boundedness of variation operators and oscillation operators for certain semigroups

Nonlinear Analysis 106 (2014) 124–137

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Nonlinear Analysis

journal homepage: www.elsevier.com/locate/na

Boundedness of variation operators and oscillation operatorsfor certain semigroupsThe Anh Bui ∗Department of Mathematics, Macquarie University, NSW 2109, AustraliaDepartment of Mathematics, University of Pedagogy, HoChiMinh City, Viet Nam

a r t i c l e i n f o

Article history:Received 9 July 2013Accepted 22 April 2014Communicated by Enzo Mitidieri

MSC:42B2042B25

Keywords:Variation operatorOscillation operatorHeat semigroupSchrödinger operator

a b s t r a c t

In this paper, we study the boundedness of variation operators and oscillation operators forcertain semigroups. As applications, this theory can be applied to various settings such asSchrödinger operators, degenerate Schrödinger operators onRn and Schrödinger operatorson Heisenberg groups and connected and simply connected nilpotent Lie groups.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Let (X, d, µ) be a metric space endowed with a nonnegative Borel measure µ. Denote B(x, r) := {y ∈ X : d(x, y) < r}.In this paper we assume the following two conditions, see [1–3]:

(D) Doubling property condition: There exists a constant C1 > 0 such thatµ(B(x, 2r)) ≤ C1µ(B(x, r)) (1)

for all x ∈ X and r > 0. This condition implies that there exists a constant n ≥ 0 so that

µ(B(x, λr)) ≤ C2λnµ(B(x, r)) (2)

for all x ∈ X, r > 0 and λ ≥ 1.(RD) Reverse doubling property condition: There exist positive constants 0 < θ ≤ n and C2 such that

λθµ(B(x, r)) ≤ C3µ(B(x, λr)) (3)for all x ∈ X , 0 < r < diamX/2 and 1 ≤ λ < diamX/2.

Let (Tt)t>0 be a family of operators defined on Lp(X), 1 ≤ p < ∞. A classical method of measuring the speed of conver-gence for the family (Tt)t>0 is to study the boundedness of variation operators and oscillation operators associated to thefamily (Tt)t>0. Recall that for ρ > 2, the ρ-variation operator associated with (Tt)t>0 is defined by

Vρ(Tt)f (x) = suptj↘0

∞j=1

|Ttj f (x) − Ttj+1 f (x)|ρ1/ρ

,

where the supremum is taken over all the positive decreasing sequences {tj}j which converge to 0.

∗ Correspondence to: Department of Mathematics, Macquarie University, NSW 2109, Australia. Tel.: +61 433878198.E-mail addresses: [email protected], [email protected].

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T.A. Bui / Nonlinear Analysis 106 (2014) 124–137 125

The oscillation operator associated with (Tt)t>0 is defined by

O(Tt)f (x) =

∞j=1

suptj+1≤ϵj+1<ϵj≤tj

|Tϵj f (x) − Tϵj+1 f (x)|21/2

,

where {tj}j is a fixed positive decreasing sequence which converges to 0.The oscillation and variation for martingales and some families of operators have been investigated intensively in proba-

bility, ergodic theory, and harmonic analysis. For further results on oscillation and variation operators in harmonic analysisand ergodic theory and their applications, we refer to [4–11] and the references therein. In [10], the authors proved thefollowing result:

Theorem A. Let (Σ, dµ) be a positive measure space. Let T = {Tt}t be a symmetric diffusion semigroup satisfying TtTs = Tt+s,T0 = I , limt→0 Tt f = f on L2(Σ, dµ) and(i) ∥Tt f ∥Lp(Σ,dµ) ≤ ∥f ∥Lp(Σ,dµ) for all 1 ≤ p ≤ ∞;(ii) Tt is self-adjoint on L2(Σ, dµ) for all t;(iii) Tt f ≥ 0 for all f ≥ 0 and all t;(iv) Tt(1) = 1 for all t.Then the operators Vρ(Tt) and O(Tt) are bounded on Lp(Σ, dµ), 1 < p < ∞.

More recently, the boundedness of variation operators of semigroups related to Schrödinger operators has been studiedin [12,13]. It was proved in [12,13] that:

Theorem B. Let L = ∆ + V be a Schrödinger operator on Rn, n ≥ 3 and let ρ > 2. If V ∈ RHq(Rn), q > n/2 then we have(i) the variation operator Vρ(e−tL) is bounded on Lp(Rn), 1 < p < ∞;(ii) the variation operator Vρ(e−tL) is bounded on BMOL(Rn), where BMOL(Rn) is a BMO space associated to L.

The aim of this paper is to extend the results in [12,13] to the more general setting which is introduced by D. Yang andY. Zhou (See [14]). Before coming to the details, we would like to introduce the setting we will work in this paper.

Let ∆ be a nonnegative self-adjoint operator on L2(X) which generates semigroups {e−t∆}t>0. Denote by p∆

t (x, y) andq∆t (x, y) the kernels associated with e−t∆ and t∆e−t∆, respectively. We assume that there exist two positive constants C

and c so that for all x, y ∈ X and t > 0,(∆1) |p∆

t (x, y)| ≤C

V (x,√t)exp

−c d(x,y)2

t

;

(∆2)X p∆

t (x, y)dµ(x) =X p∆

t (x, y)dµ(y) = 1.Obviously, {e−t∆

}t>0 is a symmetric diffusion semigroup and hence by Theorem A, the variation operators Vρ(e−tL), ρ >

2 and the oscillation operators O(e−tL) are bounded on Lp(X) for all 1 < p < ∞.Letγ be a critical function (See Section 2.1). In this paper,we assume that L is a nonnegative self-adjoint operator on L2(X).

