# Marcinkiewicz Integral Operators and Commutators on Herz Spaces with Variable Exponents

Authors: Liwei Wang
Journal: Journal of Function Spaces
 (2014)

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## Abstract

Our aim in this paper is to give the boundedness of the Marcinkiewicz integralμΩ on Herz spaces K˙p(·)α(·),q(Rn) and Kp(·)α(·),q(Rn), where the two main indices are variable. Meanwhile, we consider the boundedness of the higher order commutator μΩ,bm generated by μΩ and a function b in BMO(Rn) on these spaces.


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## Body

## 1.Introduction

LetSn-1 be the unit sphere in Rn  (n≥2) equipped with the normalized Lebesgue measure dσ(x′). Suppose that Ω is homogeneous of degree zero on Rn and has mean zero on Sn-1, that is,
(1)∫Sn-1‍Ω(x′)dσ(x′)=0.
Then the Marcinkiewicz integral μΩ in higher dimension is defined by
(2)μΩ(f)(x)=(∫0∞‍|FΩ,t(f)(x)|2dtt3)1/2,
where
(3)FΩ,t(f)(x)=∫|x-y|≤t‍Ω(x-y)|x-y|n-1f(y)dy.Denote byN the set of all positive integer numbers. Let m∈N and b∈BMO(Rn); the higher order commutator μΩ,bm is defined by
(4)μΩ,bm(f)(x)=(∫0∞‍|FΩ,b,tm(f)(x)|2dtt3)1/2,
where
(5)FΩ,b,tm(f)(x)=∫|x-y|≤t‍Ω(x-y)|x-y|n-1(b(x)-b(y))mf(y)dy.Stein [1] defined the operator μΩ and proved that if Ω∈Lip⁡γ(Sn-1)  (0<γ≤1), then μΩ is of type (p,p)  (1<p≤2) and of weak type (1,1). Benedek et al. [2] showed that μΩ is of type (p,p)  (1<p<∞) with Ω∈C1(Sn-1). Ding et al. [3] improved the previous results to the case of Ω∈H1(Sn-1), where H1(Sn-1) denotes the Hardy space on Sn-1. Obviously, μΩ,b1=[b,μΩ], which was defined by Torchinsky and Wang in [4]; moreover, they proved that if Ω∈Lip⁡γ(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 a rough kernel and showed that if Ω∈Lq(Sn-1)  (1<q≤∞), then [b,μΩ] is of type (p,p)  (1<p<∞). Ding et al. [6] established the weighted weak LlogL type estimates for μΩ,bm when Ω∈Lip⁡γ(Sn-1)  (0<γ≤1). Recently, Zhang [7] improved the previous result and proved that μΩ,bm enjoys the same weighted weak LlogL type estimates when the kernel Ω satisfies a kind of Dini’s conditions. For further details on recent developments on this field, we refer the readers to [8, 9] and references therein.Function spaces with variable exponents were intensively studied during the past 20 years, due to their applications to PDE with nonstandard growth conditions and so on; we mention [10, 11], for instance. Since the fundamental paper [12] by Kováčik and Rákosník appeared in 1991, the Lebesgue spaces with variable exponent Lp(·)(Rn) have attracted a great attention and many interesting results have been obtained; see [13–15]. Izuki [16, 17] defined the Herz spaces K˙p(·)α,q(Rn) and Kp(·)α,q(Rn) with variable exponent p but fixed α∈R and q∈(0,∞]. Wang et al. [18, 19] obtained the boundedness of μΩ and [b,μΩ] on K˙p(·)α,q(Rn) and Kp(·)α,q(Rn). Almeida and Drihem [20] established the boundedness of a wide class of sublinear operators, which includes maximal, potential, and Calderón-Zygmund operators, on Herz spaces K˙p(·)α(·),q(Rn) and Kp(·)α(·),q(Rn), where the two main exponents α and p are both variable. In this paper we will give boundedness results for μΩ and μΩ,bm on Herz spaces K˙p(·)α(·),q(Rn) and Kp(·)α(·),q(Rn).For brevity,|E| denotes the Lebesgue measure for a measurable set E⊂Rn. fE denotes 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 may have different values even in the same line. f≲g means that f≤Cg, and f≈g means that f≲g≲f.


