Product Hardy, BMO spaces and iterated commutators associated with Bessel Schr\"odinger operators

0.4 MB·

by  Jorge J. Betancor

PRODUCT HARDY, BMO SPACES AND ITERATED COMMUTATORS
ASSOCIATED WITH BESSEL SCHRÖDINGER OPERATORS
JORGE J. BETANCOR, XUANTHINHDUONG,JI LI,BRETT D.WICK, AND DONGYONGYANG
Abstract. In this paper we establish the product Hardy spaces associated with the Bessel
SchrödingeroperatorintroducedbyMuckenhouptandStein,andprovideequivalentcharacter-
izations in termsof theBessel Riesz transforms, non-tangential and radial maximal functions,
andLittlewood–Paley theory,whichareconsistentwiththeclassical productHardyspacethe-
7 ory developed by Chang and Fefferman. Moreover, in this specific setting, we also provide
1
another characterization via the Telyakovskií transform, which further implies that the prod-
0
uctHardyspaceassociatedwiththisBesselSchrödingeroperatorisisomorphictothesubspace
2
of suitable “odd functions” in the standard Chang–Fefferman product Hardy space. Based on

n thecharacterizations oftheseproductHardyspaces,westudytheboundednessoftheiterated
a commutatoroftheBesselRiesztransformsandfunctionsintheproductBMOspaceassociated
J
with Bessel Schrödinger operator. We show that this iterated commutator is bounded above,

8 but does not havea lower bound.
1

]
A
1. Introduction and statement of main results
C
h. There are several motivations for the research carried out in this paper. Associated to the
t usual Laplacian ∆ on Rn there are several important function spaces: the Hardy space H1(Rn)
a
m and the space of functions with bounded mean oscillation BMO(Rn). For the Hardy space, one
has a family of equivalent norms that can be used to study the space: via maximal functions,
[

square functions, area functions, Littlewood–Paley g-functions, Riesz transforms, and atomic
1
decompositions [St93]. Similarly the space BMO(Rn) has different ways that it can be studied:
v
1 via commutators and via Riesz transforms [CRW]. It has since become clear that the role of the
6 differential operator greatly influences the harmonic analysis questions that one can consider.
1
The work of Betancor et al, [BDT], studied the Hardy space theory associated to a Bessel
5
0 operator introduced by Muckenhoupt and Stein [MSt], that serves as primary motivation for
. our paper. Let λ R := (0, ) and
1 +
∈ ∞
0 d2 λ2 λ
7 (1.1) S f(x):= f(x)+ − f(x), x > 0.
1 λ −dx2 x2
v: The operator Sλ in (1.1) is a positive self-adjoint operator on L2(R+) and it can be written in
i divergence form as
X
S = x λDx2λDx λ =:A A ,
r λ − − − ∗λ λ
a where A := xλDx λ and A := x λDxλ is the adjoint operator of A . There has since been
λ − ∗λ − − λ
numerous investigations into harmonic analysis associated to this operator. See for example
[BFBMT, BCFR, BCFR2, BFS, BDT, BHNV, DLWY, V, YY].
The main goal of this paper is to study the product theory of harmonic analysis associated
to the operator S . In particular, we establish the product Hardy spaces associated with this
λ
BesselSchrödingeroperatorandprovideequivalentcharacterizations intermsoftheBesselRiesz
transforms, non-tangential and radial maximal functions, and Littlewood–Paley theory, which
are consistent with the classical product Hardy space theory developed by Chang and Fefferman
Date: January 19, 2017.
2010 Mathematics Subject Classification. 42B35, 42B25, 42B30, 30L99.
Key words and phrases. Bessel operator, maximal function, Littlewood–Paley theory, Riesz transform,
Cauchy–Riemann typeequations, product Hardy space, product BMO space.
1
2 JORGEJ.BETANCOR,XUANTHINHDUONG,JILI,BRETTD.WICK,ANDDONGYONGYANG
[CF]. We then also show that the commutators are bounded if the symbol belongs to a certain
BMO space associated to the operator S , but conversely this BMO does not characterize the
λ
boundedness of the commutator. This last result is surprising since it is known in the classical
multi-parameter setting thattheseiterated commutators infactcharacterize theBMOofChang
and Fefferman (see [FL, LPPW]). We now state our main results more carefully.
Throughout the paper, for every interval I R , we denote it by I := I(x,t) := (x
+
t,x + t) R . In the product setting R R⊂, we define R := (R R ,dx dx ). W−e
+ + + + + + 1 2
∩ × ×
work with the domain (R R ) (R R ) and its distinguished boundary R R . For
+ + + + + +
× × × ×
x := (x ,x ) R R , denote by Γ(x) the product cone Γ(x) := Γ (x ) Γ (x ), where
1 2 + + 1 1 2 2
∈ × ×
Γ (x ):= (y ,t ) R R : x y < t for i:= 1, 2.
i i i i + + i i i
{ ∈ × | − | }
We now provide several definitions of H1 . These spaces all end up being the same, which is
Sλ
one of the main results in this paper. This requires some additional notation, but the careful
reader will notice that the spaces are distinguished notationally by a subscript to remind how
they are defined.
