Research ArticleMarcinkiewicz Integral Operators and Commutators onHerz Spaces with Variable Exponents

Liwei Wang

School of Mathematics and Physics, Anhui Polytechnic University, Wuhu 241000, China

Correspondence should be addressed to Liwei Wang; wangliwei8013@163.com

Received 26 July 2014; Accepted 21 September 2014; Published 15 October 2014

Academic Editor: Dashan Fan

Copyright © 2014 Liwei Wang. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Our aim in this paper is to give the boundedness of theMarcinkiewicz integral 𝜇ΩonHerz spaces �̇�𝛼(⋅),𝑞

𝑝(⋅)(R𝑛) and𝐾𝛼(⋅),𝑞

𝑝(⋅)(R𝑛), where

the two main indices are variable. Meanwhile, we consider the boundedness of the higher order commutator 𝜇𝑚Ω,𝑏

generated by 𝜇Ω

and a function 𝑏 in BMO(R𝑛) on these spaces.

1. Introduction

Let S𝑛−1 be the unit sphere in R𝑛 (𝑛 ≥ 2) equipped withthe normalized Lebesgue measure 𝑑𝜎(𝑥). Suppose that Ω ishomogeneous of degree zero on R𝑛 and has mean zero onS𝑛−1, that is,

∫S𝑛−1

Ω(𝑥) 𝑑𝜎 (𝑥

) = 0. (1)

Then the Marcinkiewicz integral 𝜇Ωin higher dimension is

defined by

𝜇Ω(𝑓) (𝑥) = (∫

∞

0

𝐹Ω,𝑡 (𝑓) (𝑥)2 𝑑𝑡

𝑡3)

1/2

, (2)

where

𝐹Ω,𝑡

(𝑓) (𝑥) = ∫|𝑥−𝑦|≤𝑡

Ω(𝑥 − 𝑦)

𝑥 − 𝑦𝑛−1

𝑓 (𝑦) 𝑑𝑦. (3)

Denote by N the set of all positive integer numbers. Let𝑚 ∈ N and 𝑏 ∈ BMO(R𝑛); the higher order commutator 𝜇𝑚

Ω,𝑏

is defined by

𝜇𝑚

Ω,𝑏(𝑓) (𝑥) = (∫

∞

0

𝐹

𝑚

Ω,𝑏,𝑡(𝑓) (𝑥)

2 𝑑𝑡

𝑡3)

1/2

, (4)

where

𝐹𝑚

Ω,𝑏,𝑡(𝑓) (𝑥) = ∫

|𝑥−𝑦|≤𝑡

Ω(𝑥 − 𝑦)

𝑥 − 𝑦𝑛−1

(𝑏 (𝑥) − 𝑏 (𝑦))𝑚𝑓 (𝑦) 𝑑𝑦.

(5)

Stein [1] defined the operator 𝜇Ωand proved that if Ω ∈

Lip𝛾(S𝑛−1) (0 < 𝛾 ≤ 1), then 𝜇

Ωis of type (𝑝, 𝑝) (1 < 𝑝 ≤ 2)

and of weak type (1, 1). Benedek et al. [2] showed that 𝜇Ωis

of type (𝑝, 𝑝) (1 < 𝑝 < ∞) withΩ ∈ 𝐶1(S𝑛−1). Ding et al. [3]improved the previous results to the case of Ω ∈ 𝐻1(S𝑛−1),where𝐻1(S𝑛−1) denotes the Hardy space on S𝑛−1. Obviously,𝜇

1

Ω,𝑏= [𝑏, 𝜇

Ω], which was defined by Torchinsky and Wang

in [4]; moreover, they proved that if Ω ∈ Lip𝛾(S𝑛−1) (0 <

𝛾 ≤ 1), then [𝑏, 𝜇Ω] is bounded on 𝐿𝑝(R𝑛) (1 < 𝑝 < ∞).

Ding et al. [5] weakened the smoothness of the kernel to arough kernel and showed that if Ω ∈ 𝐿𝑞(S𝑛−1) (1 < 𝑞 ≤ ∞),then [𝑏, 𝜇

Ω] is of type (𝑝, 𝑝) (1 < 𝑝 < ∞). Ding et al. [6]

established the weighted weak 𝐿log𝐿 type estimates for 𝜇𝑚Ω,𝑏

when Ω ∈ Lip𝛾(S𝑛−1) (0 < 𝛾 ≤ 1). Recently, Zhang [7]

improved the previous result and proved that 𝜇𝑚Ω,𝑏

enjoys thesame weighted weak 𝐿log𝐿 type estimates when the kernelΩ satisfies a kind of Dini’s conditions. For further details onrecent developments on this field, we refer the readers to [8, 9]and references therein.

Function spaces with variable exponents were intensivelystudied during the past 20 years, due to their applicationsto PDE with nonstandard growth conditions and so on; wemention [10, 11], for instance. Since the fundamental paper[12] by Kováčik and Rákosnı́k appeared in 1991, the Lebesguespaces with variable exponent 𝐿𝑝(⋅)(R𝑛) have attracted a greatattention and many interesting results have been obtained;

Hindawi Publishing CorporationJournal of Function SpacesVolume 2014, Article ID 430365, 9 pageshttp://dx.doi.org/10.1155/2014/430365

2 Journal of Function Spaces

see [13–15]. Izuki [16, 17] defined the Herz spaces �̇�𝛼,𝑞𝑝(⋅)

(R𝑛)

and 𝐾𝛼,𝑞𝑝(⋅)

(R𝑛) with variable exponent 𝑝 but fixed 𝛼 ∈ R and𝑞 ∈ (0,∞]. Wang et al. [18, 19] obtained the boundednessof 𝜇

Ωand [𝑏, 𝜇

Ω] on �̇�𝛼,𝑞

𝑝(⋅)(R𝑛) and 𝐾𝛼,𝑞

𝑝(⋅)(R𝑛). Almeida and

Drihem [20] established the boundedness of a wide class ofsublinear operators, which includes maximal, potential, andCalderón-Zygmund operators, on Herz spaces �̇�𝛼(⋅),𝑞

𝑝(⋅)(R𝑛)

and 𝐾𝛼(⋅),𝑞𝑝(⋅)

(R𝑛), where the two main exponents 𝛼 and 𝑝 areboth variable. In this paper we will give boundedness resultsfor 𝜇

Ωand 𝜇𝑚

Ω,𝑏on Herz spaces �̇�𝛼(⋅),𝑞

𝑝(⋅)(R𝑛) and𝐾𝛼(⋅),𝑞

𝑝(⋅)(R𝑛).

For brevity, |𝐸| denotes the Lebesgue measure for ameasurable set 𝐸 ⊂ R𝑛. 𝑓

𝐸denotes the integral average of

𝑓 on 𝐸, that is, 𝑓𝐸= |𝐸|

−1∫

𝐸𝑓(𝑥)𝑑𝑥. 𝑝(⋅) stands for the

conjugate exponent 1/𝑝(⋅) + 1/𝑝(⋅) = 1. 𝐵(𝑥, 𝑟) = {𝑦 ∈R𝑛 : |𝑥 − 𝑦| < 𝑟}. 𝐶 denotes a positive constant, which mayhave different values even in the same line. 𝑓 ≲ 𝑔means that𝑓 ≤ 𝐶𝑔, and 𝑓 ≈ 𝑔means that 𝑓 ≲ 𝑔 ≲ 𝑓.

2. Preliminaries and Main Results

Let 𝐸 ⊂ R𝑛 with |𝐸| > 0, and let 𝑝(⋅) : 𝐸 → [1,∞) be ameasurable function. Let us first recall some definitions andnotations.