Denote by pt(x, y) and qt(x, y) the kernels associatedwith e−tL and tLe−tL, respectively.We assume the following conditions:(L1) For all N > 0, there exist positive constants c and C so that

|pt(x, y)| ≤C

V (x,√t)

exp−c

d(x, y)2

t

1 +

√t

γ (x)+

√t

γ (y)

−N

for all x, y ∈ X and t > 0;(L2) There is a positive constant δ1 > 0 so that for all N > 0, there exist positive constants c and C:

|qt(x, y) − qt(x, y)| ≤C

V (x,√t)

d(x, x)d(x, y)

δ1exp

−c

d(x, y)2

t

1 +

√t

γ (x)+

√t

γ (y)

−N

whenever d(x, x) ≤√t/2 and t > 0;

(L3) There is a positive constant δ2 > 0 so that

|pt(x, y) − p∆t (x, y)| + |qt(x, y) − q∆

t (x, y)| ≤C

V (x,√t)

√t

√t + γ (x)

δ2exp

−c

d(x, y)2

t

for all x, y ∈ X and t > 0.

Note that since the conservation property Tt(1) = 1 may not hold for the semigroups e−tL, Theorem A cannot be appliedin our situation.

Remark 1.1. Note that the condition (L1) implies that for all N > 0, there exist positive constants c and C so that

|qt(x, y)| ≤C

V (x,√t)

exp−c

d(x, y)2

t

1 +

√t

γ (x)+

√t

γ (y)

−N(4)

for all x, y ∈ X and t > 0.Since the proof of (4) is quite standard, we leave to the interested reader.

126 T.A. Bui / Nonlinear Analysis 106 (2014) 124–137

The main results in this paper are formulated in the following theorems. The first theorem gives the Lp-boundedness ofthe variation operators and the oscillation operators for the semigroup e−tL. The latter addresses the boundedness of theseoperators on the BMO type spaces associated to the critical function γ .

Theorem 1.2. Let L satisfy assumptions (L1)–(L3) and let ρ > 2. Then the operatorsVρ(e−tL) andO(e−tL) are bounded on Lp(X)for all 1 < p < ∞.

Theorem 1.3. Let L satisfy assumptions (L1)–(L3) and let ρ > 2. Then the operators Vρ(e−tL) and O(e−tL) are bounded onBMOβ

γ (X) for β ∈ [0, δ0/n), where δ0 = min{δ1/2, δ2/2}.

The organization of this paper is as follows. In Section 2, we recall the definition of the spaces of BMO type associated tocritical functions and their basic properties in [15] in Section 2. The proofs of Theorems 1.2 and 1.3 are given in Section 3.Finally, some applications are also considered in Section 4.

Throughout the paper, we always use C and c to denote positive constants that are independent of the main parametersinvolved but whose values may differ from line to line. We shall write A . B if there is a universal constant C so that A ≤ CBand A ≈ B if A . B and B . A. Given a λ > 0 and a ball B := B(x, r), we will write λB for the λ-dilated ball, which is the ballwith the same center as B and with radius λr . For each ball B ⊂ X , we set

S0(B) = B and Sj(B) = 2jB \ 2j−1B for j ∈ N.

For any measurable subset E ⊂ X , we denote V (E) := µ(E). For all x ∈ X and r > 0, we also denote V (x, r) = µ(B(x, r)).

2. Preliminaries

2.1. Critical functions

Let γ be a positive function on X . The function γ is called a critical function if there exist positive constants C and k0 sothat

γ (y) ≤ C[γ (x)]1

1+k0 [γ (x) + d(x, y)]k0

k0+1 (5)

for all x, y ∈ X .Note that the concept of critical functions was introduced first to the setting of Schrödinger operators on Rn in [16] (see

also [17]) and then was extended to the spaces of homogeneous type in [14,15,18].One of the most important classes of the critical functions is the one involving the weights satisfying the reverse Hölder

inequality. Recall that a nonnegative locally integrable function w is said to be in the reverse Hölder class RHq(X), q > 1 ifthere exists a constant C > 0 so that 1

V (B)

Bw(x)qdµ(x)

1/q≤

CV (B)

Bw(x)dµ(x)

for all balls B ⊂ X .Note that if w ∈ RHq(X) then w is a Muckenhoupt weight. See [19].Let V ∈ RHq(X) for some q > 1. Set

γ (x) = supr > 0 :

r2

µ(B(x, r))

B(x,r)

V (y)dµ(y) ≤ 1. (6)

See for example [17,14].It was proved in [14] that if n ≥ 1 and V ∈ RHq(X) with q > max{1, n/2} then γ (·) is a critical function. See for example

[14,17].For x ∈ X , we call the ball B(x, γ (x)) a critical ball. We now give some basic properties for the critical functions and

critical balls.

Lemma 2.1. (a) For λ > 0 and x ∈ X, we have

(1 + λ)−k0γ (x) . γ (y) . (1 + λ)k0

k0+1 γ (x) for all y ∈ B(x, λγ (x)).

(b) For all x, y ∈ X, we have γ (x) + d(x, y) ≈ γ (y) + d(x, y).(c) There exists a constant C so that

γ (y) ≥ C[γ (x)]1+k0 [γ (y) + d(x, y)]−k0

for all x, y ∈ X.


Proof. (a) By (5), we have γ (y) . (1 + λ)k0

k0+1 γ (x) for all y ∈ B(x, λγ (x)). It remains to prove the first inequality. Indeed, ifγ (y) ≥ γ (x), there is nothing to prove. If γ (y) ≤ γ (x), by (5), we write

γ (x) . [γ (y)]1

1+k0 [γ (y) + d(x, y)]k0

k0+1 . (1 + λ)k0

k0+1 [γ (y)]1

1+k0 [γ (x)]k0

k0+1 .