## 2.Preliminaries and Main Results

LetE⊂Rn with |E|>0, and let p(·):E→[1,∞) be a measurable function. Let us first recall some definitions and notations.Definition 1.
The Lebesgue space with variable exponentLp(·)(E) is defined by
(6)Lp(·)(E)={fismeasurable:∫E(|f(x)|λ)p(x)dx<∞forsomeconstantλ>0fismeasurable:∫E(|f(x)|λ)p(x)dx<∞}.This is a Banach space with the Luxemburg norm(7)∥f∥Lp(·)(E)=inf⁡{λ>0:∫E‍(|f(x)|λ)p(x)dx≤1}.Letf∈Lloc⁡1(E); the Hardy-Littlewood maximal operator M is defined by
(8)Mf(x)=sup⁡r>0r-n∫B(x,r)∩E‍|f(y)|dy.Denote(9)p-=essinf⁡{p(x):x∈E},p+=esssup⁡{p(x):x∈E},P(E)={p(·):p->1,p+<∞},B(E)={p(·)∈P(E):Misboundedon⁡Lp(·)(E)}.LetBk={x∈Rn:|x|≤2k}, Rk=Bk∖Bk-1, and χk=χRk be the characteristic function of the set Rk for k∈Z. For m∈N, one denotes χm~=χRm if m≥1, and χ0~=χB0. By lq  (0<q≤∞), we denote the discrete Lebesgue space equipped by the usual quasinorm.Definition 2.
Let0<q≤∞, p(·)∈P(Rn), and α(·):Rn→R with α∈L∞(Rn).(1)
The homogeneous Herz spaceK˙p(·)α(·),q(Rn) is defined by
(10)K˙p(·)α(·),q(Rn)={f∈Lloc⁡p(·)(Rn∖{0}):∥f∥K˙p(·)α(·),q(Rn)<∞},where(11)∥f∥K˙p(·)α(·),q(Rn)=∥{∥2α(·)kfχk∥Lp(·)(Rn)}k=-∞∞∥lq(Z).(2)
The inhomogeneous Herz spaceKp(·)α(·),q(Rn) is defined by
(12)Kp(·)α(·),q(Rn)={f∈Lloc⁡p(·)(Rn):∥f∥Kp(·)α(·),q(Rn)<∞},where(13)∥f∥Kp(·)α(·),q(Rn)=∥{∥2α(·)mfχm~∥Lp(·)(Rn)}m=0∞∥lq(N),with the usual modification whenq=∞.Remark 3.
It is obvious that if0<q1≤q2≤∞, then K˙p(·)α(·),q1(Rn)⊂K˙p(·)α(·),q2(Rn) and Kp(·)α(·),q1(Rn)⊂Kp(·)α(·),q2(Rn). If both α(·) and p(·) are constants, then K˙p(·)α(·),q(Rn)=K˙pα,q(Rn) and Kp(·)α(·),q(Rn)=Kpα,q(Rn) are classical Herz spaces; see [21, 22].Definition 4.
A functionα(·):Rn→R is called log-Hölder continuous at the origin, if there exists a constant Clog⁡>0 such that
(14)|α(x)-α(0)|≤Clog⁡log⁡(e+1/|x|),
for all x∈Rn. If, for some α∞∈R and Clog⁡>0, there holds
(15)|α(x)-α∞|≤Clog⁡log⁡(e+|x|)
for all x∈Rn, then α(·) is called log-Hölder continuous at infinity.
Let one denote(16)∥{hk}∥l>q(Lp(·))=(∑k⩾0‍∥hk∥Lp(·)q)1/q,∥{hk}∥l<q(Lp(·))=(∑k<0‍∥hk∥Lp(·)q)1/q
for sequences {hk}k∈Z of measurable functions (with the usual modification when q=∞).Proposition 5 (see [20]).
Let0<q≤∞, p(·)∈P(Rn), and α(·)∈L∞(Rn). If α(·) is log-Hölder continuous both at the origin and at infinity, then
(17)∥f∥K˙p(·)α(·),q(Rn)≈∥{2α(0)kfχk}∥l<q(Lp(·))+∥{2α∞kfχk}∥l>q(Lp(·)).Before stating the main results of this paper, we introduce some key lemmas that will be used later.Lemma 6 (generalized Hölder’s inequality [12]).
