A <span class="mathjax-formula">$T1$</span> criterion for Hermite-Calderón-Zygmund operators on the <span class="mathjax-formula">$BMO_{H}(\protect \mathbb{R}^{n})$</span> space and applications

A T1 criterion for Hermite-Calder´on-Zygmund operators on the BMO

( R

) space and applications

JORGEJ. BETANCOR, RAQUELCRESCIMBENI, JUANC. FARINA˜ , PABLORAUL´ STINGA ANDJOSE´ L. TORREA

Abstract. In this paper we establish a T1 criterion for the boundedness of Hermite-Calder´on -Zygmund operators on the B M O_H(Rⁿ)space naturally as- sociated to the Hermite operatorH. We apply this criterion in a systematic way to prove the boundedness onB M OH(Rⁿ)of certain harmonic analysis operators related toH(Riesz transforms, maximal operators, Littlewood-Paleyg-functions and variation operators).

Mathematics Subject Classification (2010):42B20 (primary); 42B25, 42B35, 35J10, 42B15, 42C10 (secondary).

1. Introduction

It is well-known the crucial role played by T1 and its relation with the classical B M Ospace of John and Nirenberg in the analysis ofL^p-boundedness of Calder´on- Zygmund operatorsT (see [5, 10, 11] and [9, page 590]).

Moreover, T1 is an important object to understand the behavior of certain classes of integral operators in H¨older spaces. Indeed, in [17] some operators re- lated to the harmonic oscillator (also known as Hermite operator)

H = 1+ |x|², inR^{n, (1.1)}

such as the fractional harmonic oscillator H , the Hermite-Riesz transforms, the fractional integralsH , among others, are studied when they act on certain H¨older spaces C^k,↵_H (R^{n), k} 2 N^{, 0 < ↵ <} 1, adapted to H. Roughly speaking, these operatorsT can be expressed as

T f(x)= Z

RⁿK(x,y)(f(y) f(x))dy+ f(x)T1(x). (1.2) Research partially supported by MTM2007/65609, MTM2008-06621-C02-01, MTM2011- 28149-C02-1 and PCI 2006-A7-0670 from Ministerio de Ciencia e Innovaci´on, Spain, and Uni- versidad Nacional del Comahue, Argentina. This research work started when the fourth author was a Phd student at Departamento de Matem´aticas of Universidad Aut´onoma de Madrid.

Received November 29, 2010; accepted in revised form June 20, 2011.

Here the kernelK(x,y)has a singularity forx ⇠ y, so some regularity is required on f for the integral to be well defined. Looking at how the operatorT is written, it is natural to expect thatT1 will be a bounded pointwise multiplier in the class where f belongs to. This is in fact the situation in [17]. Nevertheless, the boundedness of operators like (1.2) for the case ↵ = 0 is not covered in [17] (it does not make sense to take 0 as a H¨older exponent). However, since the H¨older spacesC^↵ can be seen as spaces of B M O↵-type (see for instance [19]), it would be natural to work withB M OH(Rⁿ). Note thatB M OH(Rⁿ⁾is the natural substitute as extremal space in the Harmonic Analysis for the Hermite function expansion setting (see Section 2). The last question motivates a characterization of pointwise multipliers on B M OH(Rⁿ). We believe that such a result belongs to the folklore, but for completeness we present it here with a proof, see Proposition 3.2. Let us point out that the characterization of pointwise multipliers for the B M O space on the torus (compact support case) was proved by S. Janson [12] and for the Euclidean

B M O(Rⁿ⁾by E. Nakai and K. Yabuta [13].

To obtain the boundedness onB M OH(Rⁿ⁾for operatorsTof the form (1.2) it seems natural to impose conditions onT1. An answer in this direction is provided in our first main result.

Theorem 1.1 (T1-type criterion). LetT be a Hermite-Calder´on-Zygmund opera- tor, see Definition3.1. ThenT is a bounded operator onB M OH(Rⁿ⁾if and only if there existsC >0such that the following two conditions hold:

(i) 1

|B(x, (x))| Z

B(x, (x))|T1(y)|dy C, for everyx2R^{n, and} (ii)

✓ 1+log

✓ (x) s

◆◆ 1

|B(x,s)| Z

B(x,s)|T1(y) (T1)B(x,s)|dy  C,for every x 2R^{nands >0such that0<s} (x), where is given by

(x):=

8>

><

>>

: 1

1+ |x|, |x| 1; 1

2, |x|<1.