It implies that γ (y) ≥ (1 + λ)−k0γ (x). This completes the proof of (a).For the proofs of (b) and (c), we refer to [14, Lemma 2.1]. �

As a direct consequence of Lemma 2.1, we can imply that if B := B(x0, γ (x0)) is a critical ball then γ (x0) ≈ γ (x) for allx ∈ B.

The following result will be useful in the sequel, see [14].

Proposition 2.2. We can pick the family of critical balls {Qj}j satisfying the following conditions:(i) ∪j Qj = X;(ii) There exists a positive constant σ0 so that for each λ ≥ 1,

j χλQj . λσ0 .

2.2. BMO type spaces associated with critical functions

Let γ be a critical function. Set

Dγ = {B(x, r) : r ≥ γ (x)}.

Definition 2.3 ([15]). Let γ be a critical function and let β ≥ 0. A function f ∈ L1loc(X) is said to be in the space BMOβγ (X) if

∥f ∥BMOβγ (X)

:= supB∈Dγ

1V (B)1+β

B|f (x) − fB|dµ(x) + sup

B∈Dγ

1V (B)1+β

B|f (x)|dµ(x) < ∞,

where fB =1

V (B)

B f (x)dµ(x).

Note that if f ∈ BMOβγ (X) then for all balls B ⊂ X , we have

1V (B)1+β

B|f (x) − fB|dµ(x) . ∥f ∥BMOβ

γ (X)

and we will use the fact frequently in the sequel.We will summarize some properties involving the BMOβ

γ (X) spaces.

Proposition 2.4. Let γ be a critical function and let β ≥ 0 and p ∈ [1, ∞). Then the following statement holds:(i) A function f belongs to the BMOβ

γ (X) space if and only if

supB∈Dγ

1V (B)1+pβ

B|f (x) − fB|pdµ(x)

1/p+ sup

B∈Dγ

1V (B)1+pβ

B|f (x)|pdµ(x)

1/p< ∞. (7)

Moreover, the left hand side of (7) is comparable with ∥f ∥BMOβγ (X)

.

(ii) For all balls B := B(x0, r) with r < γ (x0) and f ∈ BMOβγ (X), we have

1V (B)

B|f (x)|dµ(x) .

γ (x0)

r

βnV (B)β∥f ∥BMOβ

γ (X), β > 0

1 + logγ (x0)

r

V (B)β∥f ∥BMOβ

γ (X), β = 0.

(iii) For all x ∈ X and 0 < r1 < r2,

|fB(x,r1) − fB(x,r2)| .

r2r1

βnV (x, r1)β∥f ∥BMOβ

γ (X), β > 0

1 + log r2r1

V (x, r1)β∥f ∥BMOβ

γ (X), β = 0.

Proof. For the proof, we refer the reader to Lemmas 2.2 and 2.4 in [15]. �

We will end this section by the following simple result which is useful in the sequel.

Proposition 2.5. Let γ be a critical function and let β ≥ 0. If f is a locally integrable function satisfying

supB∈Dγ

1V (B)1+β

B|f (x) − fB|dµ(x) < C


and

supB:critical ball

1V (B)1+β

B|f (x)|dµ(x) < C

then f ∈ BMOβγ (X). Moreover, the best constant C and ∥f ∥BMOβ

γ (X)are comparable.

Proof. The proof of this proposition is not difficult by using Proposition 2.2 and Lemma 2.1 and hence we omit detailshere. �

3. Proof of main results

In this section, we give the proofs for operators O(e−tL). The proofs for operators Vρ(e−tL) can be shown in the similarway and we leave to the interested reader.

Proof of Theorem 1.2. Let {Qj} be a family of critical balls as in Proposition 2.2. Assume thatQj = B(xj, γ (xj)). We then have

∥O(e−tL)f ∥Lp(X) .

j

Qj

|O(e−tL)f (x)|pdµ(x)1/p

.

For each j and x ∈ Qj, we set f = fj,1 + fj,2 where fj,1 = fχ2Qj . So, we have

O(e−tL)f (x) ≤ O(e−tL)fj,1(x) + O(e−tL)fj,2(x).

For x ∈ Qj we have

O(e−tL)fj,2(x) =

∞j=1


e−ϵjLfj,2(x) − e−ϵj+1Lfj,2(x)21/2

≤

∞j=1


e−ϵjLfj,2(x) − e−ϵj+1Lfj,2(x)

≤

∞j=1


ϵj

ϵj+1

−Le−tLfj,2(x)dt

≤

∞j=1


ϵj

ϵj+1

|Le−tLfj,2(x)|dt

≤

∞

0|tLe−tLfj,2(x)|

dtt

which implies

O(e−tL)fj,2(x) .

∞

0

X\2Qj

|qt(x, y)| |f (y)|dµ(y)dtt

=

γ (xj)2

0· · · +

∞

γ (xj)2. . .

= I1 + I2. (8)

Using the upper bound of qt(x, y) we have

I1 .

γ (xj)2

0

X\2Qj

1

V (x,√t)

exp−c

d(x, y)2

t

|f (y)|dµ(y)

dtt

.

∞k=0

γ (xj)2

0

2k+1Qj\2kQj

1V (x, d(x, y))

exp−c

d(x, y)2

t

|f (y)|dµ(y)

dtt

.