Definition 1. The Lebesgue space with variable exponent𝐿

𝑝(⋅)(𝐸) is defined by

𝐿𝑝(⋅)

(𝐸)

= {𝑓 is measurable : ∫𝐸

(

𝑓 (𝑥)

𝜆)

𝑝(𝑥)

𝑑𝑥 < ∞

for some constant 𝜆 > 0} .

(6)

This is a Banach space with the Luxemburg norm

𝑓𝐿𝑝(⋅)(𝐸) = inf {𝜆 > 0 : ∫

𝐸

(

𝑓 (𝑥)

𝜆)

𝑝(𝑥)

𝑑𝑥 ≤ 1} . (7)

Let 𝑓 ∈ 𝐿1loc(𝐸); the Hardy-Littlewood maximal operator𝑀 is defined by

𝑀𝑓(𝑥) = sup𝑟>0

𝑟−𝑛∫

𝐵(𝑥,𝑟)∩𝐸

𝑓 (𝑦) 𝑑𝑦. (8)

Denote

𝑝−= ess inf {𝑝 (𝑥) : 𝑥 ∈ 𝐸} ,

𝑝+= ess sup {𝑝 (𝑥) : 𝑥 ∈ 𝐸} ,

P (𝐸) = {𝑝 (⋅) : 𝑝− > 1, 𝑝+ < ∞} ,

B (𝐸) = {𝑝 (⋅) ∈ P (𝐸) : 𝑀 is bounded on 𝐿𝑝(⋅) (𝐸)} .(9)

Let 𝐵𝑘= {𝑥 ∈ R𝑛 : |𝑥| ≤ 2𝑘}, 𝑅

𝑘= 𝐵

𝑘\𝐵

𝑘−1, and 𝜒

𝑘= 𝜒

𝑅𝑘

be the characteristic function of the set 𝑅𝑘for 𝑘 ∈ Z. For𝑚 ∈

N, one denotes 𝜒𝑚= 𝜒

𝑅𝑚

if 𝑚 ≥ 1, and 𝜒0= 𝜒

𝐵0

. By ℓ𝑞 (0 <𝑞 ≤ ∞), we denote the discrete Lebesgue space equippedby the usual quasinorm.

Definition 2. Let 0 < 𝑞 ≤ ∞, 𝑝(⋅) ∈ P(R𝑛), and 𝛼(⋅) : R𝑛 →R with 𝛼 ∈ 𝐿∞(R𝑛).

(1) The homogeneous Herz space �̇�𝛼(⋅),𝑞𝑝(⋅)

(R𝑛) is definedby

�̇�𝛼(⋅),𝑞

𝑝(⋅)(R

𝑛) = {𝑓 ∈ 𝐿

𝑝(⋅)

loc (R𝑛\ {0}) :

𝑓�̇�𝛼(⋅),𝑞

𝑝(⋅)(R𝑛)

< ∞} ,

(10)

where𝑓�̇�𝛼(⋅),𝑞

𝑝(⋅)(R𝑛)

={2

𝛼(⋅)𝑘𝑓𝜒

𝑘

𝐿𝑝(⋅)(R𝑛)}

∞

𝑘=−∞

ℓ𝑞(Z). (11)

(2) The inhomogeneous Herz space𝐾𝛼(⋅),𝑞𝑝(⋅)

(R𝑛) is definedby

𝐾𝛼(⋅),𝑞

𝑝(⋅)(R

𝑛) = {𝑓 ∈ 𝐿

𝑝(⋅)

loc (R𝑛) :

𝑓𝐾𝛼(⋅),𝑞

𝑝(⋅)(R𝑛)

< ∞} , (12)

where𝑓𝐾𝛼(⋅),𝑞

𝑝(⋅)(R𝑛)

={2

𝛼(⋅)𝑚𝑓𝜒

𝑚

𝐿𝑝(⋅)(R𝑛)}

∞

𝑚=0

ℓ𝑞(N), (13)

with the usual modification when 𝑞 = ∞.

Remark 3. It is obvious that if 0 < 𝑞1≤ 𝑞

2≤ ∞, then

�̇�𝛼(⋅),𝑞1

𝑝(⋅)(R𝑛) ⊂ �̇�

𝛼(⋅),𝑞2

𝑝(⋅)(R𝑛) and 𝐾𝛼(⋅),𝑞1

𝑝(⋅)(R𝑛) ⊂ 𝐾

𝛼(⋅),𝑞2

𝑝(⋅)(R𝑛). If

both 𝛼(⋅) and 𝑝(⋅) are constants, then �̇�𝛼(⋅),𝑞𝑝(⋅)

(R𝑛) = �̇�𝛼,𝑞

𝑝 (R𝑛)

and 𝐾𝛼(⋅),𝑞𝑝(⋅)

(R𝑛) = 𝐾𝛼,𝑞

𝑝 (R𝑛) are classical Herz spaces; see

[21, 22].

Definition 4. A function 𝛼(⋅) : R𝑛 → R is called log-Höldercontinuous at the origin, if there exists a constant 𝐶log > 0such that

|𝛼 (𝑥) − 𝛼 (0)| ≤𝐶log

log (𝑒 + 1/ |𝑥|), (14)

for all 𝑥 ∈ R𝑛. If, for some 𝛼∞∈ R and 𝐶log > 0, there holds

𝛼 (𝑥) − 𝛼∞ ≤

𝐶log

log (𝑒 + |𝑥|)(15)

for all 𝑥 ∈ R𝑛, then 𝛼(⋅) is called log-Hölder continuous atinfinity.

Let one denote

{ℎ𝑘}ℓ𝑞

>(𝐿𝑝(⋅)

)= (∑

𝑘⩾0

ℎ𝑘

𝑞

𝐿𝑝(⋅))

1/𝑞

,

{ℎ𝑘}ℓ𝑞

<(𝐿𝑝(⋅)

)= (∑

𝑘

Journal of Function Spaces 3

for sequences {ℎ𝑘}𝑘∈Z ofmeasurable functions (with the usual

modification when 𝑞 = ∞).

Proposition 5 (see [20]). Let 0 < 𝑞 ≤ ∞, 𝑝(⋅) ∈ P(R𝑛),and 𝛼(⋅) ∈ 𝐿∞(R𝑛). If 𝛼(⋅) is log-Hölder continuous both at theorigin and at infinity, then

𝑓�̇�𝛼(⋅),𝑞

𝑝(⋅)(R𝑛)

≈{2

𝛼(0)𝑘𝑓𝜒

𝑘}ℓ𝑞

<(𝐿𝑝(⋅)

)

+{2

𝛼∞

𝑘𝑓𝜒

𝑘}ℓ𝑞

>(𝐿𝑝(⋅)

).

(17)

Before stating themain results of this paper, we introducesome key lemmas that will be used later.

Lemma 6 (generalized Hölder’s inequality [12]). Let 𝑝(⋅) ∈P(R𝑛); if 𝑓 ∈ 𝐿𝑝(⋅)(R𝑛) and 𝑔 ∈ 𝐿𝑝

(⋅)(R𝑛), then

∫R𝑛

𝑓 (𝑥) 𝑔 (𝑥) 𝑑𝑥 ≤ 𝑟𝑝

𝑓𝐿𝑝(⋅)(R𝑛)

𝑔𝐿𝑝(⋅)

(R𝑛), (18)

where 𝑟𝑝= 1 + 1/𝑝

−− 1/𝑝

+.

We remark that the following Lemmas 7–9 were shown inIzuki [17, 23], and Lemma 10 was considered by Wang et al.in [18].