(1.3)

Here, as usual,(T1)_B(x,s)= 1

|B(x,s)| Z

B(x,s)

T1(y)dy.

Remark 1.2 (Vector-valued setting). Theorem 1.1 can also be stated in a vector valued setting. That is, ifT f takes values in a Banach spaceXthen the result holds when we replace the absolute values appearing in hypothesis (i) and(ii) by the norm in X.

Remark 1.3 (How to apply the result). Assume thatT1 is a bounded function in R^{n. ThenT}1 satisfies the first condition of Theorem 1.1. The second condition is fulfilled whenever there exists 0<↵1 such that|T1(x) T1(y)|C|x y|^↵, x,y2Rⁿ(for instance,(ii)holds ifrT12L¹(R^n)).

We apply Theorem 1.1 in a systematic way to prove that several harmonic analysis operators related to H are bounded on B M OH(Rⁿ). The operators are the maximal operators and Littlewood-Paleyg-functions associated to the heat and Poisson semigroups for H and the Hermite-Riesz transforms (see Section 4).

Theorem 1.4 (Harmonic Analysis operators related toH). The maximal opera- tors and the Littlewood-Paley g-functions associated with the heat{W_t^H}t>0 and Poisson {P_t^H}t>0 semigroups generated by H and the Hermite-Riesz transforms are bounded from B M OH(Rⁿ⁾into itself.

We also consider variation operators. Let (X,F^{, µ)}be a measure space and {Tt}t>0 be an uniparametric family of bounded operators in L^p(X)for some 1 

p<1, such that lim

t!0⁺Ttf(x)exists for a.e.x 2X. In the last years many papers devoted their attention to analyze the speed of convergence of the limit above in terms of the boundedness properties of the ⇢-variation operatorV⇢(Tt), ⇢ > 2.

Such operator is defined by

V⇢(Tt)(f)(x)= sup

tj&0

X1 j=1

|Ttj f(x) Tt_j+1f(x)|^⇢

!1/⇢

,

where the supremum is taken over all the sequences of real numbers{tj}j2Nthat decrease to zero. The uniparametric families we are interested in are: the heat semi- group{W_t^H}t>0, the Poisson semigroup{P_t^H}t>0and the truncated integral opera- tors for the Hermite-Riesz transforms{R^"_H}">0(see Section 4 for definitions). The L^p–theory for the variation operators related to{W_t^H}t>0,{P_t^H}t>0 and{R^"_H}">0

was studied in [3] and [4].

Theorem 1.5 (Variation operators). Let⇢>2. Denote by{Tt}t>0any of the uni- parametric families of operators{W_t^H}t>0,{P_t^H}t>0or{R^"_H}">0. Then the varia- tion operatorV⇢(Tt)is bounded fromB M OH(Rⁿ⁾into itself.

It is a remarkable fact that all the operators related to H listed above can be seen as vector valued singular integral operators. Therefore Remark 1.2 will be very useful.

Some of the operators were considered by J. Dziuba´nski et al. [6] in the more general setting of Schr¨odinger operators of the formL= 1+Vand theB M O_L- spaces associated to them inR^{n, whenn} 3. In such a context, the potential V belongs to R Hs, the reverse H¨older class of exponents, for somes > n/2. Since polynomials are in R Hs for alls > 0, the Hermite case V = |x|² is included. It was proved in [6] that the maximal operators related to the heat and Poisson semi- groups and the square function defined by the heat semigroup in the Schr¨odinger context are bounded operators onB M O_L. The procedure developed in [6] exploits, in each case, the underlying relationship between the operator considered and its corresponding Euclidean counterpart. More recently, B. Bongioanni, E. Harboure

and O. Salinas studied Schr¨odinger-Riesz transforms associated to L ^{in B M O}_L^- spaces, 0  < 1, in dimensionn 3, see [2]. In particular, they showed that ifs >nthen the Schr¨odinger-Riesz transformsRi are bounded onB M O_L. When n/2 < s < nthe operators Ri fail to be bounded in L^p for all p > p₀, where p0 >1 depends ons, see the seminal paper by Z. Shen [14]. This implies that the operators Ri are not bounded on B M O_L ifn/2 < s < n. Finally, in [1] it was proved that the (generalized) square functions defined by the Poisson semigroup related toLare bounded onB M O_L, forn 3.