Letp(·)∈P(Rn); if f∈Lp(·)(Rn) and g∈Lp′(·)(Rn), then
(18)∫Rn‍|f(x)g(x)|dx≤rp∥f∥Lp(·)(Rn)∥g∥Lp′(·)(Rn),
where rp=1+1/p--1/p+.We remark that the following Lemmas7–9 were shown in Izuki [17, 23], and Lemma 10 was considered by Wang et al. in [18].Lemma 7.
Letp(·)∈B(Rn); then one has, for all balls B in Rn,
(19)1|B|∥χB∥Lp(·)(Rn)∥χB∥Lp′(·)(Rn)≲1.Lemma 8.
Letp(·)∈B(Rn); then one has, for all balls B in Rn and all measurable subsets S⊂B,
(20)∥χS∥Lp(·)(Rn)∥χB∥Lp(·)(Rn)≲(|S||B|)δ1,∥χS∥Lp′(·)(Rn)∥χB∥Lp′(·)(Rn)≲(|S||B|)δ2,
where δ1 and δ2 are constants with 0<δ1, δ2<1.Lemma 9.
Letm∈N, b∈BMO(Rn), and k>i  (k,i∈N); then one has
(21)sup⁡B⊂Rn1∥χB∥Lp(·)(Rn)∥(b-bB)mχB∥Lp(·)(Rn)≈∥b∥BMOm,∥(b-bBi)mχBk∥Lp(·)(Rn)≲(k-i)m∥b∥BMOm∥χBk∥Lp(·)(Rn).Lemma 10.
LetΩ∈Lipγ(Sn-1)  (0<γ≤1), b∈BMO(Rn), and p(·)∈B(Rn); then one has
(22)∥μΩ(f)∥Lp(·)(Rn)≲∥f∥Lp(·)(Rn),∥μΩ,bm(f)∥Lp(·)(Rn)≲∥b∥BMOm∥f∥Lp(·)(Rn).Our results in this paper can be stated as follows.Theorem 11.
LetΩ∈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<δ1, δ2<1 are the constants appearing in Lemma 8; then the operator μΩ is bounded on K˙p(·)α(·),q(Rn) and Kp(·)α(·),q(Rn).Theorem 12.
LetΩ∈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∞<nδ2, where 0<δ1, δ2<1 are the constants appearing in Lemma 8; then the higher order commutator μΩ,bm is bounded on K˙p(·)α(·),q(Rn) and Kp(·)α(·),q(Rn).Remark 13.
Ifα(·)≡α is constant, then the statements corresponding to Theorems 11 and 12 can be found in [19, 24]. We consider only 0<q<∞ in Section 3. The arguments are similar in the case q=∞.


## 3.Proofs of the Theorems

In this section, we prove the boundedness ofμΩ and μΩ,bm on K˙p(·)α(·),q(Rn) (the same arguments can be used in Kp(·)α(·),q(Rn)); some of our decomposition techniques are similar to those used by Dong and Xu in [25].Proof of Theorem11.
In view of Proposition5, we have
(23)∥μΩ(f)∥K˙p(·)α(·),q(Rn)≈∥{2α(0)kμΩ(f)χk}∥l<q(Lp(·))+∥{2α∞kμΩ(f)χk}∥l>q(Lp(·))=I<+I>.
Letf∈K˙p(·)α(·),q(Rn); write
(24)f(x)=∑i=-∞∞‍f(x)χi(x)=∑i=-∞∞‍fi(x).
Minkowski’s inequality implies that(25)I<={∑k=-∞-1‍2α(0)kq∥μΩ(f)χk∥Lp(·)(Rn)q}1/q≲{∑k=-∞-1‍2α(0)kq(∑i=-∞k-2‍∥μΩ(fi)χk∥Lp(·)(Rn))q}1/q+{∑k=-∞-1‍2α(0)kq(∑i=k-1k+1‍∥μΩ(fi)χk∥Lp(·)(Rn))q}1/q+{∑k=-∞-1‍2α(0)kq(∑i=k+2∞‍‍∥μΩ(fi)χk∥Lp(·)(Rn))q}1/q=E<+F<+G<.