We first define the product Hardy spaces associated with the Bessel operator S using the
λ
Littlewood–Paley areafunctionsandsquarefunctionsviathesemigroups T ,where T
t t>0 t t>0
{ } { }
can be the Poisson semigroup e−t√Sλ t>0 or the heat semigroup e−tSλ t>0. We note that the
{ } { }
definition via heat semigroup was covered in [CDLWY] in a more general setting.
Given a function f on L2(R ), the Littlewood–Paley area function Sf(x), x := (x ,x )
+ 1 2
∈
R R , associated with the operator S is defined as
+ + λ
×
1
2 dy1dy2dt1dt2 2
(1.2) Sf(x):= t ∂ T t ∂ T f(y ,y ) .
1 t1 t1 2 t2 t2 1 2 t2t2
(cid:18)ZZΓ(x)(cid:12) (cid:12) 1 2 (cid:19)
The square function g(f)(x), x(cid:12):= (x ,x ) R R ,(cid:12)associated with the operator S is
(cid:12) 1 2 + + (cid:12) λ
∈ ×
defined as
1
∞ ∞ 2 dt1dt2 2
(1.3) g(f)(x) := t ∂ T t ∂ T f(x ,x ) .
1 t1 t1 2 t2 t2 1 2 t t
(cid:18)Z0 Z0 (cid:12) (cid:12) 1 2 (cid:19)
We define the product Hardy space v(cid:12)ia the Littlewood–Paley s(cid:12)quare functions as follows.
(cid:12) (cid:12)
Definition 1.1. The Hardy space H1(R ) associated with S is defined as the completion of
g + λ
{f ∈ L2(R+): kg(f)kL1(R+) < ∞}
with respect to the norm kfkHg1(R+) := kg(f)kL1(R+), where g(f) is defined by (1.3) with Tt :=
e−t√Sλ or Tt := e−tSλ .
We now define the product Hardy space via the Littlewood–Paley area functions as follows.
Definition 1.2. The Hardy space H1(R ) associated with S is defined as the completion of
S + λ
{f ∈ L2(R+) :kSfkL1(R+) < ∞}
with respect to the norm kfkHS1(R+) := kSfkL1(R+), where Sf is defined by (1.2) with Tt :=
e−t√Sλ or Tt := e−tSλ .
We now define another version of the Littlewood–Paley area function. Let
:= (∂ ,∂ ), := (∂ ,∂ ).
∇t1,y1 t1 y1 ∇t2,y2 t2 y2
Then the Littlewood–Paley area function S f(x) for f L2(R ), x := (x ,x ) R R is
u + 1 2 + +
∈ ∈ ×
defined as
1
(1.4) Suf(x) := ∇t1,y1e−t1√Sλ∇t2,y2e−t2√Sλ(f)(y1,y2) 2 dy1dy2dt1dt2 2.
(cid:18)ZZΓ(x) (cid:19)
(cid:12) (cid:12)
Then naturally we have the fol(cid:12)lowing definition of the product Hardy(cid:12)space via the Littlewood–
Paley area function S f.
u
PRODUCT HARDY, BMO SPACES ASSOCIATED WITH BESSEL SCHRÖDINGER OPERATOR 3
Definition 1.3. The Hardy space H1 (R ) is defined as the completion of
Su +
{f ∈ L2(R+) :kSufkL1(R+) < ∞}
with respect to the norm kfkHS1u(R+) := kSufkL1(R+).
Next we define the product non-tangential and radial maximal functions via the heat semi-
group and Poisson semigroup associated to S . For all α (0, ), p [1, ), f Lp(R ) and
λ +
∈ ∞ ∈ ∞ ∈
x ,x R , let
1 2 +
∈
Nhαf(x1,x2):= sup e−t1Sλe−t2Sλf(y1,y2) ,
|y1−x1|<αt1
|y2−x2|<αt2(cid:12) (cid:12)
(cid:12) (cid:12)
NPαf(x1,x2):= sup e−t1√Sλe−t2√Sλf(y1,y2)
|y1−x1|<αt1
|y2−x2|<αt2(cid:12)(cid:12) (cid:12)(cid:12)
be the product non-tangential maximal functio(cid:12)ns with aperture α via t(cid:12)he heat semigroup and
Poisson semigroup associated to S , respectively. Denote 1f by f and 1f by f.
λ Nh Nh NP NP
Moreover, let
hf(x1,x2):= sup e−t1Sλe−t2Sλf(x1,x2) ,
R
t1>0,t2>0
(cid:12) (cid:12)
Pf(x1,x2):= sup (cid:12)e−t1√Sλe−t2√Sλf(x1,(cid:12)x2)
R
t1>0,t2>0
(cid:12) (cid:12)
be the product radial maximal functions via th(cid:12)e heat semigroup and Poi(cid:12)sson semigroup associ-
(cid:12) (cid:12)
ated to S , respectively.
