# BMO-Boundedness of Maximal Operators andg-Functions Associated with Laguerre Expansions

Authors: Li Cha; Heping Liu
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

Let{φnα}n∈ℕ be the Laguerre functions of Hermite type with index α. These are eigenfunctions of the Laguerre differential operator Lα=1/2(-d2/dy2+y2+(1/y2)(α2-1/4)). In this paper, we investigate the boundedness of the Hardy-Littlewood maximal function, the heat maximal function, and the Littlewood-Paley g-function associated with Lα in the localized BMO space BMOLα, which is the dual space of the Hardy space HLα1.


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## Body

## 1. Introduction

Letn∈ℕ, α>-1. The Laguerre function of Hermite type φα on (0,∞) is defined as
(1.1)φnα(y)=(Γ(n+1)Γ(n+1+α))1/2e-y2/2yαLnα(y2)(2y)1/2,y∈(0,∞),
where Lnα(x) denotes the Laguerre polynomial of degree n and order α, see [1]. It is well known that for every α>-1 the system {φnα}n=0∞ forms an orthonormal basis of L2(0,∞). Moreover, these functions are eigenfunctions of the Laguerre differential operator
(1.2)Lα=12(-d2dy2+y2+1y2(α2-14))
satisfying Lαφnα=(2n+α+1)φnα. The operator Lα can be extended to a positive self-adjoint operator on L2(0,∞) by giving a suitable domain of definition, see [2]; we also denote the extension by Lα. Let {Ttα}t≥0 be the heat-diffusion semigroup generated by -Lα. More precisely, for f∈L2(0,∞), we define
(1.3)Ttαf(x)=∫0∞‍Wtα(x,y)f(y)dy,
where
(1.4)Wtα(x,y)=(2e-t1-e-2t)1/2(2xye-t1-e-2t)1/2Iα(2xye-t1-e-2t)exp⁡(-121+e-2t1-e-2t(x2+y2)).Iα is the modified Bessel function of the first kind and order α.In [3], we introduced and developed a localized BMO space BMOLα associated with the operator Lα, which is the dual space of the Hardy space HLα1 introduced by Dziubański [4]. More precisely, let
(1.5)ρLα(x)=18min⁡(x,1x),x>0.Definition 1.1.
Letα>-1/2, Bs(y) be any ball in (0,∞) with the center y and the radius s and f a locally integrable function on (0,∞). We say f∈BMOLα if there exists a constant C≥0 independent of s and y such that
(1.6)1|Bs(y)|∫Bs(y)‍|f-fBs(y)|≤C,ifs<ρLα(y),1|Bs(y)|∫Bs(y)‍|f|≤C,ifs≥ρLα(y).
Here, fBs(y)=(1/|Bs(y)|)∫Bs(y)‍fdx. We let ∥f∥BMOLα denote the smallest C in the two inequalities above.It is readily seen thatBMOLα is a Banach space with norm ∥·∥BMOLα.In this paper, we obtain the boundedness onBMOLα of several operators including the Hardy-Littlewood maximal operator defined on (0,∞), the heat maximal function, and the Littlewood-Paley g-function associated with Ttα.These results were investigated by Dziubański et al. in [5] for Schrödinger operators on ℝd with d≥3 and with potentials satisfying a reverse Hölder's inequality. Recently, a theory of localized BMO spaces on RD-spaces associated with an admissible function ρ was investigated in [6]; the authors also established the similar results above for their BMO spaces. The admissible function ρ in [6] is required to satisfy
(1.7)1ρ(x)≤C01ρ(y)(1+d(x,y)ρ(y))k0.
Obviously, our ρLα in (1.5) does not satisfy this condition. Indeed, let x tend to zero and y=1; then the left side becomes greater than the right.It is notable the generalized square functions associated to Schrödinger operators are studied in [7]. The authors of [7] gave several of equivalent conditions for BMO-boundedness of square functions.In this paper, in order to obtain some key estimates, we will employ the differences in integral kernels (the heat kernel, theg-function kernel) associated with the Hermite operator and the Laguerre operator, respectively (see [8, 9]).The paper is organized as follows. In the next section we present some preliminary lemmas and collect some useful estimates of the kernels associated with the heat semigroups and theg-functions. In Section 3, we establish the boundedness of two maximal operators (the Hardy-Littlewood maximal operator and the heat maximal function) from BMOLα to BMOLα. In Section 4, we obtain the boundedness on BMOLα of the Littlewood-Paley g-function associated with the heat semigroup for Lα. We make some conventions. Throughout this paper by C we always denote a positive constant that may vary at each occurrence; Br(y0) stands for {y>0,|y-y0|≤r}; A~B means (1/C)A≤B≤CA, and the notation X≲Y is used to indicate that X≤CY with an independent positive constant C.


## 2. Preliminaries

Now we give the following covering lemma for(0,∞) which will be used frequently below. The proof is trivial and left to the reader.Lemma 2.1.
