Let L=-Δ+V be a Schrödinger operator, where V belongs to some reverse Hölder class. The authors establish the boundedness of Marcinkiewicz integrals associated with Schrödinger operators and their commutators on Morrey spaces.
1. Introduction
In this paper, we consider the Schrödinger operator
(1)L=-Δ+V(x),
in Rd, d≥3, where V(x) is a nonnegative potential belonging to the reverse Hölder class RHq for some exponent q>d/2.
A nonnegative locally Lq integrable function V(x) on Rd is said to belong to RHq(q>1) if there exists C>0 such that the reverse Hölder inequality,
(2)(1|B|∫BV(y)qdy)1/q≤C|B|∫BV(y)dy,
holds true for every ball B⊂Rd. We introduce the definition of the reverse Hölder index of V as q0=sup{q:V∈RHq}. It is known that V∈RHq implies V∈RHq+ϵ for some ϵ>0.
The Marcinkiewicz integral operator μ is defined by
(3)μf(x)=(∫0∞|∫|x-y|≤tΩ(x-y)|x-y|d-1f(y)dy|2dtt3)1/2.
The above operator was introduced by Stein in [1] as an extension of the notion of Marcinkiewicz function from one dimension to higher dimensions.
Similar to the classical Marcinkiewicz function μ, we define the Marcinkiewicz function μjL associated with the Schrödinger operator L by
(4)μjLf(x)=(∫0∞|∫|x-y|≤tKjL(x,y)f(y)dy|2dtt3)1/2,
where KjL(x,y)=K~jL(x,y)|x-y| and K~jL(x,y) is the kernel of RjL=(∂/∂xj)L-1/2, j=1,…,d. In particular, when V=0, KjΔ(x,y)=KjΔ~(x,y)|x-y|=((xj-yj)/|x-y|)/|x-y|d-1 and KjΔ~(x,y) is the kernel of Rj=(∂/∂xj)Δ-1/2, j=1,…,d. So, μjΔ is defined by
(5)μjΔf(x)=(∫0∞|∫|x-y|≤tKjΔ(x,y)f(y)dy|2dtt3)1/2.
Lemma 1 (see [2]).
For any l>0, there exists Cl>0 such that
(6)|KjL(x,y)|≤Cl(1+|x-y|/ρ(y))l1|x-y|d-1,|KjL(x,y)-Kj(x,y)|≤Cρ(x)|x-y|d-2,
where ρ is the auxiliary function as follows.
Gao and Tang [2] have shown that Marcinkiewicz integral μjL is bounded on Lp(Rd) for 1<p<∞ and bounded from L1(Rd) to weak L1(Rd). Meanwhile they also proved μjL are bounded on BMOL(Rd) and also mapped from HL1(Rd) to L1(Rd) under the assumption that KjL satisfy the condition in Lemma 1.
When KjL satisfies the estimates in Lemma 8 in Section 2 below, Chen and Zou [3] also proved that the Marcinkiewicz integral μjL has the same boundedness.
Now we give the definition of the commutator generalized by μjL and b by
(7)[b,μjL](f)(x)=(∫0∞|∫|x-y|≤tKjL(x,y)(b(x)-b(y))f(y)dy|2dtt3)1/2,
and the definition of the commutator generalized by μjΔ and b by
(8)[b,μjΔ](f)(x)=(∫0∞|∫|x-y|≤tKjΔ(x,y)(b(x)-b(y))f(y)dy|2dtt3)1/2.
Let ρ(x) be the auxiliary function defined by
(9)ρ(x)=1mV(x)=sup{r>0:1rd-2∫B(x,r)V≤1},x∈Rd.
Obviously, 0<mV(x)<∞ if V≠0. In particular, mV(x)=1 with V=1 and mV(x)~(1+|x|) with V=|x|2.
In this paper, we write Ψ(B)=(1+rmV(B))θ where mV(B)=(1/|B|)∫BmV(x)dx and θ>0, and r denotes the radius of B.
The maximal operator MVf(x) is defined by
(10)MVf(x)=supx∈B1Ψ(B)|B|∫B|f(x)|dx.
When V=0, we denote M0f(x) by Mf(x) (the standard Hardy-Littlewood maximal function). It is easy to see that |f(x)|≤MVf(x)≤Mf(x) for a.e x∈Rd. This information can be found in [4].
Proposition 2 (see [5]).
There exists k0>0 such that
(11)1C(1+|x-y|ρ(x))-k0≤ρ(y)ρ(x)≤C(1+|x-y|ρ(x))k0/(k0+1).
In particular, ρ(x)~ρ(y) if |x-y|<Cρ(x).
Proposition 3 (see [6]).
There exists a sequence of points xj, j≥1, in Rd, so that the family Qj=B(xj,ρ(xj)), j≥1, satisfies the following.
⋃jQj=Rd.
For every σ≥1 there exist constants C and N1 such that ∑jχσQj≤CσN1.
Tang and Dong [7] first introduced some type Morrey space associated with Schrödinger operators. Meanwhile they obtained the strong and weak boundedness of singular integral, fractional integral, and their commutators in Morrey spaces. Inspired by their work, we give another type of Morrey space, but in fact these two kinds of Morrey spaces are the same when their exponents are restricted in some exceeding. Recently, plenty of famous results on Schrödinger operators have appeared;we refer to [8–13].
