# End Point Estimate of Littlewood-Paley Operator Associated to the Generalized Schrödinger Operator

Authors: Yanping Chen; Wenyu Tao
Journal: Journal of Function Spaces
 (2021)

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## Abstract

LetL=−Δ+μ be the generalized Schrödinger operator on ℝd,d≥3, where μ≠0 is a nonnegative Radon measure satisfying certain scale-invariant Kato conditions and doubling conditions. In this work, we give a new BMO space associated to the generalized Schrödinger operator L,BMOθ,L, which is bigger than the BMO spaces related to the classical Schrödinger operators A=−Δ+V, with V a potential satisfying a reverse Hölder inequality introduced by Dziubański et al. in 2005. Besides, the boundedness of the Littlewood-Paley operators associated to L in BMOθ,L also be proved.


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## Body

## 1. Introduction

Consider the generalized Schrödinger operator(1)L=−Δ+μ,where μ is a nonnegative Radon measure on ℝd,d≥3. According to [1], there exist positive constants C0, C1, and δ such that μ satisfies the following conditions:
(2)μBx,r≤C0rRd−2+δμBx,R,(3)μBx,2r≤C1μBx,r+rd−2,for all x∈ℝd and 0<r<R, where Bx,r denotes the open ball centered at x with radius r. Condition (2) is regarded as scale-invariant Kato-condition, and from (3), we can see that the measure μ is doubling on balls satisfying μBx,r≥crd−2. We will also assume that μ≠0. If dμ=Vxdx and V≥0 are in the reverse Hölder class, that is, there exists C=Cd,V>0 such that
(4)1Bx,r∫Bx,rVyd/2dy2/d≤CBx,r∫Bx,rVydy,then μ satisfies the conditions (2) and (3) for some δ>0. However, in general, measures which satisfy (2) and (3) need not be absolutely continuous with respect to the Lebesgue measure on ℝd, the counterexample is visible in ([1], Remark 0.10). Let A=−Δ+V, it is easy to know that L is more general than A from the above.The boundedness of the operators associated to the classical operatorA such as the Gauss-semigroup and Poisson-semigroup maximal functions, the Littlewood-Paley-square function and the fractional integral operator has attracted much interest [2–4]. It is worth noting that these operators fail to be bounded in BMO, even in the classic case (i.e.V=0). In 2005, Dziubański et al. [2] identified the BMO-type space related to A, BMOA, namely,
(5)BMOA=f∈BMO:1Br∫Brfydy≤C,forallBr=Bx,r,r≥ρx,V,where ρx,V is the auxiliary function, defined by
(6)ρx,V=supr>0:1rd−2∫Bx,rVydy≤1,(see [2, 4–7]). They proved the above operators were bounded in this space. By the fact that BMOA is a subspace of BMO, we know that it is very meaningful to consider the boundedness of these operators if we can expand the BMOA space a little bit.In this paper, we shall be interested in a newBMO space associated to the generalized Schrödinger operator L. To give the definition of the new BMO space, we first recall the auxiliary function ρx,μ (see [8]),
(7)ρx,μ=supr>0:μBx,rrd−2≤C1,where C1 is the constant in (3).Here, we define the newBMO space, BMOθ,L, namely,
(8)BMOθ,L=f∈BMOθ:1Br∫Brfydy≤C1+rρx,μθ,forallBr=Bx,r,r≥ρx,μ,where
(9)BMOθ=f∈Lloc1:1Br∫Brfy−fBrdy≤C1+rρx,μθ,forallBr=Bx,r,for θ>0. The precise definition of the norm in the spaces BMOθ,L is given in Definition 1.Definition 1.
Letμ≠0 be a nonnegative Radon measure in ℝd,d≥3. Assume that μ satisfies the conditions (2) and (3). For θ>0, we shall say that a locally integrable function f belongs to BMOθ,L whenever there is a constant C≥0 so that
(10)1BR∫BRf≤C1+Rρx,μθand1Br∫Brfy−fBrdy≤C,for all balls BR=Bx,R, Br=Bx,r such that R≥ρx,μ≥r. The norm of f∈BMOθ,L, denoted by fBMOθ,L, is the smallest C in (10) above. Here and subsequently, fB=1/B∫Bfxdx.Remark 2.
From the definition ofBMOθ,L, we have BMOθ,L⊆BMOθ. Also, we emphasize that BMOθ,L is actually bigger than BMOA, which is defined by Dziubański et al. in [2], that is, BMOA⊆BMOθ,L. From the definition of BMOθ,L and the fact that L is more general than A, it is obvious that BMOA is the subset of BMOθ,L.Sinceμ is nonnegative on ℝd, the Feynman-Kac formula implies that the kernel ktx,y of the semigro up e−tL of linear operators generated by −L satisfies
(11)0≤ktx,y≤htx−y=4πt−d/2e−x−y24t,also see in [9].After Wang [10] considered the g-function defined on BMO functions, more and more scholars pay attention to the end point estimate of the Littlewood-Paley operator [3, 8, 11–13]. In this paper, we will also consider the follows Littlewood-Paley operators associated to the generality Schrödinger operator are bounded in BMOθ,L.
(12)sfx≔∫0∞tLe−tLfx2dtt1/2=∫0∞tdTtdtfx2dtt1/2,where Ttfx=e−tLfx=∫ℝdktx,yfydy.The main theorem is as follows.Theorem 3.
Letθ>0 and s defined as in (12), then s is bounded in BMOθ,L. That is, if f∈BMOθ,L, there exist a constant C>0 such that
(13)sfBMOθ,L≤CfBMOθ,L.The paper is organized as follows. In Section2, we give some necessary lemmas. In Section 3, we consider sf and give the proof of Theorem 3.Throughout this paper, give a ballB, we denote by B∗ the ball with the same center and twice radius. c and C will denote positive constants that may not be the same in each occurrence.