Letx0=1,   xj=xj-1+ρLα(xj-1) for j≥1, and xj=xj+1-ρLα(xj+1) for j<0. One defines the family of “critical balls” of ℬ={Bk}k=-∞∞, where Bk:={x∈(0,∞):|x-xk|<ρLα(xk)}. Then one has(a)
⋃k=-∞∞Bk=(0,∞),(b)
for everyk∈ℤ, Bk∩Bj=∅ provided that j∉{k-1,k,k+1},(c)
for anyy0∈(0,∞), at most three balls in ℬ have nonempty intersection with B(y0,ρLα(y0)).Corollary 2.2.
There exists a constantC>0 such that for every BR(x)⊆(0,∞) with R>ρLα(x), one has
(2.1)|BR(x)|≤∑{Bk∈B:Bk∩BR(x)≠∅}‍|Bk|≤C|BR(x)|.Corollary 2.3.
There exists a constantC such that, for f∈BMOLα, one has
(2.2)‖f‖BMOLα≤Csup⁡k(|f|Bk+‖f‖BMO(Bk*)),
where, for any ball B, the norm ∥·∥BMO(B) is given by
(2.3)‖f‖BMO(B)=sup⁡Br(x)⊂B1Br(x)∫Br(x)‍|f-fBr(x)|dy∼sup⁡Br(x)⊂Binf⁡c∈C1Br(x)∫Br(x)‍|f-c|dy.Corollary 2.4 (see [3, Corollary  3]).
LetB=Br(y0)⊂(0,∞). There exists a constant C>0 such that, for all f∈BMOLα, one has (1)
ifr≥ρLα(y0)/2, then ((1/|B|)∫B‍|f(x)|2dx)1/2≤C∥f∥BMOLα,(2)
ifr<ρLα(y0)/2, then ((1/|B|)∫B‍|f(x)-fB|2dx)1/2≤C∥f∥BMOLα. We give two elementary lemmas, which will be used frequently in next section. The proofs are trivial, and the reader also refer to Lemmas 9 and 2 in [5].Lemma 2.5.
Leth∈BMO(Bk*) and g1 and g2 be functions in L∞(0,∞). If f is any measurable function satisfying
(2.4)h+g1≤f≤h+g2,a.e.,
then f∈BMO(Bk*) and ∥f∥BMO(Bk*)≤∥h∥BMO(Bk*)+max(∥g1∥∞,∥g2∥∞).Lemma 2.6.
For allf∈BMOLα and B=Br(y0) with r<ρLα(y0). There exists a constant C>0 such that
(2.5)|fB*|≤C(1+log⁡ρLα(y0)r)‖f‖BMOLα.LetH be the Hermite operator
(2.6)H=12(-d2dx2+x2).
One considers the heat diffusion semigroup {Wt}t>0 associated with H and defined by, for every f∈L2(ℝ),
(2.7)Wtf(x)=∫R‍Wt(x,y)f(y)dy,x∈R,
where for each x,y∈ℝ and t>0,
(2.8)Wt(x,y)=[e-tπ(1-e-2t)]1/2exp⁡(-121+e-2t1-e-2t(x2+y2)+2xye-t1-e-2t)
(see [10]).Proposition 2.7.
Letα>-1/2, Wt(x,y) be in (2.8). There exists C>0 such that, for t>0,  (a)
Wtα(x,y)≤Cyα+1/2x-α-3/2,    0<y<x/2,(b)
Wtα(x,y)≤Cxα+1/2y-α-3/2,    0<2x<y,(c)
|Wtα(x,y)-Wt(x,y)|≤C(1/y),   x/2<y<2x,(d)
|Wtα(x,y)|≤C(1/t)e-|x-y|2/10t.Parts (a), (b), and (c) are the contents of Lemma 2.11 in [8]. Part (d) is from (2.6) in [4].Remark 2.8.
The ranges0<y<x/2 and 0<2x<y are not critical; Proposition 2.7 also holds when 0<y<x/c and 0<cx<y, where c>1.Now we consider the estimates of the integral kernel for theg-function, which will be defined in Section 4:
(2.9)Qt(x,y)=t2∂Wsα(x,y)∂s|s=t2,(2.10)Pt(x,y)=t2∂Ws(x,y)∂s|s=t2.Proposition 2.9.
One has,(a)
for everyt,x,y∈(0,∞) such that e-t2xy/(1-e-2t2)≤1,
(2.11)|Qt(x,y)|≤Ct2(xy)(α+1/2)e-(x2+y2)/8t2e-(α+1)t2(1-e-2t2)α+2,(b)
for everyt,x,y∈(0,∞) such that e-t2xy/(1-e-2t2)>1,
(2.12)|Qt(x,y)-Pt(x,y)|≤Ct2e-(x-y)2/2t2et2/2xy(1-e-t2)1/2.Parts (a) and (b) are contained in [9, (3.4) and (3.6)].Proposition 2.10.