We now present the definition of Morrey spaces associated with Schrödinger operators which we needed in this paper.
Definition 4.
Let f∈Llocq(Rd), 1≤p≤q<+∞; we say f∈Mpq(Rd) (Morrey spaces) provided that
(12)∥f∥Mpq(Rd)=supB⊂Rd|B|1/q-1/p(∫B|f(x)|pdx)1/p<+∞,
where B=B(x0,r).
Obviously, when p=q, the space Mpq(Rd) is the class Lebesgue space Lp(Rd).
Definition 5.
Let f∈Llocq(Rd), 1≤p≤q<+∞, α∈(-∞,+∞), and V∈RHq(q>1); we say f∈Mp,Vq,α(Rd) (Morrey space associated with Schrödinger operators) provided that
(13)∥f∥Mp,Vq,α(Rd)=supB⊂Rd(1+rρ(x0))α|B|1/q-1/p×(∫B|f(x)|pdx)1/p<+∞,
where B=B(x0,r).
In this note we will investigate the boundedness for the commutators of Marcinkiewicz inegral associated with Schrödinger operators on Morrey spaces given in Definition 5.
Our results can be formulated as follows.
Theorem 6.
Suppose α∈(-∞,+∞) and V∈RHq(q>1).
If 1<p≤q<+∞, then
(14)∥μjLf∥Mp,Vq,α(Rd)≤C∥f∥Mp,Vq,α(Rd),
where C is independent of f.
If 1=p≤q<+∞, then, for any λ>0,
(15)λ(1+rρ(x))α|{y∈B(x,r):|μjLf(y)|>λ}|≤C|B(x,r)|1-1/q∥f∥Mp,Vq,α(Rd)
holds for all balls B(x,r), where C is independent of x, r, λ, and f.
For θ>0, we define the class BMOθ(ρ) of locally integrable functions b such that
(16)1|B(x,r)|∫B(x,r)|b(y)-bB|dy≤C(1+rρ(x))θ,
for all x∈Rd and r>0, where bB=(1/|B|)∫Bb. A norm for b∈BMOθ(ρ), denoted by [b]θ, is given by the infimum of the constants in the inequalities above. Notice that if we let θ=0, we obtain the John-Nirenberg space BMO.
Theorem 7.
Suppose b∈BMOθ(ρ), θ>0, α∈(-∞,+∞).
If 1<p≤q<+∞, then
(17)∥[b,μjL]f∥Mp,Vq,α(Rd)≤C[b]θ∥f∥Mp,Vq,α(Rd),
where C is independent of f.
There exists a constant C>0, such that, for all λ>0,
(18)λ(1+rρ(x))α|{y∈Rd:|[b,μjL]f(y)|>λ}|≤C|B(x,r)|1-1/q∫Rd|f(y)|λ(1+log+|f(y)|λ)dy.
Throughout this paper, C denotes the constants that are independent of the main parameters involved but whose value may differ from line to line. By A~B, we mean that there exists a constant C>1 such that 1/C≤A/B≤C.
2. Notation and Preliminaries
Shen [5] gave the following kernel estimate that we need.
Lemma 8.
If V∈RHq(q>1), then one has
for every N there exists a constant C such that
(19)|KjL(x,z)|≤C(1+|x-z|/ρ(x))-N|x-z|d-1,
for every N and 0<δ<min{1,1-d/q0} there exists a constant C such that
(20)|KjL(x,z)-KjL(y,z)|≤C|x-y|δ(1+|x-z|/ρ(x))-N|x-z|d-1+δ,
where |x-y|<(2/3)|x-z|,
for every 0<σ<2-d/q0, one has
(21)|KjL(x,z)-KjΔ(x,z)|≤C|x-z|d-1(|x-z|ρ(z))σ,
where KjΔ(x,z)=((xj-zj)/|x-z|)/|x-z|d-1.
Lemma 9 (see [14]).
Let θ>0 and 1≤s<∞. If b∈BMOθ(ρ), then
(22)(1|B|∫B|b-bB|s)1/s≤C[b]θ(1+rρ(x))θ′,
for all B=B(x,r), with x∈Rd and r>0, where θ′=(k0+1)θ and k0 is the constant appearing in Proposition 2.
Corollary 10 (see [14]).
Let b∈BMOθ(ρ), B=B(x0,r), and s≥1; then
(23)(1|2kB|∫2kB|b-bB|s)1/s≤C[b]θk(1+2krρ(x0))θ′,
for all k∈N, with θ′ as in Lemma 9.
From Lemma 9, the author [15] proved the John-Nirenberg inequality for BMOθ(ρ).
Proposition 11 (see [15]).
Suppose that f is in BMOθ(ρ). There exist positive constants γ and C such that
(24)supB1|B|∫Bexp{γ[f]θΨθ′(B)|f(x)-fB|}dx≤C,
where fB=(1/|B|)∫Bf(y)dy and Ψθ′(B)=(1+r/ρ(x0))θ′, B=B(x0,r), and θ′=(k0+1)θ.
The dyadic maximal operator MVΔf(x) is defined by
(25)MVΔf(x)=supx∈Q1Ψ(Q)|Q|∫Q|f(x)|dx,
where Q is a dyadic cube.