∞k=0

γ (xj)2

0

2k+1Qj\2kQj

1V (x, 2kγ (xj))

td(x, y)2

|f (y)|dµ(y)

dtt

. (9)

Since x ∈ Qj, V (x, 2kγ (xj)) ≈ V (2k+1Qj). Hence,

I1 .

∞k=0

2−2k 1V (2k+1Qj)

2k+1Qj

|f (y)|dµ(y) . Mf (x),

where M is the Hardy–Littlewood maximal function.


We next take care of I2. Using (4), we write

I2 .

∞

γ (xj)2

X\2Qj

1

V (x,√t)

exp−c

d(x, y)2

t

γ (x)√t

2|f (y)|dµ(y)

dtt

.

∞

γ (xj)2

X\2Qj

1V (x, d(x, y))

γ (x)d(x, y)

γ (x)√t

|f (y)|dµ(y)

dtt

.

∞k=0

∞

γ (xj)2

2k+1Qj\2kQj

1V (x, d(x, y))

γ (x)d(x, y)

γ (x)√t

|f (y)|dµ(y)

dtt

. (10)

The similar argument to that above also gives that I2 . Mf (x) and hence

O(e−tL)fj,2(x) . Mf (x)

for all x ∈ Qj.We return to take care of O(e−tL)fj,1(x). We have

O(e−tL)fj,1(x) ≤ O(e−t∆)fj,1(x) + O(e−tL− e−t∆)fj,1(x).

As in the estimation of O(e−tL)fj,2(x), for x ∈ Qj, by the assumption (L3) we have

O(e−tL− e−t∆)fj,1(x) ≤

∞

0|(tLe−tL

− t∆e−t∆)fj,1(x)|dtt

.

∞

0

2Qj

1

V (x,√t)

√t

√t + γ (x)

δ2exp

−c

d(x, y)2

t

|f (y)|dµ(y)

dtt

.

γ (xj)2

0

2Qj

· · · +

∞

γ (xj)2

Qj

. . .

:= II1 + II2.

By the fact that2Qj

1

V (x,√t)

exp−c

d(x, y)2

t

|f (y)|dµ(y) ≤

X

1

V (x,√t)

exp−c

d(x, y)2

t

|f (y)|dµ(y) . Mf (x),

we arrive at

II1 .γ (xj)

γ (x)

δ2Mf (x).

This together with the fact that γ (x) ≈ γ (xj) for all x ∈ Qj implies that II1 . Mf (x) for x ∈ Qj.Next, using (3) to give

II2 .

∞

γ (xj)2

2Qj

1

V (x,√t)

|f (y)|dµ(y)dtt

.

∞

γ (xj)2

2Qj

1V (x, γ (xj))

γ (xj)√t

θ

|f (y)|dµ(y)dtt

which in combination with the fact that V (x, γ (xj)) ≈ V (2Qj) for all x ∈ Qj implies

II2 .

∞

γ (xj)2

1V (2Qj)

2Qj

γ (xj)√t

θ

|f (y)|dµ(y)dtt

. Mf (x).

Taking these estimates into account, we obtain that for x ∈ Qj

O(e−tL)f (x) . Mf (x) + O(e−t∆)fj,1(x).

Therefore, by Proposition 2.2 we have

∥O(e−tL)f ∥Lp(X) .

j

Qj

|O(e−tL)f (x)|pdµ(x)1/p

.

j

Qj

Mf (x) + O(e−t∆)fj,1(x)pdµ(x)

1/p.

X

Mf (x)pdµ(x)

1/p+

j

X

O(e−t∆)fj,1(x)pdµ(x)

1/p.


The Lp-boundedness of M givesX

Mf (x)pdµ(x)

1/p. ∥f ∥Lp(X).

Using Lp-boundedness of O(e−t∆) and Proposition 2.2 again, we havej

X

O(e−t∆)fj,1(x)pdµ(x)

1/p.

j

X|fj,1(x)|pdµ(x)

1/p.

j

2Qj

|f (x)|pdµ(x)1/p

. ∥f ∥Lp(X).

Hence,

∥O(e−tL)f ∥Lp(X) . ∥f ∥Lp(X). �

To prove Theorem 1.3, we need the following auxiliary lemma in [15].

Lemma 3.1. Let β ≥ 0. Assume that the operator L satisfies (L1)–(L3) . Then for all f ∈ BMOβγ (X), we have

(i) for all x ∈ X, t > 0,

|tLe−tLf (x)| . γ (x)

√t + γ (x)

δ2V (x,

√t)β∥f ∥BMOβ

γ (X);

(ii) for all x, y ∈ X and√t ≥ 2d(x, y),

|tLe−tLf (x) − tLe−tLf (y)| .

d(x, y)

√t

δ11 +

γ (x)√t

βnV (x,


γ (X), β > 0;d(x, y)

√t

δ11 + log

γ (x)√t

V (x,


γ (X), β = 0.

Proof of Theorem 1.3. We will prove the boundedness of O(e−tL) on BMOβγ (X) for β > 0. The similar argument can be

done for β = 0 but easier.Let f ∈ BMOβ

γ (X). Due to Proposition 2.5, it suffices to prove that for all critical balls Q , we have

1V (Q )1+β

Q

|O(e−tL)f (x)|dµ(x) . ∥f ∥BMOβγ (X)

(11)

and for all B = B(x0, r) with r < γ (x0), we have

1V (B)1+β

B|O(e−tL)f (x) − (O(e−tL)f )B|dµ(x) . ∥f ∥BMOβ

γ (X). (12)

We prove (11) first. Split f = f1 + f2, where f1 = fχ2Q . We then have

1V (Q )1+β

Q

|O(e−tL)f (x)|dµ(x) ≤1

V (Q )1+β

Q

|O(e−tL)f1(x)|dµ(x) +1

V (Q )1+β

Q

|O(e−tL)f2(x)|dµ(x).