For everyN≥1, there is a constant CN such that  (a)
ift>0, |Pt(x,y)|≤CNe-t2/8(1/t)e-|x-y|2/10t2(1+t|x|)-N;(b)
for|h|≤t, |Pt(x+h,y)-Pt(x,y)|≤C(|h|/t)(1/t)e-|x-y|2/20t2, C is independent of x,y,t;(c)
|∫-∞∞‍Pt(x,y)dy|≤C(t/ρLα(x))2, C is independent of x and t.Proof.
By using (2.8) we can write, for every x,y∈ℝ and s>0,
(2.13)∂∂sWs(x,y)=-12πe-((x-e-sy)2+(y-e-sx)2)/2(1-e-2s)e-s/2(1-e-2s)3/2×{(x-e-sy)2+(y-e-sx)21-e-2s1+e-2s+2e-s(y(x-e-sy)+x(y-e-sx))-2e-2s(x-e-sy)2+(y-e-sx)21-e-2s}.
By the simple fact(2.14)(x-e-sy)2+(y-e-sx)2=2(x-y)2e-s+(x2+y2)(1-e-s)2,
a straightforward manipulation leads to
(2.15)|∂∂sWs(x,y)|≤Ce-((x-y)2+(x2+y2)(1-e-s)2)/8(1-e-2s)e-s/2(1-e-2s)3/2≤CNe-s/8(1-e-2s)3/2e-c(|x-y|2/10s)(1+s|x|)-N,
which implies (a).
To prove (b), we also directly compute thex partial derivative:
(2.16)∂∂x∂sWs(x,y)=-12πe-((x-e-sy)2+(y-e-sx)2)/2(1-e-2s)e-s/2(1-e-2s)3/2×{4e-s(y-e-sx)-4e-2s(x-y)1+e-s}+12πe-((x-e-sy)2+(y-e-sx)2)/2(1-e-2s)e-s/2(1-e-2s)5/2((x-e-sy)-e-s(y-e-sx))×{(x-e-sy)2+(y-e-sx)21-e-2s1+e-2s+2e-s(y(x-e-sy)+x(y-e-sx))-2e-2s(x-e-sy)2+(y-e-sx)21-e-2s}.
By an elementary manipulation and (2.14), we have
(2.17)|∂∂x∂sWs(x,y)|≤C1s2e-(x-y)2/16s.
This together with the mean value theorem and the condition |h|≤t leads to (b).
Letϕn(y)=ϕ(y/n); ϕ(y) is a smooth function satisfying ϕ(y)=1 for |y|≤1, ϕ(y)=0 for |y|≥2 and Δϕ(y)≤1 for y∈ℝ. From the above, for fixed s and x, a straightforward manipulation shows that
(2.18)∫-∞+∞‍|∂Ws(x,y)∂s|dy<∞.
Hence, we have
(2.19)|∫-∞∞‍∂Ws(x,y)∂sdy|=|lim⁡n→∞∫-∞∞‍∂Ws(x,y)∂sϕn(y)dy|=|lim⁡n→∞∫-∞∞‍Ws(x,y)Hϕn(y)dy|≤C∫-∞∞‍Ws(x,y)y2dy.
Using (2.8) again,
(2.20)I=|∫-∞∞‍∂Ws(x,y)∂sdy|≤C∫-∞∞‍e-s/4sexp⁡(-(x-y)2e-s+(x2+y2)(1-e-s)22(1-e-2s))((y-x)2+x2)dy,
which implies (c).Lemma 2.11 (see [3, Theorem 2]).
For allf∈BMOLα and B=Br(y0)⊆(0,∞), there exists a constant C>0 such that
(2.21)1|B|∫0r‍∫B‍Qt2f(x)dxdtt≤C‖f‖BMOLα2.


## 3. Maximal Operators

First of all, we define the following notions:(3.1)M+f(x)=sup⁡x∈B⊆(0,∞)1|B|∫B‍|f(y)|dy,(3.2)Iα*f(x)=sup⁡t>0|Ttαf(x)|.In this section, we will showℐα* and M+ are bounded on BMOLα.Theorem 3.1.
There exists a constantC>0 such that, for all f∈BMOLα, M+f<∞, for a.e. x∈(0,∞), and
(3.3)‖M+f‖BMOLα≤C‖f‖BMOLα.Proof.
First of all, we show that for a.e.x∈(0,∞), M+f<∞. To do this, we only need to show that for, at almost x∈Bk⊆ℬ in Lemma 2.1, M+f(x)<∞. Let us split f=f1+f2 with f1=fχBk*. Obviously, since f is locally integrable, we have M+f1<∞ for a.e. x∈(0,∞). For f2, if x∈B and B∩Bk*=∅, since supp f2 is in the complement of Bk*, we have (1/|B|)∫B‍|f(y)|dy=0. Otherwise, by the definition of BMOLα, (1/|B|)∫B‍|f(y)|dy≤(4/|B4r(xk)|)∫B4r(xk)‍|f(y)|dy≤c∥f∥BMOLα.