Boundedness of Harmonic Analysis operators in the Hermite setting is well- developed. In particular, boundedness results inL^pfor the related Poisson integrals, the Hermite-Riesz transforms and the square functions can be found in the book by S. Thangavelu [18], see also [16].

We would like to point out that our method in this Hermite case works for every n 1. One of the main novelties of this paper is the boundedness inB M OH(R^n)of the variation operators, Theorem 1.5. Finally, and perhaps this is a more important observation, Theorem 1.1 allows us to consider all the Harmonic Analysis operators related toHin a unified way. The key ingredient will be the vector-valued approach.

Moreover, we believe that in the cases of boundedness of the maximal operators, our proofs are easier and faster than those presented in [6].

The outline of the paper is as follows. We collect in Section 2 the main def- initions and properties related to the space B M OH(Rⁿ). In Section 3, together with the definitions of Hermite-Calder´on-Zygmund operator and T1, we present the proof of Theorem 1.1 and the characterization of pointwise multipliers, Propo- sition 3.2. Applications are developed in Section 4 (proofs of Theorems 1.4 and 1.5).

Throughout this paperCandcwill always denote suitable positive constants, not necessarily the same in each occurrence. Without mentioning it, we will re- peatedly apply the inequalityr^µe ^r  Cµe ^r/2,µ 0,r > 0, and the fact that log¹₁^+s_s ⇠sfors ⇠0, and log¹₁^+s_s ⇠ log(1 s)fors ⇠1.

2. The spaceB M O_H(Rⁿ⁾

J. Dziuba´nski et al. defined in [6] the space B M O_L(Rⁿ⁾naturally associated to a Schr¨odinger operatorL= 1+V inR^n,n 3, where the nonnegative potential V satisfies a reverse H¨older inequality. It turns out that B M O_L(Rⁿ⁾is the natural replacement of L¹(Rⁿ⁾in this context. In fact, B M O_L(Rⁿ⁾ is the dual of the Hardy space H_L¹(Rⁿ⁾associated toLdefined by J. Dziuba´nski and J. Zienkiewicz in [7]. For the definition of B M OH(Rⁿ⁾we take the space of [6] in the particular case of the harmonic oscillator (1.1), i.e. V(x)= |x|², and we extend the definition to alln 1.

A locally integrable function f inR^{n belongs to B M O}H(Rⁿ⁾if there exists C >0 such that

(i) 1

|B| Z

B|f(x) fB|dx C, for every ballBinR^{n, and}

(ii) 1

|B| Z

B|f(x)|dx  C, for every B = B(x0,r0), where x0 2 R^{n andr0} (x0).

Here fB = 1

|B| Z

B

f(x)dx, for every ballBinR^{n, and the}critical radiifunction is given by (1.3). The normkfkB M OH(Rⁿ⁾of f is defined by

kfkB M OH(Rⁿ⁾ =inf{C 0: (i) and (ii) above hold}.

Applying the classical John-Nirenberg inequality it can be seen that if in (i) and (ii) L¹-norms are replaced byL^p-norms, for 1< p<1, then the space B M O_H(Rⁿ⁾ does not change and equivalent norms appear, see [6, Corollary 3].

It is not hard to check that for everyC > 0 there exists M > 0 such that if

|x y|C (x)then 1

M (x) (y)M (x).

Covering by critical balls.According to [7, Lemma 2.3] there exists a sequence of points{xk}¹_k=1inR^{nso that ifQk}denotes the ball with centerxk and radius (xk), k 2N^{, then}

(i) S₁

k=1Qk =R^{n, and}

(ii) there existsN 2 Nsuch that card{j 2N : Q^⇤⇤_j \Q^⇤⇤_k 6= ;} N, for every k 2N^.

For a ballB,B^⇤denotes the ball with the same center thanBand twice radius.

Boundedness criterion.In order to prove that an operatorSdefined onBMOH(Rⁿ⁾ is bounded from B M OH(Rⁿ⁾into itself, it suffices to see that there existsC > 0 such that, for every f 2 B M OH(R^n)andk2N^,

(Ak) 1

|Qk| Z

Qk

|S f(x)|dx CkfkB M OH(Rⁿ⁾, and

(Bk) kS fkB M O(Q^⇤_k)CkfkB M OH(Rⁿ⁾, whereB M O(Q^⇤_k)denotes the usualB M O space on the ballQ^⇤_k,

see [6, page 346].

In the following lemma we present an example of a function in B M O_H(Rⁿ⁾ that will be useful in the sequel.