Lemma 7. Let 𝑝(⋅) ∈ B(R𝑛); then one has, for all balls 𝐵 inR𝑛,

1

|𝐵|

𝜒𝐵𝐿𝑝(⋅)(R𝑛)

𝜒𝐵𝐿𝑝(⋅)

(R𝑛)≲ 1. (19)

Lemma 8. Let 𝑝(⋅) ∈ B(R𝑛); then one has, for all balls 𝐵 inR𝑛 and all measurable subsets 𝑆 ⊂ 𝐵,𝜒𝑆

𝐿𝑝(⋅)(R𝑛)𝜒𝐵

𝐿𝑝(⋅)(R𝑛)

≲ (|𝑆|

|𝐵|)

𝛿1

,

𝜒𝑆𝐿𝑝(⋅)

(R𝑛)𝜒𝐵

𝐿𝑝(⋅)

(R𝑛)

≲ (|𝑆|

|𝐵|)

𝛿2

, (20)

where 𝛿1and 𝛿

2are constants with 0 < 𝛿

1, 𝛿

2< 1.

Lemma 9. Let 𝑚 ∈ N, 𝑏 ∈ BMO(R𝑛), and 𝑘 > 𝑖 (𝑘, 𝑖 ∈ N);then one has

sup𝐵⊂R𝑛

1𝜒𝐵

𝐿𝑝(⋅)(R𝑛)

(𝑏 − 𝑏

𝐵)𝑚𝜒

𝐵

𝐿𝑝(⋅)(R𝑛)≈ ‖𝑏‖

𝑚

BMO,

(𝑏 − 𝑏

𝐵𝑖

)𝑚

𝜒𝐵𝑘

𝐿𝑝(⋅)(R𝑛)≲ (𝑘 − 𝑖)

𝑚‖𝑏‖

𝑚

BMO𝜒

𝐵𝑘

𝐿𝑝(⋅)(R𝑛).

(21)

Lemma 10. Let Ω ∈ Lip𝛾(S𝑛−1) (0 < 𝛾 ≤ 1), 𝑏 ∈ BMO(R𝑛),

and 𝑝(⋅) ∈ B(R𝑛); then one has𝜇Ω(𝑓)

𝐿𝑝(⋅)(R𝑛) ≲𝑓𝐿𝑝(⋅)(R𝑛),

𝜇

𝑚

Ω,𝑏(𝑓)

𝐿𝑝(⋅)(R𝑛)≲ ‖𝑏‖

𝑚

BMO𝑓𝐿𝑝(⋅)(R𝑛).

(22)

Our results in this paper can be stated as follows.

Theorem 11. Let Ω ∈ Lip𝛾(S𝑛−1) (0 < 𝛾 ≤ 1), 0 < 𝑞 ≤ ∞,

and 𝑝(⋅) ∈ B(R𝑛). And let 𝛼(⋅) ∈ 𝐿∞(R𝑛) be log-Hölder

continuous both at the origin and at infinity, such that −𝑛𝛿1<

𝛼(0) ≤ 𝛼∞

< 𝑛𝛿2, where 0 < 𝛿

1, 𝛿

2< 1 are the constants

appearing in Lemma 8; then the operator 𝜇Ωis bounded on

�̇�𝛼(⋅),𝑞

𝑝(⋅)(R𝑛) and 𝐾𝛼(⋅),𝑞

𝑝(⋅)(R𝑛).

Theorem 12. Let Ω ∈ Lip𝛾(S𝑛−1) (0 < 𝛾 ≤ 1), 𝑏 ∈

BMO(R𝑛), 0 < 𝑞 ≤ ∞, and 𝑝(⋅) ∈ B(R𝑛). And let 𝛼(⋅) ∈𝐿

∞(R𝑛) be log-Hölder continuous both at the origin and at

infinity, such that −𝑛𝛿1< 𝛼(0) ≤ 𝛼

∞< 𝑛𝛿

2, where 0 < 𝛿

1,

𝛿2< 1 are the constants appearing in Lemma 8; then the

higher order commutator 𝜇𝑚Ω,𝑏

is bounded on �̇�𝛼(⋅),𝑞𝑝(⋅)

(R𝑛) and𝐾

𝛼(⋅),𝑞

𝑝(⋅)(R𝑛).

Remark 13. If 𝛼(⋅) ≡ 𝛼 is constant, then the statementscorresponding toTheorems 11 and 12 can be found in [19, 24].We consider only 0 < 𝑞 < ∞ in Section 3. The arguments aresimilar in the case 𝑞 = ∞.

3. Proofs of the Theorems

In this section, we prove the boundedness of 𝜇Ωand 𝜇𝑚

Ω,𝑏on

�̇�𝛼(⋅),𝑞

𝑝(⋅)(R𝑛) (the same arguments can be used in 𝐾𝛼(⋅),𝑞

𝑝(⋅)(R𝑛));

some of our decomposition techniques are similar to thoseused by Dong and Xu in [25].

Proof of Theorem 11. In view of Proposition 5, we have

𝜇Ω (𝑓)�̇�𝛼(⋅),𝑞

𝑝(⋅)(R𝑛)

≈{2

𝛼(0)𝑘𝜇

Ω(𝑓) 𝜒

𝑘}ℓ𝑞

<(𝐿𝑝(⋅)

)

+{2

𝛼∞

𝑘𝜇

Ω(𝑓) 𝜒

𝑘}ℓ𝑞

>(𝐿𝑝(⋅)

)

= 𝐼<+ 𝐼

>.

(23)

Let 𝑓 ∈ �̇�𝛼(⋅),𝑞𝑝(⋅)

(R𝑛); write

𝑓 (𝑥) =

∞

∑

𝑖=−∞

𝑓 (𝑥) 𝜒𝑖 (𝑥) =

∞

∑

𝑖=−∞

𝑓𝑖 (𝑥) . (24)

Minkowski’s inequality implies that

𝐼<= {

−1

∑

𝑘=−∞

2𝛼(0)𝑘𝑞𝜇Ω (𝑓) 𝜒𝑘

𝑞

𝐿𝑝(⋅)

(R𝑛)}

1/𝑞

≲ {

−1

∑

𝑘=−∞

2𝛼(0)𝑘𝑞

(

𝑘−2

∑

𝑖=−∞

𝜇Ω (𝑓𝑖) 𝜒𝑘𝐿𝑝(⋅)(R𝑛))

𝑞

}

1/𝑞

+ {

−1

∑

𝑘=−∞

2𝛼(0)𝑘𝑞

(

𝑘+1

∑

𝑖=𝑘−1

𝜇Ω (𝑓𝑖) 𝜒𝑘𝐿𝑝(⋅)(R𝑛))

𝑞

}

1/𝑞

+ {

−1

∑

𝑘=−∞

2𝛼(0)𝑘𝑞

(

∞

∑

𝑖=𝑘+2

𝜇Ω (𝑓𝑖) 𝜒𝑘𝐿𝑝(⋅)(R𝑛))

𝑞

}

1/𝑞

= 𝐸<+ 𝐹

<+ 𝐺

<.

(25)

4 Journal of Function Spaces

Similarly we obtain

𝐼>= {

∞

∑

𝑘=0

2𝛼∞

𝑘𝑞𝜇Ω (𝑓) 𝜒𝑘

𝑞

𝐿𝑝(⋅)

(R𝑛)}

1/𝑞

≲ {

∞

∑

𝑘=0

2𝛼∞

𝑘𝑞(

𝑘−2

∑

𝑖=−∞

𝜇Ω (𝑓𝑖) 𝜒𝑘𝐿𝑝(⋅)(R𝑛))

𝑞

}

1/𝑞

+ {

∞

∑

𝑘=0

2𝛼∞

𝑘𝑞(

𝑘+1

∑

𝑖=𝑘−1

𝜇Ω (𝑓𝑖) 𝜒𝑘𝐿𝑝(⋅)(R𝑛))

𝑞

}

1/𝑞

+ {

∞

∑

𝑘=0

2𝛼∞

𝑘𝑞(

∞

∑

𝑖=𝑘+2

𝜇Ω (𝑓𝑖) 𝜒𝑘𝐿𝑝(⋅)(R𝑛))

𝑞

}

1/𝑞

= 𝐸>+ 𝐹

>+ 𝐺

>.