We turn to the boundedness inBMOLα. Let M denote the Hardy-Littlewood function on ℝ; it is well known in [11] that M is bounded on BMO(ℝ). Let f0 be a function defined on ℝ which is f on (0,∞) and 0 on (-∞,0]. Notice that M+f=Mf0, for x∈(0,∞), so
(3.4)‖M+f‖BMO(Bk*)=‖Mf0‖BMO(Bk*)≤C‖f0‖BMO.
Now, we need to show that ∥f0∥BMO≤C∥f∥BMOLα. Indeed, if B⊆(0,∞), it is obvious that (1/|B|)∫B‍|f0-(f0)B|dy≤∥f∥BMOLα. If B∩(0,∞)=∅, then (1/|B|)∫B‍|f0-(f0)B|dy=0. If B∩(0,∞)≠∅ and B∩(-∞,0)≠∅, let B=B1∪B2, here B1=B∩(-∞,0) and B2=B∩(0,∞), then
(3.5)1|B|∫B‍|f0-(f0)B|dy≤21|B|∫B2‍|f0|dy≤2‖f‖BMOLα.
On the other hand, we again split f=f1+f2 with f1=fχBk*, from the argument above, M+f2(x)≤c∥f∥BMOLα, for a.e x∈Bk. So
(3.6)1|Bk|∫Bk‍|M+f|dy≤1|Bk|∫Bk‍|M+f1|dy+1|Bk|∫Bk‍|M+f2|dy≲(1|Bk|∫Bk‍|M+f1|2dy)1/2+‖f‖BMOLα≲‖f‖BMOLα,
where in the last inequality we have used Corollary 2.4.Theorem 3.2.
Letα>-1/2. There exists a constant C>0 such that
(3.7)‖Iα*f‖BMOLα≤C‖f‖BMOLα.Proof.
By the definition ofBMOLα and Corollary 2.3, it suffices to prove the following: for every fixed “critical ball” Bk∈ℬ (see Lemma 2.1) we have  (1)
(1/|B_k|)∫Bk‍|ℐα*f|dx≤C∥f∥BMOLα,(2)
∥ℐα*f∥BMO(Bk*)≤C∥f∥BMOLα.
Let us start to prove (1). It is immediate from Theorem3.1 and (d) of Proposition 2.7; since ℐα*f(x)≤M+f(x), for x>0, therefore,
(3.8)1|Bk|∫Bk‍|M+f|dy≤C‖f‖BMOLα.
It remains to show (2). By Lemma2.5, we split ℐα*f(x) into several parts. First, we shall show
(3.9)‖sup⁡t>ρLα(xk)2|Ttαf(x)|‖L∞(Bk*)≤C‖f‖BMOLα.
From (d) of Proposition2.7, we have
(3.10)|Ttαf(x)|≲∫0∞‍|f(y)|1t1/2(1+|x-y|t)-Ndy≲∑j=0∞12jN1t1/2∫{y>0,|y-x|<2jt}‍|f(y)|dy.
Notice that, forj≥0 and t>ρLα(xk)2, we have 2jt≥ρLα(x)~ρLα(xk), for x∈Bk*. Thus
(3.11)1t∫{y>0,|y-x|<2jt}‍|f(y)|dy≤C2j‖f‖BMOLα.
Therefore,(3.12)sup⁡t>ρLα(xk)2|Ttαf(x)|≲∑j=0∞12N-1‖f‖BMOLα≲‖f‖BMOLα.
By Lemma2.5, it suffices to show that sup0<t≤ρLα(xk)2|Ttαf(x)| satisfies (2). Write
(3.13)f=fχ{xk/2≤y≤2xk}+fχ{y<xk/2}+fχ{y>2xk}=f1+f2+f3.
By Proposition2.7, it easily follows that
(3.14)‖sup⁡0<t≤ρLα(xk)2|Ttαf2(x)|‖L∞(Bk*)≤C‖f‖BMOLα,‖sup⁡0<t≤ρLα(xk)2|Ttαf3(x)|‖L∞(Bk*)≤C‖f‖BMOLα.
Indeed, sincex~xk, for x∈Bk*, by (a) of Proposition 2.7 and Remark 2.8, we have
(3.15)|Ttαf2(x)|≲∫0xk/2‍yα+1/2xk-α-3/2|f(y)|dy≲1xk∫0xk/2‍|f(y)|dy≲‖f‖BMOLα.
Similarly,(3.16)|Ttαf3(x)|≲∑n=1∞‍(2nxk)-α-3/2(xk)α+1/2∫2nxk2n+1xk‍|f(y)|dy≲∑n=1∞‍(2n)-α-1/2‖f‖BMOLα≲‖f‖BMOLα.
Now, we come to treatf1. We make further decompositions. Split
(3.17)Ttαf1=(Ttαf1-Wtf1)+(Wtf1-Htf1)+Htf1,
where(3.18)Htg(x)=∫0∞‍[e-tπ(1-e-2t)]1/2exp⁡(-121+e-2t1-e-2t(x-y)2)g(y)dy.