Lemma 2.1. There exists a positive constantC > 0such that, for everyx0 2Rⁿ and0<s (x0), the function f(x;s,x0)defined by

f(x;s,x0)= [0,s](|x x0|)log

✓ (x0) s

◆

+ (s, (x0)](|x x0|)log

✓ (x0)

|x x0|

◆

, x2R^n, belongs toB M OH(R^n)andkf(·;s,x0)kB M OH(Rⁿ⁾ C.

Proof. Recall that the functionh(x)=log⇣

1

|x|

⌘

[0,1](|x|)is inB M O(R^{n), see [9,} page 520]. Hence, for every R>0, the functionhR given by

h_R(x)=h(x/R), x 2R^n,

is in B M O(R^n)andkhRkB M O(Rⁿ⁾ C, whereCis independent of R. Moreover, for every R,S>0, the functionhR,Sdefined by

hR,S(x)=min{S,h(x/R)}, x2R^n,

belongs to B M O(R^n)andkhR,SkB M O(Rⁿ⁾  C, whereC does not depend on R andS. Then, since for everyx02R^{nand 0<s}  (x0),

f(x;s,x₀)=h

(x0),log ^(x_s⁰⁾(x x₀), x 2R^n,

the function f(·;s,x0)2B M O(R^n)andkf(·;s,x0)kB M O(Rⁿ⁾C,x02R^nand 0 <s < (x0). It only remains to control the means of f(·;s,x0)onlargeballs.

For that let us first note that

|f|B(x0, (x0))= 1

|B(x0, (x0))| Z

B(x0, (x0))

f(x;s,x0)dx

 C (x0)ⁿ

 sⁿlog

✓ (x₀) s

◆ +

Z

s<|z|< (x0)

log

✓ (x₀)

|z|

◆

dz C, whereCis independent ofs andx0. Let B = B(z0,r0),x02R^{n andr}0 (z0).

We can always assume that B\B(x0, (x0)) 6= ;, since the support of f is the closure of the ball B(x₀, (x₀)). Consider first the easier case: whenr₀ (x₀).

Then we clearly have|f|B |f|B(x0, (x0))and the computation above applies. On the other hand, ifr₀ (x₀), we have that|x₀ z₀|2 (x₀)and by the properties of given above, (x0)⇠ (z0). Using this last fact and the previous observation, we get|f|B  |B(x0, (x0))|

|B(z0, (z0))||f|B(x0, (x0))C. The proof is complete.

3. Proof of Theorem 1.1 and characterization of pointwise multipliers

3.1. On theT1-criterion: Theorem 1.1

Before proving Theorem 1.1 we need to precise the definition of the operatorT we are considering.

We denote byL²_c(Rⁿ⁾the set of functions f 2L²(Rⁿ⁾whose support supp(f) is a compact subset ofR^n.

Definition 3.1. LetT be a bounded linear operator onL²(R^{n)such that} T f(x)=

Z

RⁿK(x,y)f(y)dy, f 2L²_c(R^{n) and a.e. x} 2^/supp(f).

We shall say thatT is a Hermite-Calder´on-Zygmund operator if

(1) |K(x,y)| C

|x y|ⁿ e ^{c⇥|x||x y|+|x y|2⇤}, for allx,y2R^nwithx 6=y, (2) |K(x,y) K(x,z)| + |K(y,x) K(z,x)|C |y z|

|x y|ⁿ⁺¹, when|x y|>

2|y z|.

Note that every Hermite-Calder´on-Zygmund operator is also a classical Calder´on- Zygmund operator, see [9]. Examples of Hermite-Calder´on-Zygmund operators are given in Section 4.

Definition of T f for f 2 B M OH(R^n). Suppose firstly that f 2 L²(R^{n). For} every R>0, let BR :=B(0,R). We can write

T f =T f _BR +T⇣ f _B^c_R⌘

=T f _BR + lim

n!1T⇣

f _B^c_R\Bn⌘

where the limit is understood in L²(Rⁿ). This last identity suggests to define the operatorT onB M OH(Rⁿ⁾as follows. Assume that f 2B M OH(R^n)andR>1.