(26)

Thus we get

𝜇Ω(𝑓)�̇�𝛼(⋅),𝑞

𝑝(⋅)(R𝑛)

≲ 𝐸 + 𝐹 + 𝐺, (27)

where 𝐸 = 𝐸<+ 𝐸

>, 𝐹 = 𝐹

<+ 𝐹

>, and 𝐺 = 𝐺

<+ 𝐺

>.

For 𝐹, Lemma 10 yields

𝐹 = 𝐹<+ 𝐹

>

≲ {

−1

∑

𝑘=−∞

2𝛼(0)𝑘𝑞𝑓𝑘

𝑞

𝐿𝑝(⋅)

(R𝑛)}

1/𝑞

+ {

∞

∑

𝑘=0

2𝛼∞

𝑘𝑞𝑓𝑘

𝑞

𝐿𝑝(⋅)

(R𝑛)}

1/𝑞

≲{2

𝛼(0)𝑘𝑓𝜒

𝑘}ℓ𝑞

<(𝐿𝑝(⋅)

)+{2

𝛼∞

𝑘𝑓𝜒

𝑘}ℓ𝑞

>(𝐿𝑝(⋅)

)

≈𝑓�̇�𝛼(⋅),𝑞

𝑝(⋅)(R𝑛)

.

(28)

Nowwe turn to estimate𝐸. Observe that if 𝑥 ∈ 𝑅𝑘, 𝑦 ∈ 𝑅

𝑖,

and 𝑖 ≤ 𝑘 − 2, then |𝑥 − 𝑦| ≈ |𝑥| ≈ 2𝑘 and

1

𝑥 − 𝑦2−

1

|𝑥|2

≲

𝑦

𝑥 − 𝑦3. (29)

Since Ω ∈ Lip𝛾(S𝑛−1) ⊂ 𝐿∞(S𝑛−1), by Minkowski’s

inequality and Lemma 6, we have

𝜇Ω (𝑓𝑖) (𝑥)

≲ (∫

|𝑥|

0

∫|𝑥−𝑦|≤𝑡

Ω(𝑥 − 𝑦)

|𝑥 − 𝑦|𝑛−1𝑓

𝑖(𝑦)𝑑𝑦

2

𝑑𝑡

𝑡3)

1/2

+ (∫

∞

|𝑥|

∫|𝑥−𝑦|≤𝑡

Ω(𝑥 − 𝑦)

|𝑥 − 𝑦|𝑛−1𝑓

𝑖(𝑦)𝑑𝑦

2

𝑑𝑡

𝑡3)

1/2

≲ ∫𝑅𝑖

𝑓𝑖 (𝑦)

𝑥 − 𝑦𝑛−1

(∫|𝑥−𝑦|≤𝑡,|𝑥|≥𝑡

𝑑𝑡

𝑡3)

1/2

𝑑𝑦

+ ∫𝑅𝑖

𝑓𝑖 (𝑦)

𝑥 − 𝑦𝑛−1

(∫

∞

|𝑥|

𝑑𝑡

𝑡3)

1/2

𝑑𝑦

≲ ∫𝑅𝑖

𝑓𝑖 (𝑦)

𝑥 − 𝑦𝑛−1

⋅

𝑦1/2

𝑥 − 𝑦3/2

𝑑𝑦 + ∫𝑅𝑖

𝑓𝑖 (𝑦)

𝑥 − 𝑦𝑛−1

⋅1

|𝑥|𝑑𝑦

≲ 2(𝑖−𝑘)/2

2−𝑘𝑛𝑓𝑖

𝐿𝑝(⋅)(R𝑛)𝜒𝑖

𝐿𝑝(⋅)

(R𝑛)

+ 2−𝑘𝑛𝑓𝑖

𝐿𝑝(⋅)(R𝑛)𝜒𝑖

𝐿𝑝(⋅)

(R𝑛)

≲ 2−𝑘𝑛𝑓𝑖

𝐿𝑝(⋅)(R𝑛)

𝜒

𝐵𝑖

𝐿𝑝(⋅)

(R𝑛).

(30)

Lemmas 7 and 8 lead to

𝜇Ω(𝑓𝑖)(𝑥)𝜒𝑘𝐿𝑝(⋅)(R𝑛)

≲ 2−𝑘𝑛𝑓𝑖

𝐿𝑝(⋅)(R𝑛)

𝜒

𝐵𝑖

𝐿𝑝(⋅)

(R𝑛)

𝜒

𝐵𝑘

𝐿𝑝(⋅)(R𝑛)

≲𝑓𝑖

𝐿𝑝(⋅)(R𝑛)

𝜒

𝐵𝑖

𝐿𝑝(⋅)

(R𝑛)𝜒

𝐵𝑘

𝐿𝑝(⋅)

(R𝑛)

≲ 2(𝑖−𝑘)𝑛𝛿

2𝑓𝑖

𝐿𝑝(⋅)(R𝑛).

(31)

Thus we get

𝐸<≲ {

−1

∑

𝑘=−∞

2𝑘𝛼(0)𝑞

(

𝑘−2

∑

𝑖=−∞

2(𝑖−𝑘)𝑛𝛿

2𝑓𝑖

𝐿𝑝(⋅)(R𝑛))

𝑞

}

1/𝑞

≈ {

−1

∑

𝑘=−∞

(

𝑘−2

∑

𝑖=−∞

2𝛼(0)𝑖𝑓𝑖

𝐿𝑝(⋅)(R𝑛)2(𝑖−𝑘)(𝑛𝛿

2−𝛼(0))

)

𝑞

}

1/𝑞

.

(32)

If 1 < 𝑞 < ∞, since 𝑛𝛿2− 𝛼(0) > 0, Hölder’s inequality

implies that

𝐸<≲{

{

{

−1

∑

𝑘=−∞

(

𝑘−2

∑

𝑖=−∞

2𝛼(0)𝑖𝑞𝑓𝑖

𝑞

𝐿𝑝(⋅)

(R𝑛)2

(𝑖−𝑘)(𝑛𝛿2−𝛼(0))𝑞/2

)

×(

𝑘−2

∑

𝑖=−∞

2(𝑖−𝑘)(𝑛𝛿

2−𝛼(0))𝑞

/2)

𝑞/𝑞

}

}

}

1/𝑞

≲ {

−3

∑

𝑖=−∞

2𝛼(0)𝑖𝑞𝑓𝑖

𝑞

𝐿𝑝(⋅)

(R𝑛)

−1

∑

𝑘=𝑖+2

2(𝑖−𝑘)(𝑛𝛿

2−𝛼(0))𝑞/2

}

1/𝑞

≲{2

𝛼(0)𝑘𝑓𝜒

𝑘}ℓ𝑞

<(𝐿𝑝(⋅)

)

≲𝑓�̇�𝛼(⋅),𝑞

𝑝(⋅)(R𝑛)

.