Similarly we obtain(26)I>={∑k=0∞‍2α∞kq∥μΩ(f)χk∥Lp(·)(Rn)q}1/q≲{∑k=0∞‍2α∞kq(∑i=-∞k-2‍∥μΩ(fi)χk∥Lp(·)(Rn))q}1/q+{∑k=0∞‍2α∞kq(∑i=k-1k+1‍∥μΩ(fi)χk∥Lp(·)(Rn))q}1/q+{∑k=0∞‍2α∞kq(∑i=k+2∞‍∥μΩ(fi)χk∥Lp(·)(Rn))q}1/q=E>+F>+G>.
Thus we get(27)∥μΩ(f)∥K˙p(·)α(·),q(Rn)≲E+F+G,
where E=E<+E>, F=F<+F>, and G=G<+G>.
ForF, Lemma 10 yields
(28)F=F<+F>≲{∑k=-∞-1‍2α(0)kq∥fk∥Lp(·)(Rn)q}1/q+{∑k=0∞‍2α∞kq∥fk∥Lp(·)(Rn)q}1/q≲∥{2α(0)kfχk}∥l<q(Lp(·))+∥{2α∞kfχk}∥l>q(Lp(·))≈∥f∥K˙p(·)α(·),q(Rn).
Now we turn to estimateE. Observe that if x∈Rk, y∈Ri, and i≤k-2, then |x-y|≈|x|≈2k and
(29)|1|x-y|2-1|x|2|≲|y||x-y|3.
SinceΩ∈Lip⁡γ(Sn-1)⊂L∞(Sn-1), by Minkowski’s inequality and Lemma 6, we have
(30)|μΩ(fi)(x)|≲(∫0|x|‍|∫|x-y|≤t‍Ω(x-y)|x-y|n-1fi(y)dy|2dtt3)1/2+(∫|x|∞‍|∫|x-y|≤t‍Ω(x-y)|x-y|n-1fi(y)dy|2dtt3)1/2≲∫Ri‍|fi(y)||x-y|n-1(∫|x-y|≤t,|x|≥t‍dtt3)1/2dy+∫Ri‍|fi(y)||x-y|n-1(∫|x|∞‍dtt3)1/2dy≲∫Ri‍|fi(y)||x-y|n-1·|y|1/2|x-y|3/2dy+∫Ri‍|fi(y)||x-y|n-1·1|x|dy≲2(i-k)/22-kn∥fi∥Lp(·)(Rn)∥χi∥Lp′(·)(Rn)+2-kn∥fi∥Lp(·)(Rn)∥χi∥Lp′(·)(Rn)≲2-kn∥fi∥Lp(·)(Rn)∥χBi∥Lp′(·)(Rn).
Lemmas7 and 8 lead to(31)∥μΩ(fi)(x)χk∥Lp(·)(Rn)≲2-kn∥fi∥Lp(·)(Rn)∥χBi∥Lp′(·)(Rn)∥χBk∥Lp(·)(Rn)≲∥fi∥Lp(·)(Rn)∥χBi∥Lp′(·)(Rn)∥χBk∥Lp′(·)(Rn)≲2(i-k)nδ2∥fi∥Lp(·)(Rn).
Thus we get(32)E<≲{∑k=-∞-1‍2kα(0)q(∑i=-∞k-2‍2(i-k)nδ2∥fi∥Lp(·)(Rn))q}1/q≈{∑k=-∞-1‍(∑i=-∞k-2‍2α(0)i∥fi∥Lp(·)(Rn)2(i-k)(nδ2-α(0)))q}1/q.
If1<q<∞, since nδ2-α(0)>0, Hölder’s inequality implies that
(33)E<≲{∑k=-∞-1‍(∑i=-∞k-2‍2α(0)iq∥fi∥Lp(·)(Rn)q2(i-k)(nδ2-α(0))q/2)×(∑i=-∞k-2‍2(i-k)(nδ2-α(0))q′/2)q/q′×(∑i=-∞k-2‍2(i-k)(nδ2-α(0))q′/2)q/q′}1/q≲{∑i=-∞-3‍2α(0)iq∥fi∥Lp(·)(Rn)q∑k=i+2-1‍2(i-k)(nδ2-α(0))q/2}1/q≲∥{2α(0)kfχk}∥l<q(Lp(·))≲∥f∥K˙p(·)α(·),q(Rn).