## 2. Preliminaries

We first recall some basic properties of the auxiliary functionρx,μ in (7) (see [1], Proposition 1.8). In the sequel, C0,C1, and δ are positive constants in (2) and (3).Lemma 4.
Supposeμ satisfies (2) and (3). Then
(i)
Ifr=ρx,μ, then rd−2≤μBx,r≤C1rd−2(ii)
If∣x−y∣≤Cρx,μ, then ρx,μ~ρy,μ(iii)
There exist constantsC>0 such that for x,y∈ℝd(14)C−1ρx,μ1+∣x−y∣ρx,μ−k0≤ρy,μ≤Cρx,μ1+x−yρx,μk0/k0+1,with k0=C2/δ>0 and C2=log2C1+2d−2.Next, we recall some results about coveringℝd by critical balls, which can be found in (see [14], Lemma 2.3).Lemma 5.
There exists a sequence of pointsxkk=1∞ in ℝd, so that the family of critical balls Qk=Bxk,ρxk,μ,k≥1, satisfies
(i)
⋃kQk=ℝd(ii)
There existsN such that for every k∈ℕ, cardj:4Qj∩4Qk≠∅≤NThe kernelktx,y of the semigroup e−tL satisfies following upper bound in ([15], Theorem 1.1).Lemma 6.
Letμ≠0 be a nonnegative Radon measure in ℝd,d≥3. Assume that μ satisfies the conditions (2) and (3). There exist constants c, C, the constant k0 in (14), the heat kernel ktx,y of e−tL satisfies
(15)ktx,y≤Ct−d/2e−cx−y2/te−c1+t/ρx,μ1/k0+1.Using the inequality (15) and repeating the proof process in ([2], Proposition 4), we can obtain the estimates for the integral kernel Qtx,y=t∂ktx,y/∂t.Lemma 7.
There exist constantsc,σ>0 such that for k0 in (14) and any N, there is a constant C so that
(i)
Qtx,y≤Ct−d/2e−cx−y2/te−c1+t/ρx,μ1/k0+1(ii)
Qtx+h,y−Qtx,y≤Ch/tσt−d/2e−cx−y2/te−c1+t/ρx,μ1/k0+1, for all h≤t(iii)
∫ℝdQtx,ydy≤Ct/ρx,μσ/1+t/ρx,μNFinally, following ([16], Proposition 3), we recall some basic properties about the norm of BMOθ.Lemma 8.
Forθ>0, let f∈BMOθ, B=Bx,r, and m≥1, then
(i)(16)1B∫Bfy−fBmdy1/m≤CfBMOθ′1+rρx,μθ′,and for all k∈ℕ(ii)(17)12kB∫2kBfy−fBmdy1/m≤CfBMOθ′k1+2krρx,μθ′,where θ′=k0+1θ and k0 the constant appearing in (14).