The dyadic sharp maximal operator MV♯f(x) is defined by
(26)MV♯f(x)=supx∈Qx0,r<ρ(x0)1|Qx0|∫Qx0|f(y)-fQx0|dy+supx∈Qx0,r≥ρ(x0)1Ψ(Qx0)|Qx0|∫Qx0|f(y)|dy≃supx∈Qx0,r<ρ(x0)infC1|Qx0|∫Qx0|f(y)-C|dy+supx∈Qx0,r≥ρ(x0)1Ψ(Qx0)|Qx0|∫Qx0|f(y)|dy,
where Qx0 denotes dyadic cubes Q(x0,r) and fQ=(1/|Q|)∫Qf(x)dx.
A variant of dyadic maximal operator and dyadic sharp maximal operator is defined as following:(27)Mδ,VΔf(x)=MVΔ(|f|δ)1/δ(x),Mδ,V♯f(x)=MV♯(|f|δ)1/δ(x).
In our paper, we need the following proposition when ω=1.
Proposition 12 (see [15]).
Let 1<p<∞ and suppose that ω∈Apρ. If p<p1<∞, then the equality
(28)∫Rd|MVf(x)|p1ω(x)dx≤Cp∫Rd|f(x)|p1ω(x)dx.
Further, let 1≤p<∞, ω∈Apρ, if and only if
(29)ω({x∈Rd:MVf(x)>λ})≤Cpλp∫Rd|f(x)|pω(x)dx.
A function A:[0,∞)→[0,∞) is said to be a Young function if it is continuous, convex, and increasing satisfying A(0)=0 and A(t)→∞ as t→∞. We defined the A-average of a function f over a cube Q by means of the following Luxemburg norm:
(30)∥f∥A,Q=inf{λ>0:1|Q|∫QA(|f(y)|λ)dy≤1},
and the maximal operator MA associated to ∥·∥A,Q by MAf(x)=supx∈Q∥f∥A,Q. Then the following generalized Hölder inequality holds:
(31)1|Q|∫Q|f(y)g(y)|dy≤∥f∥A,Q∥g∥A¯,Q,
where A¯ is the complementary Young function of A.
We define the corresponding maximal function
(32)MAf(x)=supx∈Q∥f∥A,Q,MV,Af(x)=supx∈QΨ(Q)-1∥f∥A,Q.
In [16], one has given a general result that can be applied to prove the boundedness of the localized classical operators. One considers a covering of balls {Qj} such that the family of a fixed dilation of them, {Q~j}, has bounded overlapping (e.g. a covering associated to ρ like in Proposition 3).
An operator S is defined by
(33)S(f)=∑jχQj|S(fχQ~j)|.
Proposition 13 (see [16]).
Let 1≤p≤ν<∞, and a weight ω on Rd with the following property: for each j, ω|Q~j, admits an extension ωj to Rd such that
(34)S:Lp(ωj)⟼Lν(ωjν/p)
bound with a constant independent of j. Then,
(35)S0:Lp(ωj)⟼Lν(ωjν/p)
continuously. If p=1, the assumption of S is changed by weak type (1,ν), and the corresponding weak type can be concluded for S0.
In our paper, set S=μjΔ and ω=1. As we know, μjΔ is bounded from Lp to Lp. So, Proposition 13 also holds for μjΔ and ω=1.
From [16], we have the following result.
Lemma 14.
Let (μjΔ)loc(f)(x)=μjΔ(fχB(x,ρ(x)))(x). Assume that 1<p<∞. Then there exists a constant C>0 such that
(36)∫Rd|(μjΔ)loc(f)(x)|pdx≤C∫Rd|f(x)|pdx.
Furthermore, when p=1, there exists a constant C such that, for all λ>0,
(37)|{x∈Rd:(μjΔ)loc(f)(x)>λ}|≤Cλ∫Rd|f(x)|dx.
Proof.
Let σ=C2k0/(k0+1), with C and k0 as in Proposition 2. Let {Qj} be the family given by Proposition 3 and set Q~j=σQj.
Clearly, we have
(38)⋃x∈QjB(x,ρ(x))⊂Q~j.
First, for x∈Qj, Minkowski’s inequality says that
(39)|(μjΔ)loc(f)(x)-μjΔ(χQ~jf)(x)|≤C(×(χQ~j-χB(x,ρ(x)))(y)dy∫|x-y|≤t|2∫0∞|∫|x-y|≤tKjΔ(x,y)f(y)×(χQ~j-χB(x,ρ(x)))(y)dy∫|x-y|≤t|2dtt3)1/2≤C∫Rd|f(y)(χQ~j-χB(x,ρ(x)))(y)|000000000×|KjΔ(x,y)|(∫|x-y|≤tdtt3)1/2≤C∫Q~j∖B(x,ρ(x))|f(y)||xj-yj||x-y|d|x-y|-1dy≤C∫Q~j∖B(x,ρ(x))|f(y)||x-y|ddy≤C1|Q~j|∫Q~j|f(y)|dy,
where we have used |x-y|≥ρ(x)≃ρ(xj) when x∈Qj and y∉B(x,ρ(x)).
Therefore, for 1<p<∞,
(40)∥(μjΔ)loc(f)∥Lpp≤C∑j∫Qj(1|Q~j|∫Q~j|f(y)|dy)pdx+∥(μjΔ)0(f)∥Lpp≤C∑j∫Qj1|Q~j|∫Q~j|f(y)|pdydx+∥(μjΔ)0(f)∥Lpp≤C∑j∫Q~j|f(y)|pdy+∥(μjΔ)0(f)∥Lpp≤C∫Rd|f(y)|pdy+∥(μjΔ)0(f)∥Lpp.