If0<q≤1, by the well-known inequality
(34)(∑i=1∞‍ai)q≤∑i=1∞‍aiq(ai>0,i=1,2,…),
we obtain
(35)E<≲{∑k=-∞-1‍∑i=-∞k-2‍2α(0)iq∥fi∥Lp(·)(Rn)q2(i-k)(nδ2-α(0))q}1/q≈{∑i=-∞-3‍2α(0)iq∥fi∥Lp(·)(Rn)q∑k=i+2-1‍2(i-k)(nδ2-α(0))q}1/q≲∥{2α(0)kfχk}∥l<q(Lp(·))≲∥f∥K˙p(·)α(·),q(Rn).
Similarly we have(36)E>≲{∑k=0∞‍2kα∞q(∑i=-∞k-2‍2(i-k)nδ2∥fi∥Lp(·)(Rn))q}1/q≈{∑k=0∞‍(∑i=-∞k-2‍2α∞i∥fi∥Lp(·)(Rn)2(i-k)(nδ2-α∞))q}1/q.
If1<q<∞, since α∞+nδ2>2α∞>2α(0), then we get
(37)E>≲{∑k=0∞‍(∑i=-∞k-2‍2α∞iq∥fi∥Lp(·)(Rn)q2(i-k)(nδ2-α∞)q/2)×(∑i=-∞k-2‍2(i-k)(nδ2-α∞)q′/2)q/q′×(∑i=-∞k-2‍2(i-k)(nδ2-α∞)q′/2)q/q′}1/q≲{∑k=0∞‍(∑i=-∞k-2‍2α∞iq∥fi∥Lp(·)(Rn)q2(i-k)(nδ2-α∞)q/2)}1/q≈{∑i=-∞-2‍2α∞iq∥fi∥Lp(·)(Rn)q∑k=0∞‍2(i-k)(nδ2-α∞)q/2+∑i=-1∞‍2α∞iq∥fi∥Lp(·)(Rn)q∑k=i+2∞‍2(i-k)(nδ2-α∞)q/2}1/q≲{∑i=-∞-2‍2α(0)iq2(α∞+nδ2-2α(0))iq/2∥fi∥Lp(·)(Rn)q+∑i=-1∞‍2α∞iq∥fi∥Lp(·)(Rn)q}1/q≲{∑i=-∞-2‍2α(0)iq∥fi∥Lp(·)(Rn)q+∑i=-1∞‍2α∞iq∥fi∥Lp(·)(Rn)q}1/q≲∥{2α(0)kfχk}∥l<q(Lp(·))+∥{2α∞kfχk}∥l>q(Lp(·))≈∥f∥K˙p(·)α(·),q(Rn).
If0<q≤1, since α(0)≤α∞, we obtain
(38)E>≲{∑k=0∞‍∑i=-∞k-2‍2α∞iq∥fi∥Lp(·)(Rn)q2(i-k)(nδ2-α∞)q}1/q≈{∑i=-∞-2‍2α∞iq∥fi∥Lp(·)(Rn)q∑k=0∞‍2(i-k)(nδ2-α∞)q+∑i=-1∞‍2α∞iq∥fi∥Lp(·)(Rn)q∑k=i+2∞‍2(i-k)(nδ2-α∞)q}1/q≲{∑i=-∞-2‍2α(0)iq∥fi∥Lp(·)(Rn)q+∑i=-1∞‍2α∞iq∥fi∥Lp(·)(Rn)q}1/q≲∥{2α(0)kfχk}∥l<q(Lp(·))+∥{2α∞kfχk}∥l>q(Lp(·))≈∥f∥K˙p(·)α(·),q(Rn).
Thus, we arrive at(39)E=E<+E>≲∥f∥K˙p(·)α(·),q(Rn).