## 3. Proof of Theorem3

Before we prove Theorem3, we first recall some basic facts about the nonnegative Radon measure μ.Lemma 9.
(i)
For all ballsBx,r in ℝd, let C1 as in (3), then
(18)∫Bx,2rdμ≤C1∫Bx,rdμ+rd−2.(ii)
For0<r<R<∞, let δ and C0 as in (3), then
(19)1rd−2∫Bx,rdμ≤C0rRδ1Rd−2∫Bx,Rdμ.(iii)
Ifr~ρx,μ, then
(20)∫Bx,rdμ~rd−2.Proof.
The proof of (i) and (ii) can be obtained directly from (2) and (3), respectively. From Lemma 4(i), the proof of (iii) can be obtained.Next, we give the following result, which is similar to the proof of ([17], Corollary 1).Lemma 10.
A functionf belong to BMOθ,L with θ>0 if and only if
(21)1B∫Bfy−fBdy≤C,
for every ballB=Bx,r with x∈ℝd and r<ρx,μ and
(22)1Bx,ρx,μ∫Bx,ρx,μfydy≤C,for all x∈ℝd.Proof.
From the Definition1, it is easy to see that if f∈BMOθ,L, then f satisfies (21) and (22), furthermore, to prove f∈BMOθ,L, we need (22) and
(23)1B∫Bfydy≤C1+rρx,μθ,for every ball B=Bx,r with x∈ℝd and r≥ρx,μ.
Obviously, we just have to prove that iff satisfies (22) then f satisfies (23). Let xkk=1∞ be a sequence as in Lemma 5, it follows from (22) that
(24)1Bk∫Bkfydy≤C,where Bk=Bxk,ρxk,μ for all k≥1. According to Lemma 5, we have the set N=k:B∩Bk≠∅ is finite.
For everyk∈N, and z∈B∩Bk, by Lemma 4(iii) we get
(25)ρxk≤Cρz1+xk−zρxk,μk0≤C2k0ρz≤C2k0ρx,μ1+x−zρx,μk0/k0+1≤C2k0ρx,μ1+rρx,μ≤C2k0+1r,then Bk⊆CB for every k∈N. Thus
(26)∫Bfydy≤∑k∈N∫B∩Bkfydy≤C∑k∈NBk≤CB.
That is forr≥ρx,μ(27)1∣B∣∫Bfydy≤C≤C1+rρx,μθ.
Now, we turn to estimate the boundedness of the operators from BMOθ,L to BMOθ,L. According to Lemma 10, we only need to check that
(1)
1/Bx0,ρx0,μ∫Bx0,ρx0,μsfxdx≤CfBMOθ,L(2)
1/B∫Bsf−sfB≤CfBMOθ,L,forB=Bx0,randr<ρx0,μProof of Theorem 3.
Without loss of generality, fixf∈BMOθ,L with fBMOθ,L=1, we first to show that
(28)1Bx0,ρx0,μ∫Bx0,ρx0,μsfxdx≤C.
We split(29)sfx2≤∫0ρx0,μ2tLe−tLfx2dtt+∫ρx0,μ2∞tLe−tLfx2dtt≕s1fx2+s2fx2.