(33)

Journal of Function Spaces 5

If 0 < 𝑞 ≤ 1, by the well-known inequality

(

∞

∑

𝑖=1

𝑎𝑖)

𝑞

≤

∞

∑

𝑖=1

𝑎𝑞

𝑖(𝑎

𝑖> 0, 𝑖 = 1, 2, . . .) , (34)

we obtain

𝐸<≲ {

−1

∑

𝑘=−∞

𝑘−2

∑

𝑖=−∞

2𝛼(0)𝑖𝑞𝑓𝑖

𝑞

𝐿𝑝(⋅)

(R𝑛)2

(𝑖−𝑘)(𝑛𝛿2−𝛼(0))𝑞

}

1/𝑞

≈ {

−3

∑

𝑖=−∞

2𝛼(0)𝑖𝑞𝑓𝑖

𝑞

𝐿𝑝(⋅)

(R𝑛)

−1

∑

𝑘=𝑖+2

2(𝑖−𝑘)(𝑛𝛿

2−𝛼(0))𝑞

}

1/𝑞

≲{2

𝛼(0)𝑘𝑓𝜒

𝑘}ℓ𝑞

<(𝐿𝑝(⋅)

)

≲𝑓�̇�𝛼(⋅),𝑞

𝑝(⋅)(R𝑛)

.

(35)

Similarly we have

𝐸>≲ {

∞

∑

𝑘=0

2𝑘𝛼∞

𝑞(

𝑘−2

∑

𝑖=−∞

2(𝑖−𝑘)𝑛𝛿

2𝑓𝑖

𝐿𝑝(⋅)(R𝑛))

𝑞

}

1/𝑞

≈ {

∞

∑

𝑘=0

(

𝑘−2

∑

𝑖=−∞

2𝛼∞

𝑖𝑓𝑖𝐿𝑝(⋅)(R𝑛)2

(𝑖−𝑘)(𝑛𝛿2−𝛼∞

))

𝑞

}

1/𝑞

.

(36)

If 1 < 𝑞 < ∞, since 𝛼∞+𝑛𝛿

2> 2𝛼

∞> 2𝛼(0), then we get

𝐸>≲{

{

{

∞

∑

𝑘=0

(

𝑘−2

∑

𝑖=−∞

2𝛼∞

𝑖𝑞𝑓𝑖

𝑞

𝐿𝑝(⋅)

(R𝑛)2

(𝑖−𝑘)(𝑛𝛿2−𝛼∞

)𝑞/2)

× (

𝑘−2

∑

𝑖=−∞

2(𝑖−𝑘)(𝑛𝛿

2−𝛼∞

)𝑞

/2)

𝑞/𝑞

}

}

}

1/𝑞

≲ {

∞

∑

𝑘=0

(

𝑘−2

∑

𝑖=−∞

2𝛼∞

𝑖𝑞𝑓𝑖

𝑞

𝐿𝑝(⋅)

(R𝑛)2

(𝑖−𝑘)(𝑛𝛿2−𝛼∞

)𝑞/2)}

1/𝑞

≈ {

−2

∑

𝑖=−∞

2𝛼∞

𝑖𝑞𝑓𝑖

𝑞

𝐿𝑝(⋅)

(R𝑛)

∞

∑

𝑘=0

2(𝑖−𝑘)(𝑛𝛿

2−𝛼∞

)𝑞/2

+

∞

∑

𝑖=−1

2𝛼∞

𝑖𝑞𝑓𝑖

𝑞

𝐿𝑝(⋅)

(R𝑛)

∞

∑

𝑘=𝑖+2

2(𝑖−𝑘)(𝑛𝛿

2−𝛼∞

)𝑞/2}

1/𝑞

≲ {

−2

∑

𝑖=−∞

2𝛼(0)𝑖𝑞

2(𝛼∞

+𝑛𝛿2−2𝛼(0))𝑖𝑞/2𝑓𝑖

𝑞

𝐿𝑝(⋅)

(R𝑛)

+

∞

∑

𝑖=−1

2𝛼∞

𝑖𝑞𝑓𝑖

𝑞

𝐿𝑝(⋅)

(R𝑛)}

1/𝑞

≲ {

−2

∑

𝑖=−∞

2𝛼(0)𝑖𝑞𝑓𝑖

𝑞

𝐿𝑝(⋅)

(R𝑛)+

∞

∑

𝑖=−1

2𝛼∞

𝑖𝑞𝑓𝑖

𝑞

𝐿𝑝(⋅)

(R𝑛)}

1/𝑞

≲{2

𝛼(0)𝑘𝑓𝜒

𝑘}ℓ𝑞

<(𝐿𝑝(⋅)

)+{2

𝛼∞

𝑘𝑓𝜒

𝑘}ℓ𝑞

>(𝐿𝑝(⋅)

)

≈𝑓�̇�𝛼(⋅),𝑞

𝑝(⋅)(R𝑛)

.

(37)

If 0 < 𝑞 ≤ 1, since 𝛼(0) ≤ 𝛼∞, we obtain

𝐸>≲ {

∞

∑

𝑘=0

𝑘−2

∑

𝑖=−∞

2𝛼∞

𝑖𝑞𝑓𝑖

𝑞

𝐿𝑝(⋅)

(R𝑛)2

(𝑖−𝑘)(𝑛𝛿2−𝛼∞

)𝑞}

1/𝑞

≈ {

−2

∑

𝑖=−∞

2𝛼∞

𝑖𝑞𝑓𝑖

𝑞

𝐿𝑝(⋅)

(R𝑛)

∞

∑

𝑘=0

2(𝑖−𝑘)(𝑛𝛿

2−𝛼∞

)𝑞

+

∞

∑

𝑖=−1

2𝛼∞

𝑖𝑞𝑓𝑖

𝑞

𝐿𝑝(⋅)

(R𝑛)

∞

∑

𝑘=𝑖+2

2(𝑖−𝑘)(𝑛𝛿

2−𝛼∞

)𝑞}

1/𝑞

≲ {

−2

∑

𝑖=−∞

2𝛼(0)𝑖𝑞𝑓𝑖

𝑞

𝐿𝑝(⋅)

(R𝑛)+

∞

∑

𝑖=−1

2𝛼∞

𝑖𝑞𝑓𝑖

𝑞

𝐿𝑝(⋅)

(R𝑛)}

1/𝑞

≲{2

𝛼(0)𝑘𝑓𝜒

𝑘}ℓ𝑞

<(𝐿𝑝(⋅)

)+{2

𝛼∞

𝑘𝑓𝜒

𝑘}ℓ𝑞

>(𝐿𝑝(⋅)

)

≈𝑓�̇�𝛼(⋅),𝑞

𝑝(⋅)(R𝑛)

.

(38)

Thus, we arrive at

𝐸 = 𝐸<+ 𝐸

>≲𝑓�̇�𝛼(⋅),𝑞

𝑝(⋅)(R𝑛)

. (39)

For 𝐺, observe that if 𝑥 ∈ 𝑅𝑘, 𝑦 ∈ 𝑅

𝑖, and 𝑖 ≥ 𝑘 + 2, then

|𝑥 − 𝑦| ≈ |𝑦| ≈ 2𝑖 and

1

𝑥 − 𝑦2−

1

𝑦2

≲|𝑥|

𝑥 − 𝑦3. (40)

From Minkowski’s inequality and Lemma 6, it followsthat𝜇Ω (𝑓𝑖) (𝑥)

≲ (∫

|𝑦|

0

∫|𝑥−𝑦|≤𝑡

Ω(𝑥 − 𝑦)

𝑥 − 𝑦𝑛−1

𝑓𝑖(𝑦)𝑑𝑦

2

𝑑𝑡

𝑡3)

1/2

+ (∫

∞

|𝑦|

∫|𝑥−𝑦|≤𝑡

Ω(𝑥 − 𝑦)

𝑥 − 𝑦𝑛−1

𝑓𝑖(𝑦)𝑑𝑦

2

𝑑𝑡

𝑡3)

1/2

≲ ∫𝑅𝑖

𝑓𝑖 (𝑦)

𝑥 − 𝑦𝑛−1

(∫|𝑥−𝑦|≤𝑡,|𝑦|≥𝑡

𝑑𝑡

𝑡3)