Hence from Proposition 13 we have done it 1<p<∞.
For the case p=1, using the estimate of |(μjΔ)loc(f)(x)-μjΔ(χQ~jf)(x)| again, we have, for each j,
(41)|{x∈Qj:|(μjΔ)loc(f)(x)-μjΔ(χQ~jf)(x)|>λ}|≤C1λ∫Q~j|f(y)|dy.
Besides, from Proposition 13,
(42)|{x∈Qj:|(μjΔ)0(χQ~jf)(x)|>λ}|≤C1λ∫Q~j|f(y)|dy.
Therefore, summing over j we have the weak type (1,1).
Lemma 15.
Let b∈BMOθ(ρ), and (k0+1)≤η<∞. Set [b,μjΔ]loc(f)(x)=[b,μjΔ](fχB(x,ρ(x)))(x). Let 0<2δ<ϵ<1. Then,
(43)Mδ,η♯([b,μjΔ]loc(f))(x)≤C[b]θ(Mϵ,ηΔ((μjΔ)loc(f))(x)0000000000+MLlogL,V,η(f)(x)((μjΔ)loc(f))),a.e.x∈Rd,
holds for any f∈C0∞(Rd).
Proof.
We fix x∈Rd and let x∈Q=Q(x0,r) (dyadic cube). We consider two cases about r; that is, r<ρ(x0) and r≥ρ(x0).
Case1. When r<ρ(x0), decomposing f=f1+f2, where f1=fχQ*, where Q*=Q(x0,4dr). Let CQ a constant be fixed along the proof. Since 0<δ<1, we then have
(44)(1|Q|∫Q||[b,μjΔ]loc(f)(y)|δ-|CQ|δ|dy)1/δ≤(1|Q|∫Q|[b,μjΔ]loc(f)(y)-CQ|δdy)1/δ≤C(1|Q|∫Q|(b(y)-bQ*)(μjΔ)loc(f)(y)|δdy)1/δ+C(1|Q|∫Q|(μjΔ)loc((b-bQ*)f1)(y)|δdy)1/δ+C(1|Q|∫Q|(μjΔ)loc((b-bQ*)f2)(y)-CQ|δdy)1/δ=I+II+III,
where bQ*=(1/|Q*|)∫Q*b(y)dy.
We start with I. For any 1<γ<ϵ/δ, note that MV(x)~MV(x0) for any x∈Q* and Ψ(Q*)~1; by Lemma 9, we obtain
(45)I≤(1|Q*|∫Q*|b(y)-bQ*|δγ′dy)1/(δγ′)×(1|Q|∫Q|(μjΔ)loc(f)(y)|δγdy)1/(δγ)≤C[b]θMϵ,ηΔ((μjΔ)loc(f))(x),
where 1/γ+1/γ′=1.
For II, note that MV(x)~MV(x0) for any x∈Q* and Ψ(Q*)~1; by Kolmogorov’s inequality, Proposition 11, and Lemma 14, we have
(46)II≤C|Q|∥(μjΔ)loc((b-bQ*)f1)∥L1,∞≤C|Q*|∫Q*|(b(y)-bQ*)f(y)|dy≤C[b]θMLlogL,V,η(f)(x).
To deal with III, we first fix the value of CQ by taking CQ=(μjΔ)loc((b-bQ*)f2)(y0) with y0∈Q; we have
(47)|(μjΔ)loc((b-bQ*)f2)(y)-(μjΔ)loc((b-bQ*)f2)(y0)|≤(×f2(z)χB(y,ρ(y))KjΔ|dz∫|z-y|≤t<|z-y0||KjΔ(y,z)(b(z)-bQ*)|2∫0∞|∫|z-y|≤t<|z-y0||KjΔ(y,z)(b(z)-bQ*)00000000000000000.000×f2(z)χB(y,ρ(y))KjΔ|dz∫|z-y|≤t<|z-y0||KjΔ(y,z)(b(z)-bQ*)|2dtt3)1/2+(×f2(z)χB(y0,ρ(y0))KjΔ|dz∫|z-y0|≤t<|z-y||2∫0∞|∫|z-y0|≤t<|z-y||KjΔ(y0,z)(b(z)-bQ*)00000000000000.0000000×f2(z)χB(y0,ρ(y0))KjΔ|dz∫|z-y0|≤t<|z-y||2dtt3)1/2+(×χB(y0,ρ(y0))|dz∫{|z-y0|≤t,|z-y|≤t}|2∫0∞|∫{|z-y0|≤t,|z-y|≤t}|KjΔ(y,z)-KjΔ(y0,z)|000000000000000000000×|χB(y0,ρ(y0))(b(z)-bQ*)f2(z)000000000000000000000000×χB(y0,ρ(y0))|dz∫{|z-y0|≤t,|z-y|≤t}|2dtt3)1/2+(×f2(z)χB(y0,ρ(y0))KjΔ|dz∫{|z-y0|≤t,|z-y|≤t}|2∫0∞|∫{|z-y0|≤t,|z-y|≤t}|KjΔ(y,z)(b(z)-bQ*)000000000000000000000000×f2(z)χB(y0,ρ(y0))KjΔ|dz∫{|z-y0|≤t,|z-y|≤t}|2dtt3)1/2+(×f2(z)χB(y,ρ(y))|dz∫{|z-y0|≤t,|z-y|≤t}|2∫0∞|∫{|z-y0|≤t,|z-y|≤t}|χB(y,ρ(y))KjΔ(y,z)(b(z)-bQ*)00000000000.000000000000×f2(z)χB(y,ρ(y))|dz∫{|z-y0|≤t,|z-y|≤t}|2dtt3)1/2=E1+E2+E3+E4+E5.