By using the Hermite-Calder´on-Zygmund condition (1) for Kwe get Z

B_2R^c |K(x,y)||f(y)|dy C X1

j=1

Z

2^jR<|y|2^j+1R

e ^{c|x y|2}

|x y|ⁿ|f(y)|dy

C X1

j=1

Z

2^jR<|y|2^j+1R

1

|x y|ⁿ⁺¹|f(y)|dy

C X1

j=1

1 (2^jR)ⁿ⁺¹

Z

|y|2^j+1R|f(y)|dy

C

R kfkB M OH(Rⁿ⁾, for everyx 2BR. Moreover, if R< Swe have

T f BS (x) T f BR (x)=T f BS\BR (x)= Z

BS\BR

K(x,y)f(y)dy

= Z

B^c_R

K(x,y)f(y)dy Z

B^c_S

K(x,y)f(y)dy, a.e.x 2B_R. We define

T^f(x)=T f BR (x)+ Z

B^c_R

K(x,y)f(y)dy, a.e.x 2BR andR>1.

Note that the definition ofT^f above is consistent in the choice of R > 1 in the sense that ifS > R >1 then the definition using BScoincides almost everywhere in B_R with the one just given.

Let us derive an expression forT^{f where}T1 appears that will be useful for the proof of our main result. Letx02R^{nandr0>0. ForB}= B(x0,r0)we write

f =(f f_B) _B⇤+(f f_B) _(B⇤)^c+ f_B =: f₁+ f₂+ f₃. (3.1) Let us choose R>0 such thatB^⇤⇢ BR. Using (3.1) we get

T^f(x)=T f BR (x)+ Z

B^c_R

K(x,y)f(y)dy

=T((f fB) _B⇤) (x)+T (f fB) _BR\B⇤ (x)+fBT BR (x) +

Z

B^c_R

K(x,y)(f(y) f_B)dy+f_B Z

B^c_R

K(x,y)dy

=T((f fB) _B⇤) (x)+ Z

(B^⇤)^c

K(x,y)(f(y) fB)dy+ fBT^1(x), (3.2)

almost everywherex 2B^⇤.

Proof of Theorem1.1. First we shall see that conditions (i) and (ii) onT1 imply thatT is bounded from B M OH(Rⁿ⁾into itself. In order to do this we will show that there existsC > 0 such that the properties (Ak) and (Bk) stated in Section 2 hold for everyk2N^{and f} 2B M OH(Rⁿ⁾when the operatorTis considered.

We start with (Ak). According to (3.2), T^f(x)=T⇣

(f fQk) _Q⇤ k

⌘(x)+ Z

(Q^⇤_k)^c

K(x,y)(f(y) fQk)dy+ fQkT^1(x), almost everywherex 2 Qk. AsT maps L²(R^{n)into L2(}Rⁿ), by using H¨older’s inequality and [6, Corollary 3],

1

|Q_k| Z

Qk

T⇣

(f f_Qk) _Q⇤ k

⌘(x) dx C

✓ 1

|Q_k| Z

Qk

T⇣

(f f_Qk) _Q⇤ k

⌘(x)²dx

◆1/2

C 1

|Qk| Z

Q^⇤_k

f(x) f_Qk ²dx

!1/2

CkfkB M OH(Rⁿ⁾.

On the other hand, givenx 2 Qk, by the size condition (1) of the kernelKit can be checked in a standard way, see for instance [9], that

Z

(Q^⇤_k)^c

K(x,y) f(y) fQk dy CkfkB M OH(Rⁿ⁾.

Finally, since (i) holds, we have 1

|Qk| Z

Qk

fQkT^{1(x) dx} = |fQk| 1

|Qk| Z

Qk

|T^1(x)|dx CkfkB M OH(Rⁿ⁾. Hence, we conclude that (Ak) holds for Twith a constant C > 0 that does not depend onk.

Now we have to prove that T satisfies (Bk) for a certain C > 0 that it is independent ofk. LetB = B(x0,r0)✓ Q^⇤_k, wherex02R^{nandr0 >}0. Note that ifr0 (x0),then (x0)⇠ (xk)⇠r0,hence proceeding as above we will have

1

|B| Z

B|T^{f(x) (}T^f)B|dx  2

|B| Z

B|T^f(x)|dx CkfkB M OH(Rⁿ⁾, as soon as we have checked that 1

|B| Z

B|T1(x)|dx  C. In the definition ofT1 we can write

T1(x)=T( _Q⇤⇤

k )(x)+ Z

(Q^⇤⇤_k )^c

K(x,y)dy, x2 Q^⇤_k.