Fors1f2, we set
(30)f=f−fBx0,2ρx0,μXBx0,2ρx0,μ+f−fBx0,2ρx0,μXBx0,2ρx0,μc+fBx0,2ρx0,μ≕∑i=13fi.
Using the self-adjointness oftLe−tL, we get
(31)sfL2~fL2.
Then, combining Hölder’s inequality, (31), Lemma 8(i), and Lemma 4(ii) with ρx,μ~ρx0,μ for x∈Bx0,ρx0,μ, we have
(32)1Bx0,ρx0,μ∫Bx0,ρx0,μs1f1xdx2≤C1Bx0,ρx0,μ∫Bx0,ρx0,μs1f1x2dx≤C1Bx0,ρx0,μf1L22=C1Bx0,ρx0,μ∫Bx0,2ρx0,μfx−fBx0,2ρx0,μ2dx≤CfBMOθ21+2ρx0,μρx0,μ2θ′≤C,where the last inequality follows from Remark 2.
Next, fort<ρx0,μ2 and x∈Bx0,ρx0,μ, it follows from Lemma 7(i) and Lemma 8(ii) that
(33)tLe−tLf2x≤C∫ℝdf2yQtx,ydy≤C∫Bx0,2ρx0,μcfy−fBx0,2ρx0,μt−d/2e−c′x−y2/te−c″x−y2/tdy≤C∫Bx0,2ρx0,μcfy−fBx0,2ρx0,μt−d/2x−yt−d+1e−c″x−y2/tdy≤C∫Bx0,2ρx0,μcfy−fBx0,2ρx0,μtx−y−d+1e−c″x−y2/tdy≤C∑j=1∫Bx0,2j+1ρx0,μ\Bx0,2jρx0,μt2jρx0,μd+1fy−fBx0,2ρx0,μe−c″x0−y2/tdy≤Ctρx0,μ∑j=12−d2−je−c″22j12j+1ρx0,μd∫Bx0,2j+1ρx0,μfy−fBx0,2ρx0,μdy≤Ctρx0,μ∑j=12−d2−je−c″22jj2jθ′fBMOθ′≤Ctρx0,μ.
Then, we obtain(34)1Bx0,ρx0,μ∫Bx0,ρx0,μs1f2xdx2≤C∫0ρx0,μ2tρx0,μ2dtt≤C.
Now, we turn to estimate the term fors1f3. Using ρx,μ~ρx0,μ for x∈Bx0,ρx0,μ and Lemma 7(iii), we have
(35)1Bx0,ρx0,μ∫Bx0,ρx0,μs1f3xdx2≤1Bx0,ρx0,μ∫0ρx0,μ2∫Bx0,ρx0,μtLe−tLfBx0,2ρx0,μ2dxdtt≤CfBx0,2ρx0,μ2Bx0,ρx0,μ∫0ρx0,μ2∫Bx0,ρx0,μtρx0,μ2σdxdtt≤CfBx0,2ρx0,μ2≤C1+2ρx0,μρx02θ≤C.
Fors2f, one can easily check using Lemma 7(i), Minkowski’s inequality and ρx,μ~ρx0,μ for x∈Bx0,ρx0,μ that
(36)s2fx≤C∫ρx0,μ2∞∫ℝdQtx,yfydy2dtt1/2≤C∫ρx0,μ2∞∫ℝde−c1+tρx0,μ1/k0+1t−d2e−cx−y2tfydy2dtt1/2≤C∫ρx0,μ2∞e−c1+tρx0,μ1/k0+11td/2∫x−y≤tfydy+e−c1+tρx0,μ1/k0+1∑j=1e−22j1td/2∫x−y~2jtfydy2dtt1/2≤C∫ρx0,μ2∞e−c1+tρx0,μ2/k0+11+tρx0,μ2θdtt1/2+∑j=1e−22j2jd2j+1θ∫ρx0,μ2∞e−c1+tρx0,μ2/k0+1tρx0,μ2θdtt1/2≤C.
Next, we deal with the following case(37)1B∫Bsf−sfB≤C,forB=Bx0,randr<ρx0,μ.
In this section, inspired by the methods in [2], we decompose f as follows