For the first term, by (c) of Proposition2.7, we have
(3.19)|Ttαf1-Wtf1|≤C1xk∫x2/22xk‍|f(y)|dy≤C‖f‖BMOLα.
By (2.8),
(3.20)Wtf1(x)-Htf1(x)=∫0∞‍[e-tπ(1-e-2t)]1/2exp⁡(-121+e-2t1-e-2t(x-y)2)[e-2xy(1-e-t)2/1-e-2t-1]f1(y)dy.
Notice that|e-2xy(1-e-t)2/(1-e-2t)-1|≤ctxk2, when t≤ρLα(xk)2, xk/2≤y≤2xk and x∈Bk*. Therefore, for x∈Bk* and t≤ρLα(xk)2, we obtain
(3.21)|Wtf1(x)-Htf1(x)|≲txk2∫0∞‍1te-c0|x-y|2/t|f1(y)|dy≲txk2∑1≤2j≤ρLα(xk)/t‍2-j(N-1)12jt∫{y>0,|y-x|<2jt}‍|f(y)|dy+txk2∑ρLα(xk)/t<2j‍2-j(N-1)12jt∫{y>0,|y-x|<2jt}‍|f(y)|dy≲∑1≤2j≤ρLα(xk)/t‍2-j(N-1)txk2(1+log⁡(ρLα(xk)2jt))‖f‖BMOLα+∑j=0∞2-j(N-1)‖f‖BMOLα≲∑j=0∞2-j(N-1)‖f‖BMOLα≲‖f‖BMOLα.
Finally, by Lemma2.5 again, we need to show that sup0<t≤ρLα(xk)2|Htf1(x)| satisfies (2). Consider B=Br(x0)⊂Bk* and write
(3.22)f1=(f1-fB*)χB*+[(f1-fB*)χ(B*)c∩(0,∞)+fB*χ(0,∞)]=f11+f12.
By Corollary 2.3, we choose a constant CB=sup0<t≤ρLα(xk)2|Htf12(x0)|,
(3.23)1|B|∫B‍|sup⁡0<t≤ρLα(xk)2|Htf1(x)|-CB|dx≤1|B|∫B‍sup⁡0<t≤ρLα(xk)2|Htf1(x)-Htf12(x0)|dx≤1|B|∫B‍sup⁡0<t≤ρLα(xk)2|Htf11(x)|dx+1|B|∫B‍sup⁡0<t≤ρLα(xk)2|Htf12(x)-Htf12(x0)|dx.
For the first integral, by Corollary 2.4 it easily follows that
(3.24)1|B|∫B‍sup⁡0<t≤ρLα(xk)2|Htf11(x)|dx≤(1|B|∫B‍|f11|2dx)1/2≤C‖f‖BMOLα.
For the second integral,
(3.25)|Htf1(x)-Htf12(x0)|≤|∫(B*)c∩(0,∞)‍(Ht(x,y)-Ht(x0,y))(f1(y)-fB)dy|+|∫0∞‍(Ht(x,y)-Ht(x0,y))fBdy|=I1t(x)+I2t(x).
By the mean value theorem and the elementary inequality
(3.26)12nr∫{y>0,|y-x0|<2nr}‍|f(y)-fB|dy≤cn‖f‖BMOLα,
we have
(3.27)I1t(x)≲∫(B*)c∩(0,∞)‍1te-|y-x0|2/10t|x-x0|t|f1(y)-fB|dy≲∫(B*)c∩(0,∞)‍|x-x0||y-x0|2|f1(y)-fB|dy≲r∑n=0∞1(2nr)2∫{y>0,|y-x0|<2nr}‍(|f(y)-fB|+|f(y)-f1(y)|)dy≲∑n=0∞2-n(n+1)‖f‖BMOLα.
On the other hand, by the fact |fB|≤C(1+log(ρLα(x0)/r))∥f∥BMOLα in Lemma 2.6, we obtain
(3.28)I2t(x)≲|∫-∞0‍(Ht(x,y)-Ht(x0,y))fBdy|≲∫{y<0}‍|x-x0||y-x0|2|fB|dy≲rρLα(x0)(1+log⁡ρLα(x0)r)‖f‖BMOLα≲‖f‖BMOLα.
Therefore, we obtain
(3.29)1|B|∫B‍sup⁡0<t≤ρLα(xk)2|Htf12(x)-Htf12(x0)|dx≲‖f‖BMOLα,
which establishes the proof.


## 4.g-Function

For allf∈Lloc1(0,∞) and x∈(0,∞), define the Littlewood-Paley g-function by
(4.1)g(f)(x)≡(∫0∞‍|Qtf(x)|2dtt)1/2,
where, {Qt}t>0 is a family of operators with the integral kernels
(4.2)Qt(x,y)=t2∂Wsα(x,y)∂s|s=t2.Theorem 4.1.
Letα>-1/2. There exists a constant C>0 such that, for all f∈BMOLα, g(f)∈BMOLα and ∥g(f)∥BMOLα≤C∥f∥BMOLα.Proof.