λ
Definition 1.4. The Hardy space H1 (R ) associated to the maximal function f is defined
+
M
as the completion of the set M
{f ∈ L2(R+) :kMfkL1(R+) < ∞}
with the norm kfkHM1 (R+) := kMfkL1(R+). Here Mf is one of the following maximal functions:
f, f, f and f.
h P h P
N N R R
Next we recall the definition of the Riesz transforms associated with S . Define
λ
1/2
(1.5) RSλf =AλSλ− f.
Then we consider the definition of the product Hardy space via the Bessel Riesz transforms
R (f) and R (f) on the first and second variable, respectively.
Sλ,1 Sλ,2
Definition 1.5. The product Hardy space H1 (R ) is defined as the completion of
Riesz +
f L1(R ) L2(R ) : R f, R f, R R f L1(R )
∈ + ∩ + Sλ,1 Sλ,2 Sλ,1 Sλ,2 ∈ +
endowed with the norm
(cid:8) (cid:9)
kfkHR1iesz(R+) := kfkL1(R+)+kRSλ,1fkL1(R+)+kRSλ,2fkL1(R+)+kRSλ,1RSλ,2fkL1(R+).
The first main result of this paper is as follows.
Theorem 1.6. Let λ (1, ). The product Hardy spaces H1(R ), H1(R ), H1 (R ),
∈ ∞ g + S + Su +
H1 (R ),H1 (R ),H1 (R ), H1 (R )andH1 (R )coincideandhaveequivalent norms.
Nh + Rh + RP + NP + Riesz +
Because we have a family of equivalent norms we now choose to use H1 (R ) to denote the
Sλ +
product Hardy space associated to S . Based on the atomic decomposition of H1 (R ), we see
λ Sλ +
that we can identify H1 (R ) as a closed subspace of L1(R ).
Sλ + +
Based on the characterization of product Hardy space H1 (R ) via Bessel Riesz transforms
Sλ +
and theduality ofH1 (R )with BMO (R ), we directly have thesecondresult asacorollary:
Sλ + Sλ +
the decomposition of BMO (R ). For the definition of BMO (R ), we refer to Section 7.
Sλ + Sλ +
The proof of this result is similar to the classical setting.
4 JORGEJ.BETANCOR,XUANTHINHDUONG,JILI,BRETTD.WICK,ANDDONGYONGYANG
Corollary 1.7. The following two statements are equivalent:
(i) ϕ BMO (R );
(ii) T∈here exiSstλ g + L (R ), i= 1,2,3,4, such that
i ∞ +
∈
ϕ =g +R (g )+R (g )+R R (g ).
1 Sλ,1 2 Sλ,2 3 Sλ,1 Sλ,2 4
The second main result of this paper is to understand the structure of the space H1 (R ).
Sλ +
Given a function f L1(R ), we now introduce the “product odd extension” as follows
+
∈
f(x ,x ), x > 0,x > 0;
1 2 1 2
f( x ,x ), x < 0,x > 0;
(1.6) f (x ,x ):= − − 1 2 1 2
o 1 2
 f(−x1,−x2), x1 < 0,x2 < 0;
f(x , x ), x > 0,x < 0.
1 2 1 2
− −


Note that for this odd extension, wehave, for any fixed x2 R, fo(x1,x2) = fo( x1,x2); and
∈ − −
for any fixed x R, f (x ,x ) = f (x , x ).
1 o 1 2 o 1 2
Then we defin∈e the Hardy space−H1(R −) as follows:
o +
H1(R ) := f L1(R ) :f H1(R R)
o + { ∈ + o ∈ × }
with the norm kfkHo1(R+) = kfokH1(R×R), where H1(R×R) is the standard Chang–Fefferman
product Hardy space (see [CF]). This leads to the second main theorem:
Theorem 1.8. H1 (R ) and H1(R ) coincide and they have equivalent norms.
Sλ + o +
As corollary and application of our main Theorems 1.6 and 1.8, we also have the following
results. The first one is the comparison of the classical standard Hardy space H1(R ) and our
+
Hardy space H1 (R ). For the definition and properties of H1(R ), we refer to the classical
Sλ + +
Hardy space H1(R R) studied by Chang–Fefferman [CF] and modify and adapt it to R , in
+
×
terms of Littlewood–Paley area functions and atomic decomposition. We also refer to [HLL2] if
we consider R as a product space of homogeneous type. We use BMO(R ) to denote the dual
+ +
space of H1(R ). For the definition of BMO(R ) we refer to Section 7.
+ +
Theorem 1.9. The classical product Hardy space H1(R ) is a proper subspace of H1 (R ). As
+ Sλ +
a consequence, we obtain that BMO (R ) is a proper subspace of the classical product BMO
Sλ +
space BMO(R ).