(38)f=f−fB∗XB∗+f−fB∗XB∗c+fB∗≕∑i=13f~ix.
Set(39)sB=∫0ρx0,μ2tΔe−tΔf~2x02dtt1/2.
It is easy to see thatsB is a finite constant and
(40)sfx−sB≤∫ρx0,μ2∞tLe−tLfx2dtt1/2+∫0ρx0,μ2tLe−tLfx−tΔe−tΔfx2dtt1/2+∫0ρx0,μ2tΔe−tΔfx−tΔe−tΔf~2x02dtt1/2≕S1fx+S2fx+S3fx.
To complete the proof, it suffices to show that(41)1B∫BSifxdx≤Cfori=1,2,3.
Note thatρx,μ~ρx0,μ for any x∈Bx0,r, by (36), we get for x∈Bx0,r(42)S1fx≤∫ρx0,μ2∞∫ℝdQtx,yfydy2dtt1/2≤C.
It remains to estimateS2f. Let Gtx,y=t∂tktx,y−t∂thtx,y. We claim
(43)Gtx,y≤Cs−d/2e−cx−y2/ttρx,μσ,for t<ρx,μ. It follows from the perturbation formula (see ([18], Chapter 9, (2.3))) that
(44)Gtx,y=∫ℝdtht/2w,xkt/2w,ydμw+∫t/2t∫ℝdthtw,x∂tkt−sw,ydμwds+∫0t/2∫ℝdt∂tht−sw,xktw,ydμwds≕I1+I2+I3.
By Lemma9, we have for t<ρx,μ,
(45)I1=∫x−w<x−y/2tht/2w,xkt/2w,ydμw+∫x−w≥x−y/2tht/2w,xkt/2w,ydμw≤Ct−d2+1e−cx−y2t∫x−w<x−y/2t−d2e−cx−w2tdμw+Ct−d2+1∫x−w≥x−y/2t−d2e−cx−w2+x−y2tdμw≤Ct−d2e−cx−y2t∫ℝdt−d−22e−cx−w2tdμw≤Ct−d2e−cx−y2t1td−2∫x−w<tdμw+∑k=0∞2−2k1td−2∫2kt<x−w<2k+1tdμw≤Ct−d2e−cx−y2ttρx,μσ.
By the same argument as estimatingI1, combining Lemma 7, we also obtain
(46)I2=∫t/2t∫ℝdtt−shtw,xQt−sw,ydμwds≤∫t/2t∫w−y<x−y/2tt−shtw,xQt−sw,ydμwds+∫t/2t∫w−y≥x−y/2tt−shtw,xQt−sw,ydμwds≤Ct−d2e−cx−y2t∫t/2t∫w−y<x−y/2s−d2e−cy−w2sdμwds+Ct−d2∫t/2t∫w−y≥x−y/2s−d2e−cx−y2+y−w2sdμwds≤Ct−d2e−cx−y2t∫t/2t1ssρx,μσds=Ct−d2e−cx−y2ttρx,μσ.
The same estimate asI2, we have
(47)I3≤Ct−d/2e−cx−y2/ttρx,μσ.
The proof of (43) is finished. For x∈Bx0,r, ρx0,μ~ρx,μ, it follows from Definition 1 and Lemma 8(ii) that
(48)S2fx≤∫0ρx0,μ2∫ℝdGtx,yfydy2dtt1/2≤C∫0ρx0,μ2∫ℝdt−d/2e−cx−y2/ttρx,μσfydy2dtt1/2≤C∫0ρx0,μ2∫Bx,ρx,μρx,μ−σx−y−d−σ2fydy2dtt1−σ/2+∫0ρx0,μ2∫Bx,ρx,μce−c″x−y2/t·x−y−d+1fydy2dt1/2≤C∑j=0∞ρx,μ−d2j+1d−σ/2∫Bx,2−jρx,μ\Bx,2−j+1ρx,μfydy+C∑j=0∞ρx,μ−d2−jd+1e−c″22j∫Bx,2j+1ρx,μ\Bx,2jρx,μfydy≤C∑j=0∞2d2−j+1δ/212−jρx,μd∫Bx,2−jρx,μfydy+C∑j=0∞2j+1d2−jd+1e−c″22j12j+1ρx,μd∫Bx,2j+1ρx,μfydy≤C∑j=0∞2d2−j+11Bx,2−jρx,μ∫Bx,2−jρx,μfy−fBx,2−jρx,μdy+∑i=0j−1fBx,2−j+iρx,μ−fBx,2−j+i+1ρx,μ+fBx,ρx,μ+C∑j=0∞2j+1d2−jd+1e−c″22j1+2j+1ρx,μρx,μθ≤C∑j=0∞2d2−j+1δ/2j2θ′+2θ+C∑j=0∞2d2−j2j+1θe−c″22j≤C.