ForG, observe that if x∈Rk, y∈Ri, and i≥k+2, then |x-y|≈|y|≈2i and
(40)|1|x-y|2-1|y|2|≲|x||x-y|3.
From Minkowski’s inequality and Lemma6, it follows that
(41)|μΩ(fi)(x)|≲(∫0|y|‍|∫|x-y|≤t‍Ω(x-y)|x-y|n-1fi(y)dy|2dtt3)1/2+(∫|y|∞‍|∫|x-y|≤t‍Ω(x-y)|x-y|n-1fi(y)dy|2dtt3)1/2≲∫Ri‍|fi(y)||x-y|n-1(∫|x-y|≤t,|y|≥t‍dtt3)1/2dy+∫Ri‍|fi(y)||x-y|n-1(∫|y|∞‍dtt3)1/2dy≲∫Ri‍|fi(y)||x-y|n-1·|x|1/2|x-y|3/2dy+∫Ri‍|fi(y)||x-y|n-1·1|y|dy≲2(k-i)/22-in∥fi∥Lp(·)(Rn)∥χi∥Lp′(·)(Rn)+C2-in∥fi∥Lp(·)(Rn)∥χi∥Lp′(·)(Rn)≲2-in∥fi∥Lp(·)(Rn)∥χBi∥Lp′(·)(Rn).
By Lemmas7 and 8, we have
(42)∥μΩ(fi)(x)χk∥Lp(·)(Rn)≲2-in∥fi∥Lp(·)(Rn)∥χBi∥Lp′(·)(Rn)∥χBk∥Lp(·)(Rn)≲∥fi∥Lp(·)(Rn)∥χBk∥Lp(·)(Rn)∥χBi∥Lp(·)(Rn)≲2(k-i)nδ1∥fi∥Lp(·)(Rn).
Thus we get(43)G<≲{∑k=-∞-1‍2kα(0)q(∑i=k+2∞‍2(k-i)nδ1∥fi∥Lp(·)(Rn))q}1/q≈{∑k=-∞-1‍(∑i=k+2∞‍2α(0)i∥fi∥Lp(·)(Rn)2(k-i)(nδ1+α(0)))q}1/q,G>≲{∑k=0∞‍(∑i=k+2∞‍2α∞i∥fi∥Lp(·)(Rn)2(k-i)(nδ1+α∞))q}1/q.
Using the same arguments as that forE< and E>, we get
(44)G=G<+G>≲∥f∥K˙p(·)α(·),q(Rn).
Hence the proof of Theorem11 is completed.Proof of Theorem12.
We apply Proposition5 again and get
(45)∥μΩ,bm(f)∥K˙p(·)α(·),q(Rn)≈∥{2α(0)kμΩ,bm(f)χk}∥l<q(Lp(·))+∥{2α∞kμΩ,bm(f)χk}∥l>q(Lp(·))=J<+J>.
Letf∈K˙p(·)α(·),q(Rn), and write
(46)f(x)=∑i=-∞∞‍f(x)χi(x)=∑i=-∞∞‍fi(x).
By Minkowski’s inequality, we have(47)J<={∑k=-∞-1‍2α(0)kq∥μΩ,bm(f)χk∥Lp(·)(Rn)q}1/q≲{∑k=-∞-1‍2α(0)kq(∑i=-∞k-2‍∥μΩ,bm(fi)χk∥Lp(·)(Rn))q}1/q+{∑k=-∞-1‍2α(0)kq(∑i=k-1k+1‍∥μΩ,bm(fi)χk∥Lp(·)(Rn))q}1/q+{∑k=-∞-1‍2α(0)kq(∑i=k+2∞‍∥μΩ,bm(fi)χk∥Lp(·)(Rn))q}1/q=U<+V<+W<.
By the same way, we obtain(48)J>={∑k=0∞‍2α∞kq∥μΩ,bm(f)χk∥Lp(·)(Rn)q}1/q≲{∑k=0∞‍2α∞kq(∑i=-∞k-2‍∥μΩ,bm(fi)χk∥Lp(·)(Rn))q}1/q+{∑k=0∞‍2α∞kq(∑i=k-1k+1‍∥μΩ,bm(fi)χk∥Lp(·)(Rn))q}1/q+{∑k=0∞‍2α∞kq(∑i=k+2∞‍∥μΩ,bm(fi)χk∥Lp(·)(Rn))q}1/q=U>+V>+W>.