+
We also can provide the following result regarding the product BMO space BMO (R ) and
Sλ +
the iterated commutator [[b,R ],R ].
Sλ,1 Sλ,2
Theorem 1.10. Let b BMO (R ). Then we have
∈ Sλ +
k[[b,RSλ,1],RSλ,2]kL2(R+)→L2(R+) . kbkBMOSλ(R+),
where the implicit constant is independent of b. However, the lower bound is NOT true. In
particular, there exists a locally integrable function b BMO (R ) such that
0 6∈ Sλ +
k[[b0,RSλ,1],RSλ,2]kL2(R+)→L2(R+) ≤ Cb0 < ∞,
where the constant C is related to b .
b0 0
At this point we remark that this is a novel and surprising result. In the classical multi-
parameter setting, it was shown by Ferguson and Lacey, [FL], and Lacey and Terwilleger, [LT],
that these iterated commutators characterize the product BMO of Chang and Fefferman. See
also[LPPW,LPPW2]forthecaseofRiesztransforms. Whereas inthiscase, thatnatural BMO
is sufficient for the boundedness of these commutators, but is not necessary.
The outline of the paper is as follows. In Section 2 we study the pointwise upper bound
[λ] [λ]
of the heat kernel W (x,y) and Poisson kernel P (x,y) of the operator S . We point out
t t λ
PRODUCT HARDY, BMO SPACES ASSOCIATED WITH BESSEL SCHRÖDINGER OPERATOR 5
[λ]
that although the potential here can be negative, W (x,y) still satisfies the standard Gaussian
t
[λ]
upper bound and P (x,y) satisfies the standard Poisson upper bound.
t
In Section 3, we prove that for f L1(R ),
+
∈
(1.7) kg(f)kL1(R+) ≈ kS(f)kL1(R+)
and the implicit constants are independent of f. Here the Littlewood–Paley g-function and area
functions can be defined using both heat semigroups and Poisson semigroups of S . The main
λ
strategy here is the atomic decomposition, especially the direction that the g-function implies
the atomic decomposition, where we apply the Moser type inequality of the Poisson semigroup.
In Section 4, we prove that for f L1(R ),
+
∈
(1.8) f . f . f . f . f . f
k kHS1(R+) k kHS1u(R+) k kHN1P(R+) k kHR1P(R+) k kHR1h(R+) k kHN1h(R+)
. f
k kHS1(R+)
and the implicit constants are independent of f. The main approach here is the use Merryfield’s
Lemma, atomic decomposition, and the Moser type inequality of the Poisson semigroup.
In Section 5, we prove that for f L1(R ),
+
∈
(1.9) f . f . f ,
k kHR1P(R+) k kHR1iesz(R+) k kHS1(R+)
and the implicit constants are independent of f. These inequalities, together with the loop in
(1.8),implythatourmainresultTheorem1.6holds. ThemaintoolsweuseherearetheCauchy–
Riemann type equations associated with S and the conjugate harmonic function estimates.
λ
For the proof of our second main result Theorem 1.8 is as follows. We introduce a product
Hardy type space H1(R ) via the Telyakovskií transform on R , which is also called the local
+ +
Hilbert transform (seTe [AM] and [F]), defined by
3x
2 f(t)
f(x)= p.v. dt.
TR+ x x t
Z2 −
The product Hardy type space H1(R ) is defined as the completion of
+
T
f L1(R ) L2(R ) : f, f, f L1(R )
{ ∈ + ∩ + TR+,1 TR+,2 TR+,1TR+,2 ∈ + }
with respect to the norm
kfkHT1(R+) := kfkL1(R+)+kTR+,1fkL1(R+)+kTR+,2fkL1(R+)+kTR+,1TR+,2fkL1(R+),
where denotestheTelyakovskiítransformonthefirstvariableand thesecond. Then,
TR+,1 TR+,2
to prove Theorem 1.8, we demonstrate that
kfkHR1iesz(R+) ≈ kfkHT1(R+) ≈ kfkHo1(R+).
Finally, as applications of our main Theorems 1.6 and 1.8, in Section 6 we provide the proof
of Theorems 1.9 and 1.10.
Throughout the whole paper, we denote by C positive constant which is independent of the
main parameters, but it may vary from line to line. If f Cg, we then write f . g or g & f;
≤
and if f . g . f, we write f g.
≈
2. Heat kernel and Poisson kernel estimates
[λ]
The heat semigroup W generated by S is defined by
{ t }t>0 − λ
[λ] ∞ [λ]
W (f)(x) = W (x,y)f(y)dy,
t t
Z0
where
1
[λ] (xy)2 xy x2+y2
Wt (x,y) = 2t Iλ−12 2t e− 4t , t,x,y ∈ (0,∞),
(cid:16) (cid:17)
6 JORGEJ.BETANCOR,XUANTHINHDUONG,JILI,BRETTD.WICK,ANDDONGYONGYANG
see [BDT]. Here I represents the modified Bessel function of the first kind and order ν (see
ν
[Le] for the properties of I ).