Hence, by hypothesis (i) onT1, H¨older’s inequality and the size condition (1) on the kernelK,

1

|B| Z

B|T1(x)|dx  C

|Q_k| Z

Qk

|T1(x)|dx+ C

|Q^⇤_k| Z

Q^⇤_k\Qk

|T1(x)|dx

C+ C

|Q^⇤_k| Z

Q^⇤_k|T( _Q⇤⇤

k )(x)|²dx

!1/2

+ Z

Q^⇤_k\Qk

Z

(Q^⇤⇤_k )^c

K(x,y)dy dx C.

Assume thatr0 < (x0). Using (3.2) we have that 1

|B| Z

B|T^{f(x) (}T^f)B|dx  1

|B| Z

B

1

|B| Z

B|T f1(x) T f1(z)|dz dx + 1

|B| Z

B

1

|B| Z

B|F2(x) F2(z)|dz dx + 1

|B| Z

B|T^f3(x) (T^f3)_B|dx

=:L1+L2+L3, where we defined

F2(x)= Z

(B^⇤)^c

K(x,y)f2(y)dy, x 2B,

and f = f1+ f2+ f3as in (3.1). By H¨older’s inequality and the boundedness in L²(R^n)ofT,

L1 2

|B| Z

B|T f1(x)|dx C

✓ 1

|B| Z

B^⇤|f(x) fB|²dx

◆1/2

CkfkB M OH(Rⁿ⁾.

It is well-known, see for instance [9], that the smoothness property (2) of the kernel K implies that

|F2(x) F2(z)|CkfkB M OH(Rⁿ⁾, x,z2 B. (3.3) Therefore, L₂  CkfkB M OH(Rⁿ⁾. Finally, by using the assumption (ii) on T¹ and [6, Lemma 2], it follows that

L3= |fB| 1

|B| Z

B|T^{1(x) (}T^1)B|dx

CkfkB M OH(Rⁿ⁾

✓

1+log (x0) r0

◆ 1

|B| Z

B|T^{1(x) (}T^1)B|dx

CkfkB M OH(Rⁿ⁾.

Hence, we conclude that 1

|B| Z

B|T^{f(x) (}T^f)B|dx  CkfkB M OH(Rⁿ⁾ for all B⇢Q^⇤_k and (Bk) is proved.

Let us now prove the converse statement. Suppose thatTis a bounded operator from B M OH(Rⁿ⁾ into itself. Since the functiong(x) = 1, x 2 Rⁿ, belongs to B M O_H(R^n),T^{1 is inB M O}H(Rⁿ). Then property (i) holds and there existsC>0 such that, for every ball B,

1

|B| Z

B|T^{1(y) (}T^1)B|dy C.

Let x0 2 R^{n and 0 < s < (x}0). Consider the function f(·;s,x0) defined in Lemma 2.1. Following the argument used in the estimate for the term L3 in the proof of the first part of this Theorem and using the fact that f(·;s,x0) 2 B M O_H(Rⁿ), we can find a constantC >0 that does not depend onsandx₀such that

log

✓ (x0) s

◆ 1

|B(x0,s)| Z

B(x0,s)|T^{1(y) (}T^1)B|dyC.

Then, condition (ii) holds and the proof of Theorem 1.1 is complete.

3.2. Pointwise multipliers inBMOH(Rⁿ⁾

Proposition 3.2. Let g be a measurable function on Rⁿ. We denote by Tg the multiplier operator defined by Tg(f) = f g. Then Tg is a bounded operator in

B M OH(Rⁿ⁾if and only if

(i) g2 L¹(R^{n); and}

(ii) there existsC >0such that log

✓ (x) s

◆ 1

|B(x,s)| Z

B(x,s)|g(y) gB(x,s)|dy C,

for everyx2Rⁿand every ball B(x,s)with radius0<s (x), where is given in(1.3).

Remark 3.3. Condition(ii)in Proposition 3.2 is fulfilled, for instance, when there exists 0<↵1 such that|g(x) g(y)|C|x y|^↵,x,y2R^n.

Remark 3.4. If for some Hermite-Calder´on-Zygmund operatorT we have thatT1 defines a pointwise multiplier inB M OH(Rⁿ⁾then the proposition above and The- orem 1.1 imply thatT is a bounded operator onB M OH(R^n).

Proof of Proposition3.2. Ifgis a measurable function inRⁿsatisfying the proper- ties (i) and (ii) in Proposition 3.2 we can proceed as in the proof of Theorem 1.1 to see thatgdefines a pointwise multiplier inB M O_H(Rⁿ⁾(note that the kernel of the operatorT =T_gis zero).