Thus, we have(49)∥μΩ,bm(f)∥K˙p(·)α(·),q(Rn)≲U+V+W,
where U=U<+U>, V=V<+V>, and W=W<+W>.
ForV, by Lemma 10, we have
(50)V=V<+V>≲{∑k=-∞-1‍2α(0)kq∥fk∥Lp(·)(Rn)q}1/q+{∑k=0∞‍2α∞kq∥fk∥Lp(·)(Rn)q}1/q≲∥{2α(0)kfχk}∥l<q(Lp(·))+∥{2α∞kfχk}∥l>q(Lp(·))≈∥f∥K˙p(·)α(·),q(Rn).
ForU, observe that if x∈Rk, y∈Ri, and i≤k-2, then
(51)|μΩ,bm(fi)(x)|≲(∫0|x|‍|∫|x-y|≤t‍Ω(x-y)|x-y|n-1[b(x)-b(y)]mfi(y)dy|2dtt3)1/2+(∫|x|∞‍|∫|x-y|≤t‍Ω(x-y)|x-y|n-1[b(x)-b(y)]mfi(y)dy|2dtt3)1/2≲∫Ri‍|b(x)-b(y)|m|fi(y)||x-y|n-1(∫|x-y|≤t,|x|≥t‍dtt3)1/2dy+∫Ri‍|b(x)-b(y)|m|fi(y)||x-y|n-1(∫|x|∞‍dtt3)1/2dy≲∫Ri‍|b(x)-b(y)|m|fi(y)||x-y|n-1·|y|1/2|x-y|3/2dy+∫Ri‍|b(x)-b(y)|m|fi(y)||x-y|n-1·1|x|dy≲2(i-k)/22-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∑j=0m‍Cmj|b(x)-bBi|m-j∫Ri‍|bBi-b(y)|j|fi(y)|dy≲2-kn∥fi∥Lp(·)(Rn)∑j=0m‍Cmj|b(x)-bBi|m-j∥(bBi-b)jχi∥Lp′(·)(Rn).
An application of Lemmas7, 8, and 10 gives
(52)∥μΩ,bm(fi)χk∥Lp(·)(Rn)≲2-kn∥fi∥Lp(·)(Rn)×∑j=0m‍Cmj∥(b(x)-bBi)m-jχk∥Lp(·)(Rn)∥(bBi-b)jχi∥Lp′(·)(Rn)≲2-kn∥fi∥Lp(·)(Rn)×∑j=0m‍Cmj(k-i)m-j∥b∥BMOm-j∥χBk∥Lp(·)(Rn)∥b∥BMOj∥χBi∥Lp′(·)(Rn)≲(k-i+1)m2-kn∥fi∥Lp(·)(Rn)∥χBk∥Lp(·)(Rn)∥χBi∥Lp′(·)(Rn)≲(k-i+1)m∥fi∥Lp(·)(Rn)∥χBi∥Lp′(·)(Rn)∥χBk∥Lp′(·)(Rn)≲(k-i+1)m2(i-k)nδ2∥fi∥Lp(·)(Rn).
For convenience below we putσ=nδ2-α(0); if 1<q<∞, then we use Hölder’s inequality and obtain
(53)U<≲{∑k=-∞-1‍(∑i=-∞k-2‍2α(0)i∥fi∥Lp(·)(Rn)(k-i+1)m2(i-k)σ)q}1/q≲{×(∑i=-∞k-2‍(k-i+1)mq′2(i-k)σq′/2)q/q′∑k=-∞-1‍(∑i=-∞k-2‍2α(0)iq∥fi∥Lp(·)(Rn)q2(i-k)σq/2)×(∑i=-∞k-2‍(k-i+1)mq′2(i-k)σq′/2)q/q′×(∑i=-∞k-2‍(k-i+1)mq′2(i-k)σq′/2)q/q′}1/q≲{∑i=-∞-3‍2α(0)iq∥fi∥Lp(·)(Rn)q∑k=i+2-1‍2(i-k)σq/2}1/q≲∥{2α(0)kfχk}∥l<q(Lp(·))≲∥f∥K˙p(·)α(·),q(Rn).
