLp·q+supκ∈ℤ2−κσq∑ℓ=−∞κ2ℓα·∑j=ℓ+2∞MΩ,ΛiϱfjχℓLp·q≕Y1+Y2+Y3.Let us first estimateY1. From Proposition 6 and the boundedness of MΩρ on Lp·(see [30]), and using the similar methods as that for T1, it is not difficult to see that
(55)Y1≍maxsupκ<0κ∈ℤ2−κσq∑ℓ=−∞κ2ℓα0q∑j=ℓ−1ℓ+1MΩ,ΛiϱfjχℓLp·q,supκ≥0κ∈ℤ2−κσq∑ℓ=−∞−12ℓα0q∑j=ℓ−1ℓ+1MΩ,ΛiϱfjχℓLp·q+supκ≥0κ∈ℤ2−κσq∑ℓ=0κ2ℓα∞q∑j=ℓ−1ℓ+1MΩ,ΛiϱfjχℓLp·q≲Λ∗imaxsupκ<0κ∈ℤ2−κσq∑ℓ=−∞κ2ℓα0fχℓLp·q,supκ≥0κ∈ℤ2−κσq∑ℓ=−∞−12ℓα0fχℓLp·q+supκ≥0κ∈ℤ2−κσq∑ℓ=0κ2ℓα∞fχℓLp·q≲Λ∗ifMK̇q,p·α·,σq.Now, let us turn to the estimates ofY2. We consider 2ℓαxMΩ,Λϱfjx(56)2ℓαxMΩ,Λϱfjx·χℓx≲2ℓαx∫0xt−ϱ∫x−y≤tΛx−ΛyiΩx−yx−yn−ρfjydy2dtt1/2·χℓx+2ℓαx∫x∞t−ϱ∫x−y≤tΛx−ΛyiΩx−yx−yn−ρfjydy2dtt1/2·χℓx.It is clear that ifx∈Rℓ,j+2≤ℓ,andy∈Rj, then x−y≍x. Thus, for ρ∈0,n, we use the Minkowski’s inequality to get
(57)2ℓαxMΩ,Λϱfjx·χℓx≲2ℓαx∫ℝnΛx−ΛyiΩx−yfjyx−yn−ρ·∫x−yxdtt2ρ+11/2dy·χℓx+2ℓαx∫ℝnΛx−ΛyiΩx−yfjyx−yn−ρ·∫x∞dtt2ρ+11/2dy·χℓx≲2ℓαx∫RjΛx−ΛyiΩx−yfjyx−yn−ρ·y1/2x−yρ+1/2dy·χℓx+2ℓαx∫RjΛx−ΛyiΩx−yfjyx−yndy·χℓx≲2ℓαxxn∫RjΩx−yΛx−Λyifjydy·χℓx.Using Lemma7 and inequality (12), it follows that
(58)2ℓαxMΩ,Λiϱfjx·χℓx≲1xn∫Rj2ℓαxΛx−ΛyiΩx−yfydy·χℓx≲1xn∫Rj2ℓαx2j−jαyΛx−ΛyiΩx−yfydy·χℓx2ℓ−jα+xnΛx−ΛBji∫Rj2jαyΩx−yfydy+∫Rj2jαyΩx−yΛy−ΛBjifydy·χℓx.Applying Hölder’s inequality (12), the inequality (35), and Lemmas 4–8, we obtain
(59)2ℓα·MΩ,ΛiϱfjxχℓLp·≲x−n2ℓ−jα+Λ−ΛBjiχℓLp·Ωx−·χjLp′·+ΛBj−ΛiΩx−·χjLp′·χℓLp·2jα·fjLp·≲x−n2ℓ−jα+Λ−ΛBjiχℓLp·Ωx−·χjLsχjLp∗·+ΛBj−ΛiχjLp∗·Ωx−·χjLsχℓLp·2jα·fjLp·≲x−n2ℓ−jα+Λ−ΛBjiχℓLp·Ωx−·χjLsχjLp∗·+Λ∗iχjLp∗·Ωx−·χjLsχℓLp·2jα·fjLp·2−ℓn2ℓ−jα++n−1/s·ℓ−jiΛ∗iχℓLp·+Λ∗iχℓLp·χℓLp′·2jα·fjLp·≲2ℓ−jα++n−1/s−nδ2ℓ−ji2jα·fjLp·Λ∗i.Hence, combining the above estimate and using the same approach as the one used for estimatingT2, we conclude that
(60)Y2≲Λ∗ifMK̇q,p·α·,σ.Finally, we estimateY3. It is clear that if x∈Rℓ,j+2≤ℓ,andy∈Rj, then x−y≍y≍2jn. By (31), the Minkowski’s inequality, and the inequality (12), we deduce, for ρ∈0,n,
(61)MΩ,Λiϱfjx·χℓx≲∫0yt−ρ∫x−y≤tΛx−ΛyiΩx−yx−yn−ρfjydy2dtt1/2·χℓx+∫y∞t−ρ∫x−y≤tΛx−ΛyiΩx−yx−yn−ρfjydy2dtt1/2·χℓx≲∫RjΛx−ΛyiΩx−yfjyx−yn−ρ·x1/2x−yρ+1/2dy·χℓx+∫RjΛx−ΛyiΩx−yfjyx−yndy·χℓx≲y−n∫RjΛx−ΛyiΩx−yfjydy·χℓx≲y−nΛ−ΛBjiΩx−·χjLp′·+ΛBj−ΛiΩx−·χjLp′·χℓxfjLp·.From this, Lemmas4–8 and (38), we deduce
(62)MΩ,ΛiϱfjxχℓLp·≲y−nΛ−ΛBjiχℓLp·Ωx−·χjLp′·+ΛBj−ΛiΩx−·χjLp′·χℓLp·fjLp·≲y−nΩLsSn−1j−ℓiχℓLp·+Λ∗iχℓLp·χjLp·fjLp·≲ΩLsSn−1fjLp·Λ∗i2l−jnδ1j−ℓi.Thus, combining the above estimates and using the same approach as for theT2 estimate, we deduce that
(63)Y2≲Λ∗ifMK̇q,p·α·,σq.Summing up the estimates ofY1,Y2, and Y3, we conclude that
(64)MΩ,ΛiϱfMK̇q,p·α·,σ≲Y1+Y2+Y3≲Λ∗ifMK̇q,p·α·,σ.The proof for Theorem11 is finished.



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*Source: 7209977-2022-09-14.xml*