1/2

𝑑𝑦

+ ∫𝑅𝑖

𝑓𝑖 (𝑦)

𝑥 − 𝑦𝑛−1

(∫

∞

|𝑦|

𝑑𝑡

𝑡3)

1/2

𝑑𝑦

≲ ∫𝑅𝑖

𝑓𝑖 (𝑦)

𝑥 − 𝑦𝑛−1

⋅|𝑥|

1/2

𝑥 − 𝑦3/2

𝑑𝑦 + ∫𝑅𝑖

𝑓𝑖 (𝑦)

𝑥 − 𝑦𝑛−1

⋅1𝑦

𝑑𝑦

6 Journal of Function Spaces

≲ 2(𝑘−𝑖)/2

2−𝑖𝑛𝑓𝑖

𝐿𝑝(⋅)(R𝑛)𝜒𝑖

𝐿𝑝(⋅)

(R𝑛)

+ 𝐶2−𝑖𝑛𝑓𝑖

𝐿𝑝(⋅)(R𝑛)𝜒𝑖

𝐿𝑝(⋅)

(R𝑛)

≲ 2−𝑖𝑛𝑓𝑖

𝐿𝑝(⋅)(R𝑛)

𝜒

𝐵𝑖

𝐿𝑝(⋅)

(R𝑛).

(41)

By Lemmas 7 and 8, we have

𝜇Ω(𝑓𝑖)(𝑥)𝜒𝑘𝐿𝑝(⋅)(R𝑛)

≲ 2−𝑖𝑛𝑓𝑖

𝐿𝑝(⋅)(R𝑛)

𝜒

𝐵𝑖

𝐿𝑝(⋅)

(R𝑛)

𝜒

𝐵𝑘

𝐿𝑝(⋅)(R𝑛)

≲𝑓𝑖

𝐿𝑝(⋅)(R𝑛)

𝜒

𝐵𝑘

𝐿𝑝(⋅)(R𝑛)𝜒

𝐵𝑖

𝐿𝑝(⋅)(R𝑛)

≲ 2(𝑘−𝑖)𝑛𝛿

1𝑓𝑖

𝐿𝑝(⋅)(R𝑛).

(42)

Thus we get

𝐺<≲ {

−1

∑

𝑘=−∞

2𝑘𝛼(0)𝑞

(

∞

∑

𝑖=𝑘+2

2(𝑘−𝑖)𝑛𝛿

1𝑓𝑖

𝐿𝑝(⋅)(R𝑛))

𝑞

}

1/𝑞

≈ {

−1

∑

𝑘=−∞

(

∞

∑

𝑖=𝑘+2

2𝛼(0)𝑖𝑓𝑖

𝐿𝑝(⋅)(R𝑛)2(𝑘−𝑖)(𝑛𝛿

1+𝛼(0))

)

𝑞

}

1/𝑞

,

𝐺>≲ {

∞

∑

𝑘=0

(

∞

∑

𝑖=𝑘+2

2𝛼∞

𝑖𝑓𝑖𝐿𝑝(⋅)(R𝑛)2

(𝑘−𝑖)(𝑛𝛿1+𝛼∞

))

𝑞

}

1/𝑞

.

(43)

Using the same arguments as that for 𝐸<and 𝐸

>, we get

𝐺 = 𝐺<+ 𝐺

>≲𝑓�̇�𝛼(⋅),𝑞

𝑝(⋅)(R𝑛)

. (44)

Hence the proof of Theorem 11 is completed.

Proof of Theorem 12. We apply Proposition 5 again and get

𝜇

𝑚

Ω,𝑏(𝑓)

�̇�𝛼(⋅),𝑞

𝑝(⋅)(R𝑛)

≈{2

𝛼(0)𝑘𝜇

𝑚

Ω,𝑏(𝑓) 𝜒

𝑘}ℓ𝑞

<(𝐿𝑝(⋅)

)

+{2

𝛼∞

𝑘𝜇

𝑚

Ω,𝑏(𝑓) 𝜒

𝑘}ℓ𝑞

>(𝐿𝑝(⋅)

)

= 𝐽<+ 𝐽

>.

(45)

Let 𝑓 ∈ �̇�𝛼(⋅),𝑞𝑝(⋅)

(R𝑛), and write

𝑓 (𝑥) =

∞

∑

𝑖=−∞

𝑓 (𝑥) 𝜒𝑖 (𝑥) =

∞

∑

𝑖=−∞

𝑓𝑖 (𝑥) . (46)

By Minkowski’s inequality, we have

𝐽<= {

−1

∑

𝑘=−∞

2𝛼(0)𝑘𝑞

𝜇𝑚

Ω,𝑏(𝑓) 𝜒

𝑘

𝑞

𝐿𝑝(⋅)

(R𝑛)}

1/𝑞

≲ {

−1

∑

𝑘=−∞

2𝛼(0)𝑘𝑞

(

𝑘−2

∑

𝑖=−∞

𝜇

𝑚

Ω,𝑏(𝑓

𝑖) 𝜒

𝑘

𝐿𝑝(⋅)(R𝑛))

𝑞

}

1/𝑞

+ {

−1

∑

𝑘=−∞

2𝛼(0)𝑘𝑞

(

𝑘+1

∑

𝑖=𝑘−1

𝜇

𝑚

Ω,𝑏(𝑓

𝑖) 𝜒

𝑘

𝐿𝑝(⋅)(R𝑛))

𝑞

}

1/𝑞

+ {

−1

∑

𝑘=−∞

2𝛼(0)𝑘𝑞

(

∞

∑

𝑖=𝑘+2

𝜇

𝑚

Ω,𝑏(𝑓

𝑖) 𝜒

𝑘

𝐿𝑝(⋅)(R𝑛))

𝑞

}

1/𝑞

= 𝑈<+ 𝑉

<+𝑊

<.

(47)

By the same way, we obtain

𝐽>= {

∞

∑

𝑘=0

2𝛼∞

𝑘𝑞𝜇

𝑚

Ω,𝑏(𝑓) 𝜒

𝑘

𝑞

𝐿𝑝(⋅)

(R𝑛)}

1/𝑞

≲ {

∞

∑

𝑘=0

2𝛼∞

𝑘𝑞(

𝑘−2

∑

𝑖=−∞

𝜇

𝑚

Ω,𝑏(𝑓

𝑖) 𝜒

𝑘

𝐿𝑝(⋅)(R𝑛))

𝑞

}

1/𝑞

+ {

∞

∑

𝑘=0

2𝛼∞

𝑘𝑞(

𝑘+1

∑

𝑖=𝑘−1

𝜇

𝑚

Ω,𝑏(𝑓

𝑖) 𝜒

𝑘

𝐿𝑝(⋅)(R𝑛))

𝑞

}

1/𝑞

+ {

∞

∑

𝑘=0

2𝛼∞

𝑘𝑞(

∞

∑

𝑖=𝑘+2

𝜇

𝑚

Ω,𝑏(𝑓

𝑖) 𝜒

𝑘

𝐿𝑝(⋅)(R𝑛))

𝑞

}

1/𝑞

= 𝑈>+ 𝑉

>+𝑊

>.

(48)

Thus, we have

𝜇

𝑚

Ω,𝑏(𝑓)

�̇�𝛼(⋅),𝑞

𝑝(⋅)(R𝑛)

≲ 𝑈 + 𝑉 +𝑊, (49)

where 𝑈 = 𝑈<+ 𝑈

>, 𝑉 = 𝑉

<+ 𝑉

>, and𝑊 = 𝑊

<+𝑊

>.