For E1, since ρ(y)~ρ(x0) and |z-y|~|z-y0|~|z-x0|, using Minkowski’s inequality and Proposition 11, we obtain
(48)E1≤C∫Rd|KjΔ(y,z)(b(z)-bQ*)f2(z)χB(y,ρ(y))|×(∫|z-y|≤t<|z-y0|dtt3)1/2dz≤Cr1/2∫(Q*)c⋂B(y,ρ(y))|(b(z)-bQ*)f(z)|×|zj-yj|/|z-y||z-y|d-11|z-y|3/2dz≤Cr1/2∫(Q*)c⋂B(y,ρ(y))|(b(z)-bQ*)f(z)|×1|z-y|d+1/2dz≤Cr1/2∫4dr<|z-x0|≤ρ(x0)|(b(z)-bQ*)f(z)|×1|z-x0|d+1/2dz≤C∑k=2N02-k/2(2kdr)d∫|z-x0|≤2k+1dr|(b(z)-bQ*)f(z)|dz≤C[b]θMLlogL,V,η(f)(x),
where the integer k0 satisfies 2N0dr≤ρ(x0)≤2N0+1dr.
Similarly,
(49)E2≤C[b]θMLlogL,V,η(f)(x).
For E3, since ρ(y0)~ρ(x0) and |z-y|~|z-y0|~|z-x0|, by Minkowski’s inequality and Proposition 11, we have
(50)E3≤C∫Rd|KjΔ(y,z)-KjΔ(y0,z)|×|(b(z)-bQ*)f2(z)χB(y0,ρ(y0))|×(∫{|z-y0|≤t,|z-y|≤t}dtt3)1/2dz≤C∫(Q*)c⋂B(y0,ρ(y0))|(b(z)-bQ*)f(z)|×|yj-y0j||z-y0|d1|z-y0|dz≤C∫4dr<|z-x0|≤ρ(x0)|(b(z)-bQ*)f(z)||y-y0||z-x0|d+1dz≤C∑k=2N0r(2kdr)d+1∫|z-x0|≤2k+1dr|(b(z)-bQ*)f(z)|dz≤C∑k=2N02-k(2kdr)d∫|z-x0|≤2k+1dr|(b(z)-bQ*)f(z)|dz≤C[b]θMLlogL,V,η(f)(x),
where the integer N0 is the same as above.
For E4, since ρ(y0)~ρ(x0) and |z-y|~|z-y0|~|z-x0|, using Minkowski’s inequality and Proposition 11, we obtain
(51)E4≤C∫Rd|KjΔ(y,z)(b(z)-bQ*)f2(z)χB(y0,ρ(y0))|×(∫{|z-y0|≤t,|z-y|≤t}dtt3)1/2dz≤C∫(Q*)c⋂B(y0,ρ(y0))|(b(z)-bQ*)f(z)|000000000000000×|zj-yj|/|z-y||z-y|d-11|z-y|dz≤C∫4dr<|z-x0|≤ρ(x0)|(b(z)-bQ*)f(z)|1|z-x0|ddz≤C∑k=2N01(2kdr)d∫|z-x0|≤2k+1dr|(b(z)-bQ*)f(z)|dz≤C[b]θMLlogL,V,η(f)(x),
where the integer N0 is the same as above, and we know it is finite.
Similarly,
(52)E5≤C[b]θMLlogL,V,η(f)(x).
So,
(53)III≤C|Q|∫Q|(μjΔ)loc((b-bQ*)f2)(y)00000000-(μjΔ)loc((b-bQ*)f2)(y0)|dy≤C[b]θMLlogL,V,η(f)(x).Case2. When r≥ρ(x0), decomposing f=f1+f2, where f1=fχQ*, where Q*=Q(x0,C02k0+4dr). Since 0<2δ<ϵ<1, so a=η/δ and ϵ/δ>2; then,
(54)1Ψ(Q)a(1|Q|∫Q|[b,μjΔ]loc(f)(y)|δdy)1/δ≤1Ψ(Q)a(1|Q|∫Q|(b(y)-bQ*)(μjΔ)loc(f)(y)00000000000000000+(μjΔ)loc((b-bQ*)f)(y)|δdy1|Q|∫Q|(b(y)-bQ*)(μjΔ)loc(f)(y))1/δ≤C1Ψ(Q)a(1|Q|∫Q|(b(y)-bQ*)(μjΔ)loc(f)(y)|δdy)1/δ+C1Ψ(Q)a(1|Q|∫Q|(μjΔ)loc((b-bQ*)f1)(y)|δdy)1/δ+C1Ψ(Q)a(1|Q|∫Q|(μjΔ)loc((b-bQ*)f2)(y)|δdy)1/δ=I+II+III,
where bQ*=(1/|Q*|)∫Q*b(y)dy.