By Hölder’s inequality, the L2-boundedness of O(e−tL) and Proposition 2.4, we obtain that

1V (Q )1+β

Q

|O(e−tL)f1(x)|dµ(x) ≤

1V (Q )1+2β

Q

|O(e−tL)f1(x)|2dµ(x)1/2

. 1V (Q )1+2β

X|f1(x)|2dµ(x)

1/2. 1V (2Q )1+2β

2Q

|f (x)|2dµ(x)1/2

. ∥f ∥BMOβγ (X)

.

As in (8), we have

O(e−tL)f2(x) .

∞

0

X\2Q

|qt(x, y)| |f (y)|dµ(y)dtt

=

γ (x0)2

0· · · +

∞

γ (x0)2. . .

= E1(x) + E2(x),

for all x ∈ Q .


For any N > nβ , the similar argument to that in (9) gives

E1(x) .

∞k=0

2−Nk 1V (2k+1Q )

2k+1Q

|f (y)|dµ(y)

which implies

1V (Q )1+β

Q

|E1(x)|dµ(x) .

∞k=0

2−k(N−nβ) 1V (2k+1Q )1+β

2k+1Q

|f (y)|dµ(y) . ∥f ∥BMOβγ (X)

.

Using the same argument as in (10), we arrive at

1V (Q )1+β

Q

|E2(x)|dµ(x) . ∥f ∥BMOβγ (X)

.

This gives (11).We next take care of (12). To do this, we will claim that for any ball B := B(x0, r) with r < γ (x0), we have

1V (B)1+β

B|O(e−tL)f (x) − cB|dµ(x) . ∥f ∥BMOβ

γ (X),

where cB is a constant which can be determined as follows. Let j0 be an index so that tj0−1 ≥ 8r2 > tj0 . We then choose

cB =

j0−2j=1


|(e−ϵj+1L − e−ϵjL)f (x0)|2 + suptj0≤ϵj0<ϵj0−1≤tj0−1

|Tϵj0(f , B) − Tϵj0−1(f , B)|

21/2

,

where Tt(f , B) = e−tLfB if t ≤ 8r2 and Tt(f , B) = e−tLf (x0) if t > 8r2.Hence,

|O(e−tL)f (x) − cB| . ∞j=j0


|(e−ϵj+1L − e−ϵjL)f (x)|21/2

+

j0−2j=1


|(e−ϵj+1L − e−ϵjL)(f (x) − f (x0))|21/2

+ sup0<t≤8r2

|e−tL(f (x) − fB)| + supt>8r2

|e−tL(f (x) − f (x0))|

:= F1(x) + F2(x) + F3(x) + F4(x)

which implies

1V (B)1+β

B|O(e−tL)f (x) − cB|dµ(x) .

4i=1

1V (B)1+β

BFi(x)dµ(x)

:=

4i=1

Gi. (13)

Estimation of G1:We now break f = (f − fB)χ2B + (f − fB)χ(2B)c + fB := f1 + f2 + f3. Hence,

G1 .

3k=1

1V (B)1+β

B

∞j0


|(e−ϵj+1L − e−ϵjL)fk(x)|21/2

dµ(x)

:= G1,1 + G1,2 + G1,3.

By Hölder’s inequality and the L2-boundedness of O(e−tL), we have

G1,1 ≤

1V (B)1+2β

B|O(e−tL)f1(x)|2dµ(x)

1/2. 1V (B)1+2β

B|f1(x)|2dµ(x)

1/2. ∥f ∥BMOβ

γ (X).


Next, by Gaussian upper bounds of qt(x, y) we have

G1,2 ≤1

V (B)1+2β

B

8r2

0|tLe−tLf2(x)|

dttdµ(x)

.

∞k=1

1V (B)1+2β

B

8r2

0

2k+1B\2kB

1V (x, d(x, y))

exp−c

d(x, y)2

t

|f2(y)|dµ(y)

dttdµ(x)

.

∞k=1

e−c22k 22knβ

V (2k+1B)1+2β

2k+1B

|f2(y)|dµ(y)


.

Using the assumption (L3) and the fact that t∆e−t∆f3 = 0, we have

G1,3 ≤1

V (B)1+2β

B

8r2

0|tLe−tLf3(x)|

dttdµ(x) =

1V (B)1+2β

B

r2

0|(tLe−tL

− t∆e−t∆)f3(x)|dttdµ(x)

.1

V (B)1+2β

B

8r2

0

X

√t

γ (x)

δ2 1

V (x,√t)

exp−c

d(x, y)2

t

|fB|dµ(y)

dttdµ(x)

.1

V (B)1+2β

B

rγ (x)

δ2|fB|dµ(x)

.1

V (B)1+2β

B

rγ (x0)

δ2|fB|dµ(x)

which together with Proposition 2.4 gives

G1,3 . rγ (x0)

δ2−βn∥f ∥BMOβ

γ (X). ∥f ∥BMOβ

γ (X)

provided β < δ2/n. Therefore, G1 . ∥f ∥BMOβγ (X)

.

Estimation of G2:We have

G2 .1

V (B)1+β

B

j0−2j=1


(e−ϵj+1L − e−ϵjL)(f (x) − f (x0))dµ(x)

.1

V (B)1+β

B

∞

8r2|tLe−tLf (x) − tLe−tLf (x0)|

dttdµ(x)

.1

V (B)1+β

B

8γ (x0)2

8r2· · · +

1V (B)1+β

∞

8γ (x0)2. . .