By Proposition2.9 and (a) of Proposition 2.10, we have
(4.3)Qt(x,y)≤c1te-c1|x-y|2/t2.
For f∈BMOLα, because of this and the integrability of (1+|x|)-2f(x) (see [12, page 141]),
(4.4)Qtf(x)=∫0∞‍Qt(x,y)f(y)dy
is well defined absolutely convergent integral for all (x,t)∈(0,∞)×(0,∞). Similar to the proof of Theorem 3.2, we will try to show that, for Bk⊂ℬ in Lemma 2.1,  (1)
(1/|Bk|)∫Bk‍|g(f)(x)|dx≤C∥f∥BMOLα,(2)
∥g(f)(x)∥BMO(Bk*)≤C∥f∥BMOLα. We split
(4.5)[g(f)(x)]2=[g1(f)(x)]2+[g2(f)(x)]2=∫020ρLα(xk)‍|Qtf(x)|2dtt+∫20ρLα(xk)∞‍|Qtf(x)|2dtt.
By Lemma 2.11 and Hölder inequality, assertion (1) holds for g1(f)(x). To finish the proof of (1), it suffices to show that
(4.6)‖g2(f)‖L∞(Bk*)≤c‖f‖BMOLα.
In the next proof, for the sake of brevity we introduce the additional notations:(4.7)X1t(x)={y∈(0,∞):e-t2xy1-e-2t2≤1},(4.8)X2t(x)=&{y∈(0,∞):e-t2xy1-e-2t2>1}.
By X1t(x) and X1t(x), we split Qtf(x) as
(4.9)|Qtf(x)|≤∫X1t(x)‍|f(y)||Qt(x,y)|dy+∫X2t(x)‍|f(y)||Qt(x,y)-Pt(x,y)|dy+∫X2t(x)‍|f(y)||Pt(x,y)|dy=I1t(x)+I2t(x)+I3t(x).
ForI1t(x) and x∈Bk*, we shall first show the inequality
(4.10)J1(x)=∫20ρLα(xk)∞‍|I1tf(x)|2dtt≤c‖f‖BMOLα2.
Using (a) of Proposition 2.9, if xk≤1, ρLα(xk)~xk, we get
(4.11)|Qt(x,y)|≤Ct2α+3(xyt2)α+1/2e-(x2+y2)/8t2e-(α+1)t2(1-e-2t2)α+2≤C(xkt)α+1/21te-|x-y|2/16t2.
Ifxk≥1, ρLα(xk)~1/xk, we have
(4.12)|Qt(x,y)|≤C1te-(x2+y2)/8t2e-((α+1)/2)t2≤C1txk1te-|x-y|2/16t2.
The previous two inequalities above imply
(4.13)J1(x)≲∫20ρLα(xk)∞‍|∫0∞‍(ρLα(xk)t)σ1te-|x-y|2/16t2|f(y)|dy|2dtt≲∫20ρLα(xk)∞‍(ρLα(xk)t)2σ|∑j=0∞‍2-j(N-1)12jt∫{y>0,|y-x|<2jt}‍|f(y)|dy|2dtt≲‖f‖BMOLα2.
ForI2t(x) and x∈Bk*, we shall also prove the inequality
(4.14)J2(x)=∫20ρLα(xk)∞‍|I2tf(x)|2dtt≤c‖f‖BMOLα2.
We split this integral as
(4.15)J2(x)=∫20ρLα(xk)1‍|I2tf(x)|2dtt+∫1∞‍|I2tf(x)|2dtt=J3(x)+J4(x).
To deal with J3(x), we discuss two cases. In the first case of xk≤1, notice that y>x, when x∈Bk*, y∈X2t(xk) and t≥20ρLα(xk). According to (b) of Proposition 2.9,
(4.16)J3(x)≲∫20ρLα(xk)1‍|∫(1-e-2t2)et2/x∞‍1te-|x-y|2/4t2xt(1-e-2t2)et2-x2|f(y)|dy|2dtt≲∫20ρLα(xk)∞‍xk2t2|∫0∞‍1te-|x-y|2/4t2|f(y)|dy|2dtt≲‖f‖BMOLα2.
The last inequality is from the same proof of J1(x). In the second case of xk>1, using (b) of Proposition 2.9 again, for t>20ρLα(xk) we obtain
(4.17)|I2t(x)|≲1xk∫xk/22xk‍txk|f(y)|dy+∫(0,∞)∩(xk/2,2xk)c‍1te-|x-y|2/2t2|f(y)|dy≲txk‖f‖BMOLα+txk∫0∞‍1te-|x-y|2/4t2|f(y)|dy≲txk‖f‖BMOLα.
The last inequality is also from the same proof of J1(x). Inserting this into J3(x) leads to J3(x)≤C∥f∥BMOLα2. Now, it remains to show J4(x)≤c∥f∥BMOLα2. Using (b) of Proposition 2.9, by the standard argument it easily follows that
(4.18)J4(x)≲∫1∞‍e-t/10t|∫0∞‍1te-|x-y|2/2t2|f(y)|dy|2dt≲‖f‖BMOLα2.