For 𝑉, by Lemma 10, we have

𝑉 = 𝑉<+ 𝑉

>

≲ {

−1

∑

𝑘=−∞

2𝛼(0)𝑘𝑞𝑓𝑘

𝑞

𝐿𝑝(⋅)

(R𝑛)}

1/𝑞

+ {

∞

∑

𝑘=0

2𝛼∞

𝑘𝑞𝑓𝑘

𝑞

𝐿𝑝(⋅)

(R𝑛)}

1/𝑞

≲{2

𝛼(0)𝑘𝑓𝜒

𝑘}ℓ𝑞

<(𝐿𝑝(⋅)

)+{2

𝛼∞

𝑘𝑓𝜒

𝑘}ℓ𝑞

>(𝐿𝑝(⋅)

)

≈𝑓�̇�𝛼(⋅),𝑞

𝑝(⋅)(R𝑛)

.

(50)

Journal of Function Spaces 7

For 𝑈, observe that if 𝑥 ∈ 𝑅𝑘, 𝑦 ∈ 𝑅

𝑖, and 𝑖 ≤ 𝑘 − 2, then

𝜇

𝑚

Ω,𝑏(𝑓

𝑖) (𝑥)

≲ (∫

|𝑥|

0

∫|𝑥−𝑦|≤𝑡

Ω(𝑥 − 𝑦)

𝑥 − 𝑦𝑛−1

[𝑏 (𝑥) − 𝑏 (𝑦)]𝑚𝑓

𝑖(𝑦)𝑑𝑦

2

𝑑𝑡

𝑡3)

1/2

+ (∫

∞

|𝑥|

∫|𝑥−𝑦|≤𝑡

Ω(𝑥 − 𝑦)

𝑥 − 𝑦𝑛−1

[𝑏 (𝑥) − 𝑏 (𝑦)]𝑚𝑓

𝑖(𝑦)𝑑𝑦

2

𝑑𝑡

𝑡3)

1/2

≲ ∫𝑅𝑖

𝑏 (𝑥) − 𝑏 (𝑦)𝑚 𝑓𝑖 (𝑦)

𝑥 − 𝑦

𝑛−1

(∫|𝑥−𝑦|≤𝑡,|𝑥|≥𝑡

𝑑𝑡

𝑡3)

1/2

𝑑𝑦

+ ∫𝑅𝑖

𝑏 (𝑥) − 𝑏 (𝑦)𝑚 𝑓𝑖 (𝑦)

𝑥 − 𝑦

𝑛−1

(∫

∞

|𝑥|

𝑑𝑡

𝑡3)

1/2

𝑑𝑦

≲ ∫𝑅𝑖

𝑏 (𝑥) − 𝑏 (𝑦)𝑚 𝑓𝑖 (𝑦)

𝑥 − 𝑦

𝑛−1

⋅

𝑦1/2

𝑥 − 𝑦3/2

𝑑𝑦

+ ∫𝑅𝑖

𝑏 (𝑥) − 𝑏 (𝑦)𝑚 𝑓𝑖 (𝑦)

𝑥 − 𝑦

𝑛−1

⋅1

|𝑥|𝑑𝑦

≲ 2(𝑖−𝑘)/2

2−𝑘𝑛

∫𝑅𝑖

𝑏 (𝑥) − 𝑏 (𝑦)𝑚 𝑓𝑖 (𝑦)

𝑑𝑦

+ 2−𝑘𝑛

∫𝑅𝑖

𝑏 (𝑥) − 𝑏 (𝑦)𝑚 𝑓𝑖 (𝑦)

𝑑𝑦

≲ 2−𝑘𝑛

∫𝑅𝑖

𝑏 (𝑥) − 𝑏 (𝑦)𝑚 𝑓𝑖 (𝑦)

𝑑𝑦

≲ 2−𝑘𝑛

𝑚

∑

𝑗=0

𝐶𝑗

𝑚

𝑏 (𝑥) − 𝑏𝐵

𝑖

𝑚−𝑗

∫𝑅𝑖

𝑏𝐵𝑖

− 𝑏 (𝑦)

𝑗 𝑓𝑖 (𝑦) 𝑑𝑦

≲ 2−𝑘𝑛𝑓𝑖

𝐿𝑝(⋅)(R𝑛)

𝑚

∑

𝑗=0

𝐶𝑗

𝑚

𝑏 (𝑥) − 𝑏𝐵

𝑖

𝑚−𝑗(𝑏

𝐵𝑖

− 𝑏)𝑗𝜒

𝑖

𝐿𝑝(⋅)

(R𝑛).

(51)

An application of Lemmas 7, 8, and 10 gives

𝜇

𝑚

Ω,𝑏(𝑓

𝑖)𝜒

𝑘

𝐿𝑝(⋅)(R𝑛)

≲ 2−𝑘𝑛𝑓𝑖

𝐿𝑝(⋅)(R𝑛)

×

𝑚

∑

𝑗=0

𝐶𝑗

𝑚

(𝑏(𝑥) − 𝑏

𝐵𝑖

)𝑚−𝑗

𝜒𝑘

𝐿𝑝(⋅)(R𝑛)

(𝑏

𝐵𝑖

− 𝑏)𝑗𝜒

𝑖

𝐿𝑝(⋅)

(R𝑛)

≲ 2−𝑘𝑛𝑓𝑖

𝐿𝑝(⋅)(𝑅𝑛)

×

𝑚

∑

𝑗=0

𝐶𝑗

𝑚(𝑘 − 𝑖)

𝑚−𝑗‖𝑏‖

𝑚−𝑗

BMO𝜒

𝐵𝑘

𝐿𝑝(⋅)(R𝑛)‖𝑏‖

𝑗

BMO𝜒

𝐵𝑖

𝐿𝑝(⋅)

(R𝑛)

≲ (𝑘 − 𝑖 + 1)𝑚2

−𝑘𝑛𝑓𝑖𝐿𝑝(⋅)(R𝑛)

𝜒

𝐵𝑘

𝐿𝑝(⋅)(R𝑛)

𝜒

𝐵𝑖

𝐿𝑝(⋅)

(R𝑛)

≲ (𝑘 − 𝑖 + 1)𝑚𝑓𝑖

𝐿𝑝(⋅)(R𝑛)

𝜒

𝐵𝑖

𝐿𝑝(⋅)

(R𝑛)𝜒

𝐵𝑘

𝐿𝑝(⋅)

(R𝑛)

≲ (𝑘 − 𝑖 + 1)𝑚2

(𝑖−𝑘)𝑛𝛿2𝑓𝑖

𝐿𝑝(⋅)(R𝑛).

(52)For convenience below we put 𝜎 = 𝑛𝛿

2− 𝛼(0); if 1 < 𝑞 <

∞, then we use Hölder’s inequality and obtain

𝑈<≲ {

−1

∑

𝑘=−∞

(

𝑘−2

∑

𝑖=−∞

2𝛼(0)𝑖𝑓𝑖

𝐿𝑝(⋅)(R𝑛)(𝑘 − 𝑖 + 1)𝑚2

(𝑖−𝑘)𝜎)

𝑞

}

1/𝑞

≲{

{

{

−1

∑

𝑘=−∞

(

𝑘−2

∑

𝑖=−∞

2𝛼(0)𝑖𝑞𝑓𝑖

𝑞

𝐿𝑝(⋅)

(R𝑛)2

(𝑖−𝑘)𝜎𝑞/2)

× (

𝑘−2

∑

𝑖=−∞

(𝑘 − 𝑖 + 1)𝑚𝑞

2(𝑖−𝑘)𝜎𝑞

/2)

𝑞/𝑞

}

}

}

1/𝑞

≲ {

−3

∑

𝑖=−∞

2𝛼(0)𝑖𝑞𝑓𝑖

𝑞

𝐿𝑝(⋅)

(R𝑛)

−1

∑

𝑘=𝑖+2

2(𝑖−𝑘)𝜎𝑞/2

}

1/𝑞

≲{2

𝛼(0)𝑘𝑓𝜒

𝑘}ℓ𝑞

<(𝐿𝑝(⋅)

)

≲𝑓�̇�𝛼(⋅),𝑞

𝑝(⋅)(R𝑛)

.