:= G2,1 + G2,2.

Using Lemma 3.1 and the fact that γ (x) ≈ γ (x0), we have

G2,2 . ∥f ∥BMOβγ (X)

1V (B)1+β

B

∞

8γ (x0)2

d(x, x0)√t

δ11 +

γ (x)√t

βnV (x,

√t)β

dttdµ(x)


1V (B)1+β

B

∞

8γ (x0)2

d(x, x0)√t

δ11 +

γ (x0)√t

βn√t

r

βnV (x, r)β

dttdµ(x)


as long as β < δ1/n.Next, we write

G2,1 .1

V (B)1+β

B

8γ (x0)2

8r2

X(qt(x, z) − qt(x0, z))f (z)dµ(z)

dttdµ(x)

.1

V (B)1+β

B

8γ (x0)2

8r2

X(qt(x, z) − qt(x0, z))(f (z) − fB)dµ(z)

dttdµ(x)

+1

V (B)1+β

B

8γ (x0)2

8r2

X(qt(x, z) − qt(x0, z))fBdµ(z)

dttdµ(x).


By the assumption (L2), we have X(qt(x, z) − qt(x0, z))fBdµ(z)

.

X

1

V (x,√t)

d(x, x0)√t

δ1exp

−c

d(x, y)2

t

|fB|dµ(z)

. r

√t

δ1|fB|.

Moreover, since t∆e−t∆1 = 0, this together with (L3) and the fact that γ (x) ≈ γ (x0) implies that X(qt(x, z) − qt(x0, z))fBdµ(z)

=

X(qt(x, z) − q∆

t (x, z) − qt(x0, z) + q∆t (x0, z))fBdµ(z)

.

X|qt(x, z) − q∆

t (x, z)| |fB|dµ(z) +

X|qt(x0, z) − q∆

t (x0, z)| |fB|dµ(z)

. √

t√t + γ (x)

δ2|fB| +

√t

√t + γ (x0)

δ2|fB|

. √

t√t + γ (x0)

δ2|fB|.

Hence, X(qt(x, z) − qt(x0, z))fBdµ(z)

. r

√t + γ (x0)

δ0|fB|

. r

√t + γ (x0)

δ0γ (x0)r


γ (X),

where in the last inequality we used Proposition 2.4.Therefore,

1V (B)1+β

B

8γ (x0)2

8r2

X(qt(x, z) − qt(x0, z))fBdµ(z)

dttdµ(x)


1V (B)1+β

B

8γ (x0)2

8r2

r√t + γ (x0)

δ0γ (x0)r

βnV (B)β

dttdµ(x)


as long as β < δ0/n.By the assumption (L2), we have

1V (B)1+β

B

8γ (x0)2

8r2

X(qt(x, z) − qt(x0, z))(f (z) − fB)dµ(z)

dttdµ(x)

.1

V (B)1+β

B

8γ (x0)2

8r2

X

1

V (x,√t)

r√t

δ1exp

−c

d(x, z)2

t

|f (z) − fB|dµ(z)

dttdµ(x)

.

∞j=0

1V (B)1+β

B

8γ (x0)2

8r2

Sj(B(x0,

√t))

e−c2−2j

V (x,√t)

r√t

δ1|f (z) − fB|dµ(z)

dttdµ(x)

.

∞j=0

1V (B)1+β

B

8γ (x0)2

8r2

B(x0,2j

√t)

e−c2−2j

V (x0, 2j√t)

r√t

δ1|f (z) − fB(x0,2j

√t)|dµ(z)

dttdµ(x)

+

∞j=0

1V (B)1+β

B

8γ (x0)2

8r2

B(x0,2j

√t)

e−c2−2j

V (x0, 2j√t)

r√t

δ1|fB − fB(x0,2j

√t)|dµ(z)

dttdµ(x). (14)

It is easy to see that the first term is dominated by

C∥f ∥BMOβγ (X)

∞j=0

1V (B)1+β

B

8γ (x0)2

8r2e−c2−2j

r√t

δ1V (x0, 2j

√t)β

dttdµ(x)


∞j=0

1V (B)1+β

B

8γ (x0)2

8r2e−c2−2j

r√t

δ12j√t

r

βnV (B)β

dttdµ(x)


as long as β < δ1/n.


Using Proposition 2.4, we can dominate the second term by

C∥f ∥BMOβγ (X)

∞j=0

1V (B)1+β

B

8γ (x0)2

8r2e−c2−2j

r√t

δ12j√t

r

βnV (B)β

dttdµ(x) . ∥f ∥BMOβ

γ (X)

provided β < δ1/n.Therefore, G2 . ∥f ∥BMOβ

γ (X).

Estimation of G3:We first have

G3 ≤1

V (B)1+β

B

sup0<t≤8r2

|e−tL((f − fB)χ2B)(x)|dµ(x) +1

V (B)1+β

B

sup0<t≤8r2

|e−tL((f − fB)χ(2B)c )(x)|dµ(x).

Since supt>0 |e−tLf | is bounded on L2 (see for example [20]), by Hölder’s inequality and Proposition 2.4, we have

1V (B)1+β

B

sup0<t≤8r2

|e−tL((f − fB)χ2B)(x)|dµ(x) . 1V (B)1+2β

2B

|f (y) − fB|2dµ(y)1/2


.

Now for 0 < t ≤ 8r2 and x ∈ Bwe have

|e−tL((f − fB)χ(2B)c )(x)| .

X\2B

1

V (x,√t)

exp−c

d(x, y)2

t

|f (y) − fB|dµ(y)

.