ν
2.1. Upper bounds of the heat kernel and Poisson kernel.
[λ] [λ]
Theorem 2.1. The kernel W (x,y) of the heat semigroup W satisfies the Gaussian
t { t }t>0
estimate. Namely, there are positive constants C and c such that for t > 0,
[λ] C −|x−y|2
(Ga) Wt (x,y) ≤ √te ct .
(cid:12) (cid:12)
Proof. First, we note that (cid:12) (cid:12)
1
[λ] (xy)2 xy x2+y2
Wt (x,y) = 2t Iλ−21 2t e− 4t
1 (cid:16) (cid:17)
xy 2 xy xy 1 (x−y)2
= 2t Iλ−12 2t e−2t · √2te− 4t .
(cid:18) (cid:19)
(cid:16) (cid:17)
Then we claim that there exists a positive constant C such that for all x,y,t > 0,
1
xy 2 xy xy
(2.1) 2t Iλ−12 2t e−2t ≤ C.
(cid:18) (cid:19)
(cid:16) (cid:17)
[λ]
This would then prove the Theorem that the heat semigroup W satisfies the Gaussian
{ t }t>0
estimate (Ga). To see the claim (2.1), we first recall that (see [Le])
(2.2) C 1xν I (x) Cxν, 0 < x 1,
− ν
≤ ≤ ≤
and that I (x) has the expansion
ν
ex µ 1 (µ 1)(µ 9) (µ 1)(µ 9)(µ 25)
I (x) 1 − + − − − − − +
ν ∼ √2πx − 8x 2!(8x)2 − 3!(8x)3 ···
(cid:26) (cid:27)
with µ = 4ν2, which gives that
ex 1
(2.3) I (x) = +Ψ(x), with Ψ(x) C exx 3/2 for x > .
ν ν −
√2πx | | ≤ 4
We now proceed by case analysis.
Case 1: 0 < xy 1. Then from (2.2) we have
2t ≤
xy xy λ 1
I −2.
λ−12 2t ≈ 2t
(cid:16) (cid:17) (cid:16) (cid:17)
So we get
x2yt 12 Iλ−12 x2yt e−x2yt ≈ x2yt 12 x2yt λ−12 e−x2yt ≤ x2yt λ e−x2yt ≤ C.
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
(cid:16) (cid:17) (cid:16) (cid:17)
Case 2: xy > 1. By (2.3) it follows
2t
xy
xy e2t xy
I = +Ψ .
λ−21 2t 2πxy 2t
(cid:16) (cid:17) 2t (cid:16) (cid:17)
So we can write q
1 1 xy 1
xy 2 xy xy xy 2 e2t xy xy 2 xy xy
(cid:18)2t(cid:19) Iλ−21 2t e−2t = (cid:18)2t(cid:19) 2πxy e−2t +(cid:18)2t(cid:19) Ψ 2t e−2t
(cid:16) (cid:17) 2t (cid:16) (cid:17)
q 1
1 +C xy 2 ex2yt xy −32 e−x2yt C.
≤ √2π 2t 2t ≤
(cid:18) (cid:19)
(cid:16) (cid:17)
(cid:3)
PRODUCT HARDY, BMO SPACES ASSOCIATED WITH BESSEL SCHRÖDINGER OPERATOR 7
As a direct consequence, we obtain the following corollary.
[λ] [λ]
Corollary 2.2. The heat kernel W (x,y) of the heat semigroup W satisfies the Davies–
t { t }t>0
Gaffney estimate. Namely, there are positive constants C and c such that for all open subsets
U , U (0, ) and all t > 0,
1 2
⊂ ∞
dist(U ,U )2
[λ] 1 2
(DG) W f ,f Cexp f f
|h t 1 2i| ≤ − ct k 1kL2(0,∞)k 2kL2(0,∞)
(cid:16) (cid:17)
for every f L2(0, ) with suppf U , i = 1,2. Here dist(U ,U ):= inf x y ..
i ∈ ∞ i ⊂ i 1 2 x∈U1,y∈U2| − |
[λ]
WenowconsidertheintegralkernelP (x,y)associatedwiththePoissonsemigroupgenerated
t
by √S .
λ
−
Corollary 2.3. There exists a positive constant C such that
t
[λ]
(2.4) P (x,y) C for all x,y,t > 0.
| t | ≤ t2+(x y)2
−
Proof. By the principle of subordination, we have
1
[λ] ∞ 1 3 [λ]
(2.5) Pt (x,y) = 2√π e−4ss−2Wt2s(x,y)ds.