To complete the proof of (4.6), we need to show that
(4.19)J5(x)=∫20ρLα(xk)∞‍|I3tf(x)|2dtt≤c‖f‖BMOLα2.
We also consider two cases of xk≤1 and xk>1. When xk≤1, repeating the above argument for J3(x) and using (a) of Proposition 2.10, we have J5≤c∥f∥BMOLα2. When xk>1, using (a) of Proposition 2.10 again, for t≥20ρLα(xk), we obtain
(4.20)|I3t(x)|≲1xkt∫0∞‍1te-c|y-x|2/t2|f(y)|dy≲1txk∑j=0∞‍2-j(N-1)12jt∫{y>0,|y-x|<2jt}‍|f(y)|dy≲ρLα(xk)t‖f‖BMOLα,
which shows that (4.19) holds.
Next, we come to prove assertion (2). By (4.6) and Lemma 2.5, we only need to show
(4.21)‖(∫020ρLα(xk)‍|Qtf(x)|2dtt)1/2‖BMO(Bk*)≤C‖f‖BMOLα.
Consider any ballB=Br(x0)⊂Bk*. By Lemma 2.11, we have
(4.22)1|B|∫B‍(∫0r‍|Qtf(x)|2dtt)1/2dx≤(1|B|∫B‍∫0r‍|Qtf(x)|2dttdx)1/2≤C‖f‖BMOLα.
Therefore, by Lemma 2.5 and Corollary 2.3, it suffices to prove
(4.23)1|B|∫B‍|(∫r20ρLα(xk)‍|Qtf(x)|2dtt)1/2-(∫r20ρLα(xk)‍|Qtf(x0)|2dtt)1/2|dx≤C‖f‖BMOLα.
To prove (4.23), we first claim that, for all f∈BMOLα, x∈Bk*, and t≤20ρLα(xk),
(4.24)|Qtf(x)|≤C‖f‖BMOLα.
We shall split into three different estimates:(4.25)∫X1t(x)‍|f(y)||Qt(x,y)|dy≤C‖f‖BMOLα,(4.26)∫X2t(x)‍|f(y)||Qt(x,y)-Pt(x,y)|dy≤C‖f‖BMOLα,(4.27)|∫X2t(x)‍f(y)Pt(x,y)dy|≤C‖f‖BMOLα.
Let us first treat (4.25). Since y≤c(ρLα(xk)2/xk), when y∈X1t(x), notice that x~xk when x∈Bk*, using (a) of Proposition 2.9, and recalling the definition of ρLα(x), we have
(4.28)∫X1t(x)‍|f(y)||Qt(x,y)|dy≤C∫0cρLα(xk)2/xk‍txk2|f(y)|dy≤Ctxk‖f‖BMOLα≤C‖f‖BMOLα.
For (4.26), using (b) of Proposition 2.9, for x∈Bk*, the left side of (4.26) is controlled by
(4.29)1xk∫xk/22xk‍txk|f(y)|dy+∫(0,∞)∩(xk/2,2xk)c‍1te-|x-y|2/2t2|f(y)|dy≲txk‖f‖BMOLα+txk∫0∞‍1te-|x-y|2/4t2|f(y)|dy≲txk‖f‖BMOLα+(txk)12‖f‖BMOLα.
The third inequality is from the same argument for dealing with (3.21) in the proof of Theorem 3.2.
For (4.27), we write
(4.30)|∫X2t(x)‍f(y)Pt(x,y)dy|≤|∫X2t(x)‍|f(y)-fBt(x)||Pt(x,y)|dy|+|∫X2t(x)‍fBt(x)Pt(x,y)dy|.
By (a), (c) of Proposition2.10 and the fact that |fBt(x)|≲(1+log20ρLα(xk)/t)∥f∥BMOLα, we have
(4.31)|∫X2t(x)‍fBt(x)Pt(x,y)dy|≤|∫-∞∞‍fBt(x)Pt(x,y)dy|+|∫-∞et2(1-e-2t2)/x‍fBt(x)Pt(x,y)dy|≲t2ρLα(xk)2(1+log⁡20ρLα(xk)t)‖f‖BMOLα+∫-∞(1-e-2t2)et2/x‍1te-(x-y)2/16t2dy(1+log⁡20ρLα(xk)t)‖f‖BMOLα≲∫-∞(1-e-2t2)et2/tx-x/t‍e(-1/16)z2dz(1+log⁡20ρLα(xk)t)‖f‖BMOLα+‖f‖BMOLα≲e-(1/20)((1-e-2t2)et2/tx-xt)(1+log⁡20ρLα(xk)t)‖f‖BMOLα+‖f‖BMOLα≲t2ρLα(xk)2(1+log⁡20ρLα(xk)t)‖f‖BMOLα+‖f‖BMOLα≲‖f‖BMOLα.