∞j=2

Sj(B)

1

V (x,√t)

exp−c

d(x, y)2

t


.

∞j=2

e−c22j

V (2jB)

2jB


.

∞j=2

e−c22j

V (2jB)

2jB

|f (y) − f2jB| + |f2jB − fB|dµ(y).

This together with Proposition 2.4 implies

|e−tL((f − fB)χ(2B)c )(x)| . ∥f ∥BMOβγ (X)

V (B)β .

Hence,1

V (B)1+β

B

sup0<t≤8r2

|e−tL((f − fB)χ(2B)c )(x)|dµ(x) . ∥f ∥BMOβγ (X)

.

Taking these two estimates into account, we arrive at G3 . ∥f ∥BMOβγ (X)

.

Estimation of G4:For t > 8r2, x, x0 ∈ B, we have

|e−tLf (x) − e−tLf (x0)| =

X(pt(x, y) − pt(x0, y))f (y)dµ(y)

.

X(pt(x, y) − pt(x0, y))(f (y) − fB(x0,

√t))dµ(y)

+

X(pt(x, y) − pt(x0, y))fB(x0,

√t)dµ(y)

.Using the assumption (L2) and the argument as in (14), we have

X(pt(x, y) − pt(x0, y))(f (y) − fB(x0,

√t))dµ(y)

.

X

1

V (x,√t)

d(x, x0)√t

δ1exp

−c

d(x, y)2

t

|f (y) − fB(x0,

√t)|dµ(y)

.

∞j=0

Sj(B(x0,

√t))

e−c22j

V (x, 2j√t)

r√t

δ1|f (y) − fB(x0,

√t)|dµ(y)


.


Note that e−t∆1 = 1. This together with the condition (L3) and the fact that γ (x) ≈ γ (x0) implies X(pt(x, y) − pt(x0, y))fB(x0,

√t)dµ(y)

=

X[(pt(x, y) − p∆

t (x, y)) − (pt(x0, y) − p∆t (x0, y))]fB(x0,

√t)dµ(y)

. √

tγ (x0)

δ2|fB(x0,

√t)|.

Moreover, the condition (L2) implies X(pt(x, y) − pt(x0, y))fB(x0,

√t)dµ(y)

. r

√t

δ1|fB(x0,

√t)|.

These two estimates and Proposition 2.4 imply that X(pt(x, y) − pt(x0, y))fB(x0,

√t)dµ(y)

. r

γ (x0)

δ0|fB(x0,

√t)|

. r

γ (x0)

δ0γ (x0)√t

βnV (x0,


γ (X)

. r

γ (x0)

δ0γ (x0)√t

βn√t

r


γ (X)


provided β < δ0/nwith δ0 = min{δ1/2, δ2/2}.This completes our proof. �

4. Applications

In this section, we apply the obtained results to study the boundedness of the variation operators and the oscillationoperators in various settings such as Schrödinger operators, degenerate Schrödinger operators on Rn and Schrödinger oper-ators on Heisenberg groups and connected and simply connected nilpotent Lie groups. Throughout this section, the functionγ will be defined as in (5).

4.1. Schrödinger operators on Rn

Let ∆ = −n

j=1∂2

∂x2jbe a Laplacian on Rn, n ≥ 3. Let L = ∆ + V be a Schrödinger operator with the nonnegative

potential V being in the reverse Hölder class RHq, q > n/2. Obviously, the Laplacian ∆ satisfies (∆1) and (∆2). Moreover, itwas proved in [21,22] that the Schrödinger operator L satisfies (L1) and (L2). It was also proved in [21] that there is a positiveconstant δ2 > 0 so that for all x, y ∈ Rn and t > 0,

|pt(x, y) − p∆t (x, y)| ≤

C

V (x,√t)

√t

√t + γ (x)

δ2exp

−c

d(x, y)2

t

. (15)

To estimate the difference qt(x, y) − q∆t (x, y), we write

qt(x, y) − q∆t (x, y) = t

Rn

V (z)pt/2(x, z)p∆t/2(z, y)dz + t

t/2

0

Rn

V (z)qt−s(x, z)p∆s (z, y)dz

dst − s

+ t t

t/2

Rn

V (z)pt−s(x, z)q∆s (z, y)dz

dss

(16)

and then using the similar argument as in [22], we arrive at

|qt(x, y) − q∆t (x, y)| ≤

C

V (x,√t)

√t

√t + γ (x)

δ2exp

−c

d(x, y)2

t

, (17)

for all x, y ∈ Rn and t > 0. Hence, L satisfies (L3).As a direct consequence of Theorems 1.2 and 1.3, we obtain that:

Theorem 4.1. Let L = ∆ + V be a Schrödinger operator on Rn, n ≥ 3 and V ∈ RHq, q > n/2. Then(a) the operators Vρ(e−tL) and O(e−tL) are bounded on Lp(Rn) for all 1 < p < ∞;(b) the operators Vρ(e−tL) and O(e−tL) are bounded on BMOβ

γ (Rn) for 0 ≤ β < δ0/n.

The assertion (a) is in line with the result in (i) of Theorem B meanwhile the assertion (b) extends the result in (ii) ofTheorem B to β > 0.


4.2. Degenerate Schrödinger operators

Let w be a weight in Muckenhoupt class A2(Rd), d ≥ 3, i.e., there exists a constant C > 0 so that 1|B|

Bw(x)dx

1|B|

Bw−1(x)dx

≤ C

for all balls B ⊂ Rd. Then the triple (Rd, | · |, w(x)dx) satisfies (1) and (3). Hence, there exist 0 < θ ≤ n < ∞ so that

λθw(B(x, r)) . w(B(x, λr)) . λnw(B(x, r))

for all x ∈ Rd, r > 0 and λ ≥ 1, where w(E) =E w(x)dx for any measurable subset E ⊂ Rd.