(53)If 0 < 𝑞 ≤ 1, then we get

𝑈<≲ {

−1

∑

𝑘=−∞

𝑘−2

∑

𝑖=−∞

2𝛼(0)𝑖𝑞𝑓𝑖

𝑞

𝐿𝑝(⋅)

(R𝑛)(𝑘 − 𝑖 + 1)

𝑚𝑞2

(𝑖−𝑘)𝜎𝑞}

1/𝑞

≈ {

−3

∑

𝑖=−∞

2𝛼(0)𝑖𝑞𝑓𝑖

𝑞

𝐿𝑝(⋅)

(R𝑛)

−1

∑

𝑘=𝑖+2

(𝑘 − 𝑖 + 1)𝑚𝑞2

(𝑖−𝑘)𝜎𝑞}

1/𝑞

≲{2

𝛼(0)𝑘𝑓𝜒

𝑘}ℓ𝑞

<(𝐿𝑝(⋅)

)

≲𝑓�̇�𝛼(⋅),𝑞

𝑝(⋅)(R𝑛)

.

(54)Similarly, we put 𝜂 = 𝑛𝛿

2− 𝛼

∞; if 1 < 𝑞 < ∞, by Hölder’s

inequality, we obtain

𝑈>≲ {

∞

∑

𝑘=0

(

𝑘−2

∑

𝑖=−∞

2𝛼∞

𝑖𝑓𝑖𝐿𝑝(⋅)(R𝑛)(𝑘 − 𝑖 + 1)

𝑚2

(𝑖−𝑘)𝜂)

𝑞

}

1/𝑞

≲{

{

{

∞

∑

𝑘=0

(

𝑘−2

∑

𝑖=−∞

2𝛼∞

𝑖𝑞𝑓𝑖

𝑞

𝐿𝑝(⋅)

(R𝑛)2

(𝑖−𝑘)𝜂𝑞/2)

× (

𝑘−2

∑

𝑖=−∞

(𝑘 − 𝑖 + 1)𝑚𝑞

2(𝑖−𝑘)𝜂𝑞

/2)

𝑞/𝑞

}

}

}

1/𝑞

≲ {

∞

∑

𝑘=0

(

𝑘−2

∑

𝑖=−∞

2𝛼∞

𝑖𝑞𝑓𝑖

𝑞

𝐿𝑝(⋅)

(R𝑛)2

(𝑖−𝑘)𝜂𝑞/2)}

1/𝑞

.

(55)

8 Journal of Function Spaces

By the same arguments as 𝐸>, we get

𝑈>≲{2

𝛼(0)𝑘𝑓𝜒

𝑘}ℓ𝑞

<(𝐿𝑝(⋅)

)+{2

𝛼∞

𝑘𝑓𝜒

𝑘}ℓ𝑞

>(𝐿𝑝(⋅)

)

≈𝑓�̇�𝛼(⋅),𝑞

𝑝(⋅)(R𝑛)

.

(56)

If 0 < 𝑞 ≤ 1, we obtain

𝑈>≲ {

∞

∑

𝑘=0

𝑘−2

∑

𝑖=−∞

2𝛼∞

𝑖𝑞𝑓𝑖

𝑞

𝐿𝑝(⋅)

(R𝑛)(𝑘 − 𝑖 + 1)

𝑚𝑞2

(𝑖−𝑘)𝜂𝑞}

1/𝑞

≈ {

−2

∑

𝑖=−∞

2𝛼∞

𝑖𝑞𝑓𝑖

𝑞

𝐿𝑝(⋅)

(R𝑛)

∞

∑

𝑘=0

(𝑘 − 𝑖 + 1)𝑚𝑞2

(𝑖−𝑘)𝜂𝑞

+

∞

∑

𝑖=−1

2𝛼∞

𝑖𝑞𝑓𝑖

𝑞

𝐿𝑝(⋅)

(R𝑛)

∞

∑

𝑘=𝑖+2

(𝑘 − 𝑖 + 1)𝑚𝑞2

(𝑖−𝑘)𝜂𝑞}

1/𝑞

≲ {

−2

∑

𝑖=−∞

2𝛼(0)𝑖𝑞𝑓𝑖

𝑞

𝐿𝑝(⋅)

(R𝑛)+

∞

∑

𝑖=−1

2𝛼∞

𝑖𝑞𝑓𝑖

𝑞

𝐿𝑝(⋅)

(R𝑛)}

1/𝑞

≲{2

𝛼(0)𝑘𝑓𝜒

𝑘}ℓ𝑞

<(𝐿𝑝(⋅)

)+{2

𝛼∞

𝑘𝑓𝜒

𝑘}ℓ𝑞

>(𝐿𝑝(⋅)

)

≈𝑓�̇�𝛼(⋅),𝑞

𝑝(⋅)(R𝑛)

.

(57)

Thus, we have𝑈 = 𝑈

<+ 𝑈

>≲𝑓�̇�𝛼(⋅),𝑞

𝑝(⋅)(R𝑛)

. (58)

For𝑊, if 𝑥 ∈ 𝑅𝑘, 𝑦 ∈ 𝑅

𝑖, and 𝑖 ≥ 𝑘+2, as in the arguments

for 𝐺 and 𝑈, we obtain𝜇

𝑚

Ω,𝑏(𝑓

𝑗)𝜒

𝑘

𝐿𝑝(⋅)(R𝑛)≲ (𝑖 − 𝑘 + 1)

𝑚2

(𝑘−𝑖)𝑛𝛿1

𝑓

𝑗

𝐿𝑝(⋅)(R𝑛). (59)

Thus we get

𝑊<

≲ {

−1

∑

𝑘=−∞

2𝑘𝛼(0)𝑞

(

∞

∑

𝑖=𝑘+2

(𝑖 − 𝑘 + 1)𝑚2

(𝑘−𝑖)𝑛𝛿1𝑓𝑖

𝐿𝑝(⋅)(R𝑛))

𝑞

}

1/𝑞

≈ {

−1

∑

𝑘=−∞

(

∞

∑

𝑖=𝑘+2

2𝛼(0)𝑖𝑓𝑖

𝐿𝑝(⋅)(R𝑛)

× (𝑖 − 𝑘 + 1)𝑚2

(𝑘−𝑖)(𝑛𝛿1+𝛼(0))

)

𝑞

}

1/𝑞

,

𝑊>

≲ {

∞

∑

𝑘=0

(

∞

∑

𝑖=𝑘+2

2𝛼∞

𝑖𝑓𝑖𝐿𝑝(⋅)(R𝑛)(𝑖 − 𝑘 + 1)

𝑚2

(𝑘−𝑖)(𝑛𝛿1+𝛼∞

))

𝑞

}

1/𝑞

.

(60)

Similar to the estimates of 𝑈<and 𝑈

>, we get

𝑊 = 𝑊<+𝑊

>≲𝑓�̇�𝛼(⋅),𝑞

𝑝(⋅)(R𝑛)

. (61)

Hence the proof of Theorem 12 is completed.

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

The author would like to thank the referees for their time andvaluable comments. This work was supported by the NSF ofChina (Grant no. 11201003) and University NSR Project ofAnhui Province (Grant no. KJ2014A087).

References

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[2] A. Benedek, A.-P. Calderón, and R. Panzone, “Convolutionoperators on Banach space valued functions,” Proceedings of theNational Academy of Sciences of the United States of America,vol. 48, pp. 356–365, 1962.

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