If0<q≤1, then we get
(54)U<≲{∑k=-∞-1‍∑i=-∞k-2‍2α(0)iq∥fi∥Lp(·)(Rn)q(k-i+1)mq2(i-k)σq}1/q≈{∑i=-∞-3‍2α(0)iq∥fi∥Lp(·)(Rn)q∑k=i+2-1‍(k-i+1)mq2(i-k)σq}1/q≲∥{2α(0)kfχk}∥l<q(Lp(·))≲∥f∥K˙p(·)α(·),q(Rn).
Similarly, we putη=nδ2-α∞; if 1<q<∞, by Hölder’s inequality, we obtain
(55)U>≲{∑k=0∞‍(∑i=-∞k-2‍2α∞i∥fi∥Lp(·)(Rn)(k-i+1)m2(i-k)η)q}1/q≲{×(∑i=-∞k-2‍(k-i+1)mq′2(i-k)ηq′/2)q/q′∑k=0∞‍(∑i=-∞k-2‍2α∞iq∥fi∥Lp(·)(Rn)q2(i-k)ηq/2)×(∑i=-∞k-2‍(k-i+1)mq′2(i-k)ηq′/2)q/q′}1/q≲{∑k=0∞‍(∑i=-∞k-2‍2α∞iq∥fi∥Lp(·)(Rn)q2(i-k)ηq/2)}1/q.
By the same arguments asE>, we get
(56)U>≲∥{2α(0)kfχk}∥l<q(Lp(·))+∥{2α∞kfχk}∥l>q(Lp(·))≈∥f∥K˙p(·)α(·),q(Rn).
If0<q≤1, we obtain
(57)U>≲{∑k=0∞‍∑i=-∞k-2‍2α∞iq∥fi∥Lp(·)(Rn)q(k-i+1)mq2(i-k)ηq}1/q≈{∑i=-∞-2‍2α∞iq∥fi∥Lp(·)(Rn)q∑k=0∞‍(k-i+1)mq2(i-k)ηq+∑i=-1∞‍2α∞iq∥fi∥Lp(·)(Rn)q∑k=i+2∞‍(k-i+1)mq2(i-k)ηq}1/q≲{∑i=-∞-2‍2α(0)iq∥fi∥Lp(·)(Rn)q+∑i=-1∞‍2α∞iq∥fi∥Lp(·)(Rn)q}1/q≲∥{2α(0)kfχk}∥l<q(Lp(·))+∥{2α∞kfχk}∥l>q(Lp(·))≈∥f∥K˙p(·)α(·),q(Rn).
Thus, we have(58)U=U<+U>≲∥f∥K˙p(·)α(·),q(Rn).
ForW, if x∈Rk, y∈Ri, and i≥k+2, as in the arguments for G and U, we obtain
(59)∥μΩ,bm(fj)χk∥Lp(·)(Rn)≲(i-k+1)m2(k-i)nδ1∥fj∥Lp(·)(Rn).
Thus we get(60)W<≲{∑k=-∞-1‍2kα(0)q(∑i=k+2∞‍(i-k+1)m2(k-i)nδ1∥fi∥Lp(·)(Rn))q}1/q≈{×(i-k+1)m2(k-i)(nδ1+α(0))∑i=k+2∞‍2α(0)i∥fi∥Lp(·)(Rn))q∑k=-∞-1‍(∑i=k+2∞‍2α(0)i∥fi∥Lp(·)(Rn)×(i-k+1)m2(k-i)(nδ1+α(0))∑i=k+2∞‍2α(0)i∥fi∥Lp(·)(Rn))q}1/q,W>≲{∑k=0∞‍(∑i=k+2∞‍2α∞i∥fi∥Lp(·)(Rn)(i-k+1)m2(k-i)(nδ1+α∞))q}1/q.
Similar to the estimates ofU< and U>, we get
(61)W=W<+W>≲∥f∥K˙p(·)α(·),q(Rn).
Hence the proof of Theorem12 is completed.



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*Source: 430365-2014-10-15.xml*