ForS3f, since the formula tΔe−tΔ1=0.
(49)S3fx≤∫0ρx0,μ2tΔe−tΔf~1x2dtt1/2+∫0ρx0,μ2tΔe−tΔf~2x−tΔe−tΔf~2x02dss1/2=S3,1x+S3,2x.
It is known that∫0∞tΔe−tΔf2dt/t1/2≕sΔf is bounded on L2ℝd ([19]). Hence, combining with Lemma 8(i), we obtain
(50)1B∫BS3,1xdx≤1B∫BsΔf~1xdx≤1B∫BsΔf~1x2dx1/2≤C1Bf~1L221/2≤C1B∫B∗f−fB∗21/2≤CfBMOθ≤C.
We set the kernel oft∂thtx,y≕Q~tx,y. We recall this kernel satisfies the estimates
(51)Q~tx,y≤Ct−d/2e−ct−1x−y2
and(52)Q~tx+h,y−Q~tx,y≤Cht−d+1/2e−ct−1x−y2ifh≤x−y2.
Forx∈Bx0,r and r≤ρx0,μ, choosing k0 such that 2k0r<ρx0,μ≤2k0+1r, it follows from Minkowski’s inequality and Lemma 8(ii) that
(53)S3,2x≤∫0ρx0,μ2∫B∗cQ~tx,y−Q~tx0,yf~2ydy2dtt1/2≤C∫0ρx0,μ2∫B∗cx−x0t−d+1/2e−ct−1x0−y2f~2ydy2dtt1/2≤C∫0r2∑j=1∞∫Bx0,2j+1r\Bx0,2jrrt−d+1/2e−c′t−1x0−y2e−c″22jf~2ydy2dtt1/2+C∫r2ρx0,μ2∑j=1k0+∑j=k0+1∞∫Bx0,2j+1r\Bx0,2jrrt−d/2+1/4e−ct−1x0−y2f~2ydy2dtt3/21/2≤C∑j=1∞e−c″22j∫0r2∫Bx0,2j+1r\Bx0,2jrrt−d/2+1e−c′t−1x0−y2f~2ydy2dt1/2+C∑j=1k0∫r2ρx0,μ2∫Bx0,2j+1r\Bx0,2jrrt−d/2+1/4e−ct−1x0−y2f~2ydy2dtt3/21/2+C∑j=k0+1∞e−c″22j−k0+1∫r2ρx0,μ2∫Bx0,2j+1r\Bx0,2jrrt−d/2+1/4e−c′t−1x0−y2f~2ydy2dtt3/21/2≤C∑j=1∞e−c″22jr4∫Bx0,2j+1r\Bx0,2jrx0−y−d+2f~2ydy21/2+C∑j=1k0r∫Bx0,2j+1r\Bx0,2jrx0−y−d+1/2f~2ydy21/2+C∑j=k0+1∞e−c″22j−k0+1r∫Bx0,2j+1r\Bx0,2jrx0−y−d+1/2f~2ydy21/2≤C∑j=1∞e−c″22j2−2j+d+∑j=1k02−1/2j+d+∑j=k0+1∞e−c″22j−k0+12−1/2j+d1Bx0,2j+1r∫Bx0,2j+1rf~2ydy≤C∑j=1∞e−c″22j2−2j+dj2jθ′+C∑j=1k02−1/2j+dj2jθ′+C∑j=k0+1∞e−c″22j−k0+12−1/2j+dj2jθ′≤C.
Therefore,(54)1B∫BS3fxdx≤C.
By (42), (48), and (54), we finish the proof of (37). Then, combining (28) and (37), we finish the proof of Theorem 3.



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