Z0
[λ]
Since W (x,y) satisfies the Gaussian upper bound (Ga), it is direct that (2.4) holds. (cid:3)
t
2.2. Finite propagation speed. Let us recall the finite propagation speed for the wave equa-
tion and spectral multipliers (see [DLY]) and adapted to the Bessel operator S . Since the heat
λ
[λ]
kernel W (x,y) satisfies the Gaussian bound (Ga), it follows from [CS, Theorem 3] that there
t
exists a finite, positive constant c with the property that the Schwartz kernel K of
0 cos(t√Sλ)
cos(t√S ) satisfies
λ
(2.6) suppK (x,y) R R : x y c t ;
cos(t√Sλ) ⊆ ∈ +× + | − |≤ 0
see also [Si]. By the Fourier inversion form(cid:8)ula, whenever F is an even, bou(cid:9)nded, Borel function
with its Fourier transform Fˆ L1(R), we can write F(√S ) in terms of cos(t√S ). More
λ λ
∈
specifically, we have
(2.7) F( S )= (2π) 1 ∞ Fˆ(t)cos(t S )dt,
λ − λ
Z
p −∞ p
which, when combined with (2.6), gives
(2.8) K (x,y) = (2π) 1 Fˆ(t)K (x,y)dt, for every x,y R .
F(√Sλ) − Z|t|≥c−01|x−y| cos(t√Sλ) ∈ +
The following two results are useful for certain estimates later. We refer to [HLMMY,
Lemma 3.5] for the
Lemma 2.4. Let ϕ C (R) be even, suppϕ ( c 1,c 1), where c is the constant in (2.6).
∈ c∞ ⊂ − −0 −0 0
Let Φ denote the Fourier transform of ϕ. Then for every κ = 0,1,2,..., and for every t > 0,
the kernel K (x,y) of the operator (t2S )κΦ(t√S ) which was defined by the spectral
(t2Sλ)κΦ(t√Sλ) λ λ
theory, satisfies
suppK (x,y) (x,y) R R : x y t .
(t2Sλ)κΦ(t√Sλ) ⊆ ∈ +× + | − |≤
n o
Lemma 2.5. Let ϕ C (R) be an even function with ϕ = 2π, suppϕ ( 1,1). For every
∈ c∞ R ⊂ −
m = 0,1,2,..., set Φ(ξ):= ϕˆ(ξ), Φ(m)(ξ) := dm Φ(ξ). Let κ,m N and κ+m 2N. Then for
dξm R ∈ ∈
any t > 0, the kernel K (x,y) of (t√S )κΦ(m)(t√L) satisfies
(t√Sλ)κΦ(m)(t√Sλ) λ
(2.9) suppK (x,y) R R : x y t
(t√Sλ)κΦ(m)(t√Sλ) ⊆ ∈ +× + | − |≤
(cid:8) (cid:9)
8 JORGEJ.BETANCOR,XUANTHINHDUONG,JILI,BRETTD.WICK,ANDDONGYONGYANG
and
(2.10) K (x,y) Ct 1
(t√Sλ)κΦ(m)(t√Sλ) ≤ −
for any x,y R+. (cid:12) (cid:12)
∈ (cid:12) (cid:12)
2.3. Moser type inequality. As the Moser type inequality, established in [BCFR2, p.454],
we mean the following: For any x R , t (0, ) and 0 < r < t ,
0 + 0 0
∈ ∈ ∞
1/p
1
(2.11) u(t ,x ) . u(t,x)pdxdt ,
| 0 0 | r2 | |
" ZB((t0,x0),r) #
where u(t,x) = P[λ](f)(x) with f L1(R ) and B((t , x ), r) := (t,x) R R : (t,x)
t ∈ + 0 0 { ∈ +× + | −
(t ,x ) < r . We mention that the inequality was proved in [BCFR2] for p = 2. However, an
0 0
| }
iteration argument shows that it also holds for general p > 0.
2.4. Kernel estimates of Riesz transform R . We next note that the Riesz transform R
Sλ Sλ
related to Bessel operator S is bounded on L2(R ), and the kernel R (x,y) of R satisfies
λ + Sλ Sλ
the following size and regularity properties, as proved in [BFBMT, Proposition 4.1]:
There exists C > 0 such that for every x,y R with x = y,
+
∈ 6
C
(i) R (x,y) ;
| Sλ | ≤ x y
| − |
∂ ∂ C
(ii) R (x,y) + R (x,y) .
∂x Sλ ∂y Sλ ≤ x y 2
(cid:12) (cid:12) (cid:12) (cid:12) | − |
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
We also recall the following version of upper bound for R (x,y), which will be very useful
Sλ
in Section 6, connecting to the Telyakovskií transforms. There exists constant C > 0 such that
yλ x
(i) R (x,y) C , 0 < y < ,
′ | Sλ | ≤ xλ+1 2
xλ+1
(ii) R (x,y) C , y > 2x,
′ | Sλ | ≤ yλ+2
1 1 1 √xy x
(iii) R (x,y) = +O 1+log , < y < 2x.