Suppose next thatgis a pointwise multiplier inB M OH(Rⁿ). For the function f(·;s,x0)defined in Lemma 2.1 and any ballB = B(x0,s)with 0 < s < ^(x₂⁰⁾, by using [6, Lemma 2], we have

log

✓ (x0) s

◆ 1

|B| Z

B|g(x)|dx = 1

|B| Z

B|f(x)g(x)|dx

 1

|B| Z

B|(f g)(x) (f g)_B|dx+(f g)_B

CkfkB M OH(Rⁿ⁾+log

✓ (x₀) s

◆

kf gkB M OH(Rⁿ⁾

Clog

✓ (x0) s

◆

kfkB M OH(Rⁿ⁾,

hence|g|B  CwithC independent of B. Therefore,gis bounded. On the other hand, ifx₀2R^{nand 0<s< (x}0)we have that

log

✓ (x₀) s

◆ 1

|B(x₀,s)| Z

B(x0,s)|g(x) g_B(x0,s)|dx

= 1

|B(x₀,s)| Z

B(x0,s)|g(x)f(x;s,x0) (g f(·;s,x0))_B(x0,s)|dx

 kg f(·;s,x0)kB M OH(Rⁿ⁾ Ckf(·;s,x0)kB M OH(Rⁿ⁾ C.

The constantsC > 0 appearing in this proof do not depend onx₀ 2 R^{n and 0 <}

s < (x₀).

4. Applications

Let us recall some definitions and properties of the operators related to the harmonic oscillator, see [18].

According to Mehler’s formula [18, page 2] the heat semigroup{W_t^H}^t>0gen- erated by H is given, for every f 2L²(R^{n), by}

W_t^Hf(x)⌘e ^{t H}f(x)= Z

RⁿW_t^H(x,y)f(y)dy, x 2R^{nand t >0, (4.1)} where

W_t^H(x,y)= e ^2t

⇡(1 e ^4t)

!n/2

e

1 2

h₁ +e 4t

1 e 4t |x|²+|y|² ₁^4e_e^2t4t x·yi

, t>0, x,y2R^n. Applying S. Meda’s change of parameterst =t(s)= ¹₂log¹₁^+s_s, 0<s<1,t >0, we obtain the following expression of the kernel ofW_t^H_(s):

W_t^H_(s)(x,y)= 1 s² 4⇡s

!n/2

e ¹⁴

h

s|x+y|²+¹s|x y|² i

, x,y2R^{n ands}2(0,1). (4.2) The semigroup{W_t^H}t>0is contractive inL^p(R^{n), 1} p 1, and selfadjoint in L²(Rⁿ⁾but it is not Markovian. Moreover, for every f 2 L^p(R^{n), 1}  p < 1,

t!lim0⁺W_t^Hf(x)= f(x)inL^p(R^{n)and a.e.x} 2R^n.

The Poisson semigroup associated to H is given byBochner’s subordination formula:

P_t^Hf(x)⌘e ^tpHf(x)= 1 0(1/2)

Z ₁

0

e ^t

2

4uHf(x)e ^u du

u^1/2, t >0. (4.3) Suppose now that f 2 B M OH(Rⁿ). Clearly for everyt 2(0,1)andx 2R^nthe

integral Z

RⁿWt(x,y)f(y)dy

is absolutely convergent. We define W_t^Hf and P_t^Hf, t > 0, by (4.1) and (4.3) respectively.

In the following subsections we prove Theorems 1.4 and 1.5.

4.1. Maximal operators for the heat and Poisson semigroups associated with the Hermite operator

Our systematic method developed in this paper (Theorem 1.1) allows us to show that the maximal operators W_⇤^H and P_⇤^H, defined byW_⇤^Hf = sup_t>0|W_t^Hf|and P_⇤^Hf =sup_t>0|P_t^Hf|, are bounded fromB M OH(Rⁿ⁾into itself, for everyn2N^.

The leading idea is to express the operators we are dealing with in such a way that the vector-valued setting can be applied, see Remark 1.2. Indeed, it is clear thatW_⇤^Hf = kW_t^HfkE, with E = L¹((0,1),dt). Hence, to see thatW_⇤^H maps

B M O_H(Rⁿ⁾into itself it is enough to show that

the operatorV(f):=(W_t^Hf)_t>0is bounded fromBMOH(R^n)intoBMOH(Rⁿ;E).