Let {ai,j}di,j=1 be a real symmetric matrix function satisfying that there exists κ > 0 so that for all x, ξ ∈ Rd,

κ−1|ξ |

2w(x) ≤

i,j

ai,j(x)ξiξ j ≤ κ|ξ |2w(x).

We consider the degenerate elliptic operator ∆ defined by

∆f (x) = −1

w(x)

i,j

∂i(ai,j(·)∂jf )(x).

Then the operator ∆ satisfies the assumptions (∆1) and (∆2). See for example [23].Let L = ∆ + V be a Schrödinger operator with V ∈ RHq(Rd, | · |, w(x)dx) with q > d/2. It was proved in [15,22] that the

generate Schrödinger operator L satisfies the conditions (L1)–(L3) for pt(x, y) − p∆t (x, y). The estimate of qt(x, y) − q∆

t (x, y)can be proven in the similar way to that of [22, Proposition 4]. Hence, as a direct consequence of Theorems 1.2 and 1.3, weconclude that:

Theorem 4.2. Let w ∈ A2(Rd). Let L = ∆+V be a degenerate Schrödinger operator onRd, d ≥ 3 and V ∈ RHq(Rd, |·|, w(x)dx)with q > d/2. Then(a) the operators Vρ(e−tL) and O(e−tL) are bounded on Lp(Rd, w(x)dx) for all 1 < p < ∞;(b) the operators Vρ(e−tL) and O(e−tL) are bounded on BMOβ

γ (Rd, w(x)dx) for 0 ≤ β < δ0/d.

4.3. Schrödinger operators on Heisenberg groups

Let Hn be a (2n+ 1)-dimensional Heisenberg group. Recall that a (2n+ 1)-dimensional Heisenberg group is a connectedand simply connected nilpotent Lie group with the underlying manifold R2n

× R. The group structure is defined by

(x, s)(y, t) =

x + y, s + t + 2

nj=1

(xd+jyj − xjyd+j)

.

The homogeneous norm on Hd is defined by

|(x, t)| = (|x|4 + |t|2)1/4 for all (x, t) ∈ Hn.

See for example [24].This norm satisfies the triangle inequality and hence induces a left-invariant metric d((x, t), (y, s)) = |(−x, −t)(y, s)|.

Moreover, there exists a positive constant C such that |B((x, t), r)| = CrQ , whereQ = 2d+2 is the homogeneous dimensionofHn and |B((x, t), r)| is the Lebesguemeasure of the ball B((x, t), r). Obviously, the triplet (Hn, d, dx) satisfies the condition(1) and (3).

A basis for the Lie algebra of left-invariant vector fields on Hd is given by

X2n+1 =∂

∂t, Xj =

∂

∂xj+ 2xn+j

∂

∂t, Xn+j =

∂

∂xn+j− 2xj

∂

∂t, j = 1, . . . , n.

The sub-Laplacian ∆Hn is defined by

∆Hn = −

2nj=1

X2j .

Furthermore, it was proved in [24] that the sub-Laplacian ∆Hn satisfies (∆1) and (∆2).We now consider the Schrödinger operator on Hn defined by L = ∆Hn + V where V ∈ RHq(Hn), q > Q/2. Then, the

Schrödinger operator L satisfies conditions (L1)–(L3). See for example [24,14]. Hence, we have:

Theorem 4.3. Let L = ∆Hn + V with V ∈ RHq(Hn) with q > Q/2. Then

(a) the operators Vρ(e−tL) and O(e−tL) are bounded on Lp(Hn) for all 1 < p < ∞;(b) the operators Vρ(e−tL) and O(e−tL) are bounded on BMOβ

γ (Hn) for 0 ≤ β < δ0/Q .


4.4. Schrödinger operators on connected and simply connected nilpotent Lie groups

For the background of the connected and simply connected nilpotent Lie groups, we follow [25,26]. Let G be a connectedand simply connected nilpotent Lie group. Let X ≡ {X1, . . . , Xk} be left invariant vector fields on G satisfying the Hömandercondition. Let d be the Carnot–Carathéodory distance on G associated to X and µ be a left invariant Haar measure on G.Then, there exist 0 < θ ≤ n < ∞ such that µ(B(x, r)) ≈ rθ when 0 < r ≤ 1, and µ(B(x, r)) ≈ rn when r ≥ 1, see forexample [26].

The sub-Laplacian is defined by ∆G = −k

j=1 X2j . Then the operator ∆G generates the analytic semigroup {e−t∆G}t>0

whose kernelspt(x, y) satisfy (∆1) and (∆2). See for example [25].Let V be a nonnegative locally integrable function onG. Assume that V ∈ RHq(G), q > n/2. Then the operator L = ∆G+V

generates the semigroup {e−tL}t>0 satisfying conditions (L1)–(L3). See for example [14,15]. Hence, from Theorems 1.2 and

1.3, we arrive at:

Theorem 4.4. Let L = ∆G + V with V ∈ RHq(G) with q > n/2. Then

(a) the operators Vρ(e−tL) and O(e−tL) are bounded on Lp(G) for all 1 < p < ∞;(b) the operators Vρ(e−tL) and O(e−tL) are bounded on BMOβ

γ (G) for 0 ≤ β < δ0/n.

Acknowledgment

The author is supported by the Australian Research Council grant no. DP110102488. He would like to thank the refereefor his/her useful comments to improve the paper.

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boundedness of variation operators and oscillation operators for certain semigroups

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