Sλ πx y x + x y 2
− (cid:18) (cid:16) | − |(cid:17)(cid:19)
3. Proof of Equation (1.7)
3.1. Product Hardy spaces H1(R ) and atoms. We first consider H1(R ) as in Definition
S + S +
1.2 via Tt = e−tSλ. Note that the kernel Wt[λ](x,y) of e−tSλ satisfies the Gaussian upper bound
(Ga). Thus, H1(R ) falls into the scope of the Hardy space theory developed in [DLY]. We
S +
recall the definition and the atomic decomposition as follows.
First we recall the dyadic intervals in R as (R ) := (R ), where for each n Z,
+ + n Z n +
(R ) := ( k , k+1] : k Z and k 0 . FoDr I,J ∪(R∈)D, we use R := I J to de∈note
Dn + { 2n 2n ∈ ≥ } ∈ D + ×
the dyadic rectangles in R R . Then we denote all the dyadic rectangles in R R by
+ + + +
× ×
(R R ) = (R ), where (R ) = R = I J : I (R ), J
D + × + ∪n1,n2∈ZDn1,n2 + Dn1,n2 + { × ∈ Dn1 + ∈
(R ) .
Dn2 + }
Suppose Ω R R is an open set of finite measure. Denote by m(Ω) the maximal dyadic
+ +
⊂ ×
subrectangles of Ω. Let m (Ω) denote those dyadic subrectangles R Ω,R = I J, that are
1
⊆ ×
maximal in the x direction. In other words if S = I J R is a dyadic subrectangle of Ω,
1 ′
× ⊇
then I = I . Define m (Ω) similarly. Let Ω := (x ,x ) R R : (χ )(x ,x ) > 1/2 ,
′ 2 1 2 + + s Ω 1 2
∈ × M
where M is the strong maximal operator defined as
s
(cid:8) (cid:9)
e 1
(f)(x ,x ) := sup f(y)dy.
s 1 2
M R: rectangles in R+, (x1,x2)∈R |R| ZR| |
PRODUCT HARDY, BMO SPACES ASSOCIATED WITH BESSEL SCHRÖDINGER OPERATOR 9
l
For any R = I ×J ∈ m1(Ω), we set γ1(R) = γ1(R,Ω) = sup |I|, where the supremum is taken
| |
over all dyadic intervals l : I l so that l J Ω. Define γ similarly. Then Journé’s lemma,
2
⊂ × ⊂
(in one of its forms, see for example [J2, P, HLLin]) says, for any δ > 0,
e
R γ δ(R) c Ω and R γ δ(R) c Ω
| | 1− ≤ δ| | | | 2− ≤ δ| |
R∈Xm2(Ω) R∈Xm1(Ω)
for some c depending only on δ, not on Ω.
δ
We now recall the definition of a (S ,2,M)-atom.
λ
Definition 3.1 ([DLY]). Let M be a positive integer. A function a(x ,x ) L2(R ) is called
1 2 +
∈
a (S ,2,M)-atom if it satisfies:
λ
1) supp a Ω, where Ω is an open set of R R with finite measure;
+ +
⊂ ×
2) a can be further decomposed into a = a where m(Ω) is the set of all maximal dyadic
R
R m(Ω)
∈
subrectangles of Ω, and there exist a series oPf functions b belonging to the domain of Sk1 Sk2
R λ ⊗ λ
in L2(R ), for each k ,k = 1, ,M, such that
+ 1 2
···
(i) a = SM SM b ;
R λ ⊗ λ R
(ii) supp Sk1 Sk2 b 10R, k ,k = 0,1, ,M;
(cid:0) λ ⊗ λ (cid:1)R ⊂ 1 2 ···
1
(iii) ||a||L2(cid:0)(R+) ≤|Ω|−(cid:1)2 and
2
ℓ(I ) 4Mℓ(J ) 4M ℓ(I )2S k1 ℓ(J )2S k2b Ω 1.
R − R − R λ R λ R −
⊗ L2(R+) ≤ | |
R=IR×XJR∈m(Ω) (cid:13)(cid:13)(cid:0) (cid:1) (cid:0) (cid:1) (cid:13)(cid:13)
(cid:13) (cid:13)
We are now able to define an atomic Hardy space H1 (R ) space for M > 0, which is
at,M +
equivalent to the space H1(R ).
S +
Definition 3.2 ([DLY]). Let M > 0. The Hardy spaces H1 (R ) is defined as follows. We
at,M +
say that f = λ a is an atomic (S ,2,M)-representation of f if λ ℓ1, each a is a
j j λ { j}∞j=0 ∈ j
(S ,2,M)-atom, and the sum converges in L2(R ). Set
λ P +
H1 (R ) := f L2(R ) :f has an atomic (S ,2,M) representation ,
at,M + ∈ + λ −
with the norm given by
(cid:8) (cid:9)
∞
(3.1) f := inf λ ,
k kH1at,M(R+) | j|
j=0
X
where the infimum is taken over sequences {λj}∞j=0 such that ∞j=0|λj| < ∞ and jλjaj is an
atomic (S ,2,M)-representation of f. The space H1 (R ) is then defined as the completion
λ at,M + P P
of H1 (R ) with respect to this norm.
at,M +
Theorem 3.3 ([DLY]). Suppose that HS1(R+) is as in Definition 1.2 via Tt = e−tSλ and that
M > 0. Then H1(R ) = H1 (R ). Moreover, f f , where the implicit
S + at,M + k kHS1(R+) ≈ k kHa1t,M(R+)
constants depend only on M.