HereB M OH(Rⁿ;E)is defined in the obvious way by replacing the absolute values

|·|by normsk·kE. It is well-known thatVis bounded fromL²(R^n)intoL2(Rⁿ;E), see [16]. The desired boundedness result can be deduced from Remark 1.3 and the following

Proposition 4.1. There exist positive constantsCandcsuch that (i) kW_t^H(x,y)kE  C

|x y|ⁿ e ^{c⇥|x y|2+|x||x y|⇤},x,y2R^n,x6=y;

(ii) rxW_t^H(x,y)

E  C

|x y|ⁿ⁺¹,x,y 2R^n,x 6= y.

(iii) Moreover,kW_t^H1kE 2L¹(R^n)and rW_t^H1

E 2 L¹(R^n).

Proof. (i) Observe that if x,y 2 R^{n, x} · y > 0, then|x + y| |y|and for all s 2(0,1),

e

1 4

hs|x+y|²+¹s|x y|²i

e ^{8s1|x y|2}e

1 8

hs|x+y|²+¹s|x y|²i

e ^{8s1|x y|2}e ^{18|x y||x+y|}

e

1 8

h₁

s|x y|²+|y||x y|i

.

(4.4)

On the other hand, ifx,y2R^n,x·y0, then|x y| |y|and for alls2(0,1) e

1 4

hs|x+y|²+¹s|x y|²i

e ^{4s1|x y|2} e ^{8s1|x y|2}e ^{8s1|y||x y|}

e ¹⁸

h1

s|x y|²+|y||x y|i

.

(4.5)

Therefore, (i) follows from (4.2), (4.4) and (4.5).

(ii) By (4.2),

rxW_t^H_(s)(x,y)  C s^n/2

✓|x y|

s +s|x+y|

◆ e

1 4

hs|x+y|²+¹s|x y|²i

 C

s^(n+1)/2 e ^{cs|x y|2}

 C

|x y|ⁿ⁺¹, x,y2R^{n, x} 6=y, ands 2(0,1).

(iii) These properties can be easily deduced from the fact that W_t^H_(s)1(x)= 1

(4p

⇡)^n/2

1 s² 1+s²

!n/2

e

s 1+s2|x|²

, x2R^{n ands} 2(0,1). (4.6) Indeed, we clearly have|W_t^H_(s)1(x)|C. Moreover,

|rW_t^H_(s)1(x)|C s

1+s² |x|e ^1+ss2|x|2C

✓ s 1+s²

◆1/2

e ^1+css2|x|2C,

for all 0<s<1 andx 2R^n.

In order to see that the maximal operator associated with the Poisson semi- groupP_⇤^Hf =sup_t>0|P_t^Hf| =kP_t^HfkE is bounded fromB M OH(Rⁿ⁾into itself we can proceed using the vector-valued setting and the boundedness for the max- imal heat semigroup as follows. Let f 2 B M OH(Rⁿ). For any ball B we have that

1

|B| Z

B

P_t^Hf(x) ⇣ P_t^Hf⌘

B E dx

= 1

|B| Z

B

1 0(1/2)

Z ₁

0

W_t^H2 4u

f(x)e ^u du u^1/2 1

|B| Z

B

1 0(1/2)

Z ₁

0

W^H_t2 4u

f(y)e ^u du u^1/2 dy

E

dx

= 1

|B| Z

B

1 0(1/2)

Z ₁

0

W_t^H2 4u

f(x)e ^u du u^1/2 1

0(1/2) Z ₁

0

1

|B| Z

B

W^H_t2 4u

f(y)dy e ^u du u^1/2 _Edx

C Z ₁

0

1

|B| Z

B

W^H_t2 4u

f(x) 1

|B| Z

B

W^H_t2 4u

f(y)dy

E

dx e ^u du u^1/2

CkW_⇤^HfkB M OH(Rⁿ⁾

Z ₁

0

e ^u du

u^1/2 CkfkB M OH(Rⁿ⁾. If B= B(x0,r0)forx02R^nandr0 (x0)then

1

|B| Z

B

P_t^Hf(x)

Edx C Z ₁

0

1

|B| Z

B

W_t^H2 4u

f(x)

E

dx e ^u du u^1/2

C W_⇤^Hf _{B M OH(R}n)

Z ₁

0

e ^u du

u^1/2 CkfkB M OH(Rⁿ⁾. Therefore P_⇤^H is bounded fromB M OH(Rⁿ⁾into itself.