To deal with I, for any 2≤γ<ϵ/δ, note that k0+1≤η; by Lemma 9, we then have
(55)I≤C1Ψθ′(Q)(1|Q*|∫Q*|b(y)-bQ*|δγ′dy)1/(δγ′)×Ψθ′(Q)Ψ(Q)a-η/(2δ)×(1Ψ(Q)η|Q|∫Q|(μjΔ)loc(f)(y)|δγdy)1/(δγ)≤C[b]θMϵ,ηΔ((μjΔ)loc(f))(x),
where 1/γ+1/γ′=1.
For II, using Kolmogorov’s inequality, Proposition 11, and Lemma 14, we have
(56)II≤CΨ(Q)a|Q|∥(μjΔ)loc((b-bQ*)f1)∥L1,∞≤CΨ(Q)a|Q*|∫Q*|(b(y)-bQ*)f(y)|dy≤C[b]θMLlogL,V,η(f)(x).
Finally, for III, notice that B(y,ρ(y))⊂Q(x0,C02k0+4dr) for any y∈Q; then III=0.
Hence the proof is finished.
The following information can be founded in [4]. Define the following maximal functions:
(57)MV,ηf(x)=supx∈B1(Ψ(B))η|B|∫B|f(y)|dy,M~V,ηf(x)=supϵ>01(1+ϵψ(B(x,ϵ)))θη×∫Rdϵ-nφ(x-yϵ)|f(y)|dy,
and their commutators:
(58)MV,ηbf(x)=supx∈B1(Ψ(B))η|B|∫B|b(x)-b(y)||f(y)|dy,M~V,ηbf(x)=supϵ>01(1+ϵψ(B(x,ϵ)))θη0000000×∫Rdϵ-nφ(x-yϵ)|b(x)-b(y)||f(y)|dy,
where ψ(B(x,ϵ))=(1/|B(x,ϵ)|)∫B(x,ϵ)ρ(y)-1dy.
Obviously, we have
(59)MV,η′bf(x)≤CM~V,ηbf(x),
where η′=(k0+1)η and η>0.
Lemma 16 (see [1]).
Let b∈BMOθ(ρ), and (k0+1)(1+1/θ)≤η<∞, η1=(k0+1)η, and η2=(k0+1)η1(1+1/θ). Let 0<2δ<ϵ<1; then,
(60)Mδ,η♯(M~V,η2b(f))(x)≤C[b]θ(Mϵ,ηΔ(M~V,η2(f))(x)000000000+MLlogL,V,η(f)(x)),a.ex∈Rd,
holds for any f∈C0∞(Rd).
In the following Lemma, we set ω=1.
Lemma 17 (see [4]).
Let 2≤η<∞, ω∈A1ρ, and A(t)=tlog(e+t). Then there exists a constant C>0 such that for all t>0(61)ω({x∈Rd:MA,V,ηf(x)>t})≤C∫RdA(|f(x)|t)ω(x)dx.
Lemma 18 (see [15]).
Let 0<η<∞ and let MV,η/2f be locally integral. Then there exist positive constants C1 and C2 independent of f and x such that
(62)C1MV,ηMV,η+1f(x)≤MLlogL,V,η+1f(x)≤C2MV,η/2MV,η/2f(x).
3. Proof of the Main TheoremsProof of Theorem 6.
(i) Without loss of generality, we may assume that α<0. Pick any x0∈Rd and r>0, and write
(63)f(x)=f0(x)+∑i=1∞fi(x),
where f0=χB(x0,2r) and fi=χB(x0,2i+1r)∖B(x0,2ir), for i≥1. Hence, we have
(64)(∫B(x0,r)|μjLf(x)|pdx)1/p≤C(∫B(x0,r)|μjLf0(x)|pdx)1/p+∑i=1∞(∫B(x0,r)|μjLfi(x)|pdx)1/p.
By the Lp boundedness of μjL, we obtain
(65)(∫B(x0,r)|μjLf0(x)|pdx)1/p≤C(1+rρ(x0))-α|B|1/p-1/q∥f∥Mp,Vq,α(Rd).
By Proposition 2, Lemma 8, and Minkowski’s inequality, we have
(66)(∫B(x0,r)|μjLfi(x)|pdx)1/p≤C(fi(y)dy∫|x-y|≤tKjL|2dtt3)p/2∫B(x0,r)(fi(y)dy∫|x-y|≤tKjL|2dtt3∫0∞|∫|x-y|≤tKjL(x,y)2222222222222222222222×fi(y)dy∫|x-y|≤tKjL|2dtt3)p/2dx)1/p≤Ck((∫|x-y|≤tdtt3)1/2dy)p∫B(x0,r)((∫|x-y|≤tdtt3)1/2(∫|x-y|≤tdtt3)1/2∫Rd|fi(y)|(1+|x-y|/ρ(x))k1|x-y|d-12222222222222222×(∫|x-y|≤tdtt3)1/2dy)pdx)1/p≤Ck(1|x-y|ddy∫B(x0,2i+1r)∖B(x0,2ir))p∫B(x0,r)(∫B(x0,2i+1r)∖B(x0,2ir)|f(y)|(1+|x-y|/ρ(x))k2222222222222222222222222×1|x-y|ddy∫B(x0,2i+1r)∖B(x0,2ir))pdx)1/p≤Ck(∫B(x0,r)(2ir)-d(1+2ir/ρ(x))kpdx22222zzz222×∫B(x0,2i+1r)|f(y)|pdy∫B(x0,r)(2ir)-d(1+2ir/ρ(x))kp)1/p≤Ck(∫B(x0,r)(2ir)-d(1+2ir/ρ(x))kpdx)1/p×(1+2irρ(x0))-α|2iB|1/p-1/q∥f∥Mp,Vq,α(Rd)≤Ck(∫B(x0,r)((1+(rρ(x0)))-k0/(k0+1))-kp(2ir)-d2222,222222222×((1+(rρ(x0)))-k0/(k0+1)1+2irρ(x0)22222z.2222222222×(1+rρ(x0))-k0/(k0+1))-kp)dx)1/p×(1+2irρ(x0))-α|2iB|1/p-1/q∥f∥Mp,Vq,α(Rd)≤Ck(2-i)d/p(1+2ir/ρ(x0))-α(1+2ir/ρ(x0))k/(k0+1)|2iB|1/p-1/q×∥f∥Mp,Vq,α(Rd).