The third and fourth inequalities come from the fact (1-e-2t2)et2/xt≲x/t and x~xk, when t≤20ρLα(xk) and x∈Bk*. It is notable that, for t≤20ρLα(xk) and x∈Bk*, the inequality above also implies
(4.32)|∫X2t(x)‍Pt(x,y)dy|≤CtρLα(xk).
On the other hand, by (a) of Proposition 2.10 and the simple fact that |f2jBt(x)-fBt(x)|≤Cj∥f∥BMOLα, we obtain
(4.33)|∫X2t(x)‍|f(y)-fBt(x)||Pt(x,y)|dy|≲∫0∞‍1te-|y-x|2/10t2|f(y)-fBt(x)|dy≲∑j=0∞2-j(N-1)12jt∫{y>0,|y-x|≤2jt}‍|f(y)-f2jBt(x)|dy+∑j=0∞2-j(N-1)|f2jBt(x)-fBt(x)|≲∑j=0∞j2-j(N-1)‖f‖BMOLα≲‖f‖BMOLα.
From (4.24), we deduce that
(4.34)|(∫r20ρLα(xk)‍|Qtf(x)|2dtt)1/2-(∫r20ρLα(xk)‍|Qtf(x0)|2dtt)1/2|≲(∫r20ρLα(xk)‍|Qtf(x)-Qtf(x0)|2dtt)1/2≲‖f‖BMOLα1/2(∫r20ρLα(xk)‍|Qtf(x)-Qtf(x0)|dtt)1/2.
By (4.28) and (4.29), for z∈Bk*, we have
(4.35)∫r20ρLα(xk)‍∫X1t(z)‍|f(y)||Qt(z,y)|dydtt≤C‖f‖BMOLα,∫r20ρLα(xk)‍∫X2t(z)‍|f(y)||Qt(z,y)-Pt(z,y)|dydtt≤C‖f‖BMOLα.
To finish the proof of (4.23), it suffices to show
(4.36)∫r20ρLα(xk)‍|∫X2t(x)‍(Pt(x,y)-Pt(x0,y))f(y)dy|dtt≤C‖f‖BMOLα,(4.37)∫r20ρLα(xk)‍|∫X2t(x)‍Pt(x0,y)f(y)dy-∫X2t(x0)‍Pt(x0,y)f(y)dy|dtt≤C‖f‖BMOLα.
It is easy to check (4.37). Indeed, without loss of generality, we assume x<x0. According to (a) of Proposition 2.10,
(4.38)|∫X2t(x)‍Pt(x0,y)f(y)dy-∫X2t(x0)‍Pt(x0,y)f(y)dy|≲∫(1-e-2t2)/et2x0(1-e-2t2)/et2x‍1t|f(y)|dytrx02‖f‖BMOLα.
Inserting this into (4.37) gives the desired result. Moreover, from the above we also have proved
(4.39)|∫X2t(x)‍Pt(x0,y)f(y)dy-∫X2t(x0)‍Pt(x0,y)dy|≤CtρLα(x0).
Forr≤t≤20ρLα(xk), we write
(4.40)|∫X2t(x)‍(Pt(x,y)-Pt(x0,y))f(y)dy|≤|∫X2t(x)‍(Pt(x,y)-Pt(x0,y))(f(y)-fB)dy|+|fB||∫X2t(x)‍(Pt(x,y)-Pt(x0,y))dy|=S1t(x)+S2t(x).
By (b) of Proposition 2.10, we obtain
(4.41)S1t(x)≲∫{y>0,|y-x0|>r}‍rt1te-c|y-x0|2/t2|f(y)-fB|dy+1t∫{y>0,|y-x0|≤r}‍|f(y)-fB|dy≲∑j=0∞‍∫2j+1B‍rt1/21(2jr)3/2|f(y)-f2j+1B+f2j+1B-fB|dy+rt‖f‖BMOLα≲(rt)1/2∑j=0∞‍2-j/2j‖f‖BMOLα+rt‖f‖BMOLα≲(rt+(rt)1/2)‖f‖BMOLα.
From the inequality above, we deduce that(4.42)∫r20ρLα(xk)‍S1t(x)dtt≤∫r20ρLα(xk)‍(rt+(rt)1/2)dtt‖f‖BMOLα≤C‖f‖BMOLα.
On the other hand, by using (b) of Proposition2.10 again, we also have
(4.43)|∫X2t(x)‍(Pt(x,y)-Pt(x0,y))dy|≤Crt.
This together with (4.32), (4.39), and |fB|≲(1+log(ρLα(x0)/r))∥f∥BMOLα gives that
(4.44)∫r20ρLα(xk)‍S2t(x)dtt≲∫r20ρLα(xk)‍(rt)2/3(tρLα(xk))1/3(1+log⁡ρLα(x0)r)dtt‖f‖BMOLα≲∫r20ρLα(xk)‍(rt)1/3dtt‖f‖BMOLα≲‖f‖BMOLα.
Thus we finish the proof of (4.36) and, hence, finish the proof of Theorem 4.1.



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*Source: 923874-2012-08-13.xml*