Second, we consider HS1(R+) as in Definition 1.2 via Tt = e−t√Sλ. Note that the kernel
P[tλ](x,y) of e−t√Sλ satisfies the Poisson upper bound. In fact, following the same approach and
techniques in the proof of [DLY, Theorem 3.4], we also obtain the following result in terms of
the Poisson semigroup.
Theorem 3.4. Suppose that HS1(R+) is as in Definition 1.2 via Tt = e−t√Sλ and that M > 0.
ThenH1(R ) = H1 (R ).Moreover, f f ,wheretheimplicitconstants
S + at,M + k kHS1(R+) ≈ k kHa1t,M(R+)
depend only on M.
10 JORGEJ.BETANCOR,XUANTHINHDUONG,JILI,BRETTD.WICK,ANDDONGYONGYANG
Based on the atomic decomposition above, we now show that the Hardy spaces H1(R ) and
g +
H1(R ) coincide and they have equivalent norms.
S +
Theorem 3.5. Suppose Hg1(R+) is as in Definition 1.1 via Tt = e−t√Sλ. Then Hg1(R+) =
HS1(R+). Moreover, kfkHS1(R+) ≈ kfkHg1(R+) where the implicit constants are independent of f.
Similar result holds for the Hardy space Hg1(R+) as in Definition 1.1 via Tt = e−t2Sλ.
Proof. Consider the Hardy space Hg1(R+) as in Definition 1.1 via Tt = e−t√Sλ. We first show
that
H1(R ) H1(R ).
S + ⊂ g +
Let f H1(R ). According to Theorem 3.4, H1(R ) = H1 for M > 0. Then, f H1 ,
∈ S + S + at,M ∈ at,M
and to see that f H1(R ), it suffices to prove that for every (S ,2,M) atom a,
∈ g + λ
(3.2) kg(a)kL1(R+) . 1.
Following the same proof as in [DLY, Lemma 3.6], we obtain that the above estimate holds for
the Littlewood–Paley g-function defined via Poisson or heat semigroups as in Definition 1.3.
Conversely, we show that
H1(R ) H1(R ).
g + ⊂ S +
To see this, we will show that we can derive atomic decomposition from the Littlewood–Paley
g-function defined via Poisson or heat semigroups as in Definition 1.3.
Let f H1(R ) L2(R ). We will first obtain the frame decomposition for f. By using the
∈ g + ∩ +
reproducing formula, setting ψ(x) := xM+1Φ(x) where Φ is defined as in Lemma 2.4, we can
write
(3.3)
dt dt
f(x1,x2) = ∞ ∞ψ(t1 Sλ)ψ(t2 Sλ)(t1 Sλe−t1√Sλ t2 Sλe−t2√Sλ)(f)(x1,x2) 1 2
⊗ t t
Z0 Z0 1 2
p p p p
∞ ∞
= K (x ,y )K (x ,y )
Z0 Z0 ZR+ ψ(t1√Sλ) 1 1 ψ(t2√Sλ) 2 2
dt dt
(t1 Sλe−t1√Sλ t2 Sλe−t2√Sλ)(f)(y1,y2)dy1dy2 1 2
⊗ t t
1 2
p p
= K (x ,y )K (x ,y )
ψ(t1√Sλ) 1 1 ψ(t2√Sλ) 2 2
R∈D(XR+×R+)ZT(R)
dt dt
(t1 Sλe−t1√Sλ t2 Sλe−t2√Sλ)(f)(y1,y2)dy1dy2 1 2
⊗ t t
1 2
p p
=: s W ,
R R
R∈D(XR+×R+)
where
sR := sup R 1/2 (t1 Sλe−t1√Sλ t2 Sλe−t2√Sλ)(f)(y1,y2)
| | | ⊗ |
(t1,y1,t2,y2) T(R)
∈ p p
and when s = 0,
R
6
1
W := K (x ,y )K (x ,y )
R s ψ(t1√Sλ) 1 1 ψ(t2√Sλ) 2 2
R ZT(R)
dt dt
(t1 Sλe−t1√Sλ t2 Sλe−t2√Sλ)(f)(y1,y2)dy1dy2 1 2.
⊗ t t
1 2
p p
I J
tHheerefoTll(oRw)in=g cIo×nd[i|2t|io,|nIs|:)×J ×[|2|,|J|), {WR}R∈D(R+×R+) is a family of frames, which satisfies