Let k=(-[α]+1)(k0+1), and we obtain
(67)∥μjLf∥Mp,Vq,α(Rd)≤C∥f∥Mp,Vq,α(Rd).
(ii) When p=1 and noting that μjL are bounded from L1(Rd) to weak L1(Rd), we have
(68)λ|{y∈B(x,r):|μjLf(y)|>λ}|≤∫B(x,r)|f(y)|dy≤C(1+rρ(x0))-α|B(x,r)|1-1/q∥f∥M1,Vq,α(Rd).
We have finished the proof of Theorem 6.
Proof of Theorem 7.
(i) Pick any x0∈Rd and r>0, as in Theorem 6; we write
(69)f(x)=f0(x)+∑i=1∞fi(x),
and, by the Lp boundedness of [b,μjL], we obtain
(70)(∫B(x0,r)|[b,μjL]f0(x)|pdx)1/p≤C(1+rρ(x0))-α|B|1/p-1/q∥f∥Mp,Vq,α(Rd).
Set br=(1/|B(x0,r)|)∫B(x0,r)b(x)dx.
We write
(71)|[b,μjL]fi|≤|b-br||μjLfi|+|μjL((b-br)fi)|.
When i≥1, by Lemmas 8 and 9, Proposition 2, Corollary 10, and Minkowski’s inequality, we have
(72)(∫B(x0,r)|[b,μjL]fi(x)|pdx)1/p≤(∫B(x0,r)|b(x)-br|p|μjLfi(x)|pdx)1/p+(∫B(x0,r)|μjL((b-br)fi)(x)|pdx)1/p≤((∫0∞|∫|x-y|≤tKjL(x,y)fi(y)dy|2dtt3)p/2∫B(x0,r)|b(x)-br|p×(∫0∞|∫|x-y|≤tKjL(x,y)fi(y)dy|2dtt3)p/2dx)1/p+(fi(y)dy∫|x-y|≤t|2dtt3)p/2∫B(x0,r)(fi(y)dy∫|x-y|≤t|2dtt3∫0∞|∫|x-y|≤tKjL(x,y)(b(y)-br)000000000000000000000000×fi(y)dy∫|x-y|≤t|2dtt3)p/2dx)1/p≤Ck(2ir)-d(1+2ir/ρ(x0))k/(k0+1)×{(∫B(x0,r)|b(x)-br|pdx)1/p∫B(x0,2i+1r)|f(y)|dy+rd/p∫B(x0,2i+1r)|b(x)-br||f(y)|dy}≤Ck(2ir)-d(1+2ir/ρ(x0))k/(k0+1)×{(∫B(x0,r)|b(x)-br|pdx)1/p×(∫B(x0,2i+1r)|f(y)|pdy)1/p(2ir)d(1-1/p)+rd/p(∫B(x0,2i+1r)|b(y)-br|p′dy)1/p′×(∫B(x0,2i+1r)|f(y)|pdy)1/p}≤Ck(2ir)-d(1+2ir/ρ(x0))k/(k0+1)×{[b]θrd/p(1+rρ(x0))θ′(2ir)d(1-1/p)0000000×(1+2irρ(x0))-α|2iB|1/p-1/q∥f∥Mp,Vq,α(Rd)0000000+rd/pi[b]θ(1+rρ(x0))θ′(2ir)d(1-1/p)0000000×(1+2irρ(x0))-α|2iB|1/p-1/q∥f∥Mp,Vq,α(Rd)[b]θrd/p(1+rρ(x0))θ′}≤Cki[b]θ(2i)-d/p(1+2ir/ρ(x0))-α(1+2ir/ρ(x0))k/(k0+1)-θ′×|2iB|1/p-1/q∥f∥Mp,Vq,α(Rd),
where 1/p+1/p′=1. Choosing k large enough such that k/(k0+1)-θ′>0, we obtain
(73)∥[b,μjL]f∥Mp,Vq,α(Rd)≤C[b]θ∥f∥Mp,Vq,α(Rd).
(ii) We adapt a similar argument to that of Theorem 1.2 in [5].
We start with [b,μjL]glob. Using Lemma 8 with N=1, we have
(75)[b,μjL]glob(f)(x)=(f(y)χBc(x,ρ(x))dy∫|x-y|≤t|2dtt3∫0∞|∫|x-y|≤tKjL(x,y)(b(x)-b(y))0000000000000×f(y)χBc(x,ρ(x))dy∫|x-y|≤t|2dtt3)1/2≤(f(y)dy∫ρ(x)<|x-y|≤t|2dtt3∫ρ(x)∞|∫ρ(x)<|x-y|≤tKjL(x,y)(b(x)-b(y))000000000000000×f(y)dy∫ρ(x)<|x-y|≤t|2dtt3)1/2≤C(∫ρ(x)∞|∫ρ(x)<|x-y|≤t(1+|x-y|/ρ(x))-1|x-y|d-10000000000000000×|(b(x)-b(y))f(y)|dy∫ρ(x)<|x-y|≤t|2dtt3)1/2≤Cρ(x)(f(y)(b(x)-b(y))|dy∑k=1[log2t/ρ(x)]+1|2∫ρ(x)∞|∑k=1[log2t/ρ(x)]+11(2kρ(x))n0000000000000000×∫|x-y|≤2kρ(x)|(b(x)-b(y))0000000000000000000000000000×f(y)(b(x)-b(y))|dy∑k=1[log2t/ρ(x)]+1|2dtt3)1/2≤Cρ(x)MV,ηbf(x)×(∫ρ(x)∞([log2tρ(x)]+1)2dtt3)1/2≤CMV,ηbf(x).
Therefore, the estimates for [b,μjL]glob follow from those for MV,ηbf(x) by Lemmas 16 and 18.
To deal with [b,μjL]loc we write
(76)[b,μjL]loc(f)(x)≤I(x)+[b,μjΔ]loc(f)(x)+II(x),
where [b,μjΔ]loc(f)(x) is defined in Lemma 15 and
(77)I(x)=((b(x)-b(y))f(y)dy∫|x-y|≤t[KjL(x,y)-KjΔ(x,y)]∫|x-y|≤t|2∫0ρ(x)|∫|x-y|≤t[KjL(x,y)-KjΔ(x,y)]0000000000×(b(x)-b(y))f(y)dy∫|x-y|≤t[KjL(x,y)-KjΔ(x,y)]∫|x-y|≤t|2dtt3∫0ρ(x))1/2,II(x)≤((b(x)-b(y))f(y)dy∫|x-y|≤ρ(x)KjL(x,y)∫|x-y|≤ρ(x)|2∫ρ(x)∞|∫|x-y|≤ρ(x)KjL(x,y)0000000000×(b(x)-b(y))f(y)dy∫|x-y|≤ρ(x)KjL(x,y)∫|x-y|≤ρ(x)|2dtt3)1/2.
For II(x), by Lemma 8 with N=0,
(78)II(x)≤C(∫ρ(x)∞|∫|x-y|≤ρ(x)1|x-y|d-1000000000000×|(b(x)-b(y))f(y)|dy∫ρ(x)∞|∫|x-y|≤ρ(x)1|x-y|d-1|2dtt3∫ρ(x)∞|∫|x-y|≤ρ(x)1|x-y|d-1)1/2≤C(∫ρ(x)∞|∑k=-∞01(2k-1ρ(x))d-10000000000×∫|x-y|≤2kρ(x)|(b(x)-b(y))f(y)|dy∑k=-∞01(2k-1ρ(x))d-1|2dtt3)1/2≤C(∫ρ(x)∞|∑k=-∞02kρ(x)(2kρ(x))d000000000000×∫|x-y|≤2kρ(x)|(b(x)-b(y))f(y)|dy∑k=-∞02kρ(x)(2kρ(x))d|2dtt3)1/2≤Cρ(x)MV,ηbf(x)(∫ρ(x)∞dtt3)1/2≤CMV,ηbf(x).
For I(x), using Lemma 8 again, we have
(79)I(x)≤C(∫0ρ(x)|∫|x-y|≤t1|x-y|d-1(|x-y|ρ(x))σ00000000000000×|(b(x)-b(y))f(y)|dy(|x-y|ρ(x))σ|2dtt3)1/2≤Cρ(x)-σ(∫0ρ(x)|∑k=-∞01(2k-1t)d-σ-100000000000000000×∫|x-y|≤2kt|(b(x)-b(y))00000000000000000000000000×(b(x)-b(y))f(y)(b(x)-b(y))|dy∑k=-∞0|2dtt3)1/2≤Cρ(x)-σ(f(y)|dy∑k=-∞0|2∫0ρ(x)|∑k=-∞0(2k)σ+1tσ+1(2kt)d00000000000000000×∫|x-y|≤2kt|(b(x)-b(y))0000000000000000000000000000×f(y)|dy∑k=-∞0(2k)σ+1tσ+1(2kt)d|2dtt3)1/2≤Cρ(x)-σ(∫0ρ(x)|∑k=-∞0(2k)σ+1tσ+1|2dtt3)1/2MV,ηbf(x)≤Cρ(x)-σ(∫0ρ(x)t2σ-1dt)1/2MV,ηbf(x)≤CMV,ηbf(x).
Using Lemmas 15, 16, 17, and 18 and Proposition 12, by adapting an argument in [2], we can obtain the desired result.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to express their hearty thanks to the referee’s comments. This paper is supported by the National Natural Science Foundation of China (10961015, 11261023, and 11326092) and the Jiangxi Natural Science Foundation of China (20122BAB201011), GJJ12203.
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