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Hardy spaces for Bessel-Schr\"odinger operators

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Hardy spaces for Bessel-Schr\"odinger operators

Abstract

Consider the Bessel operator with a potential on L^2((0,infty), x^a dx), namely Lf(x) = -f''(x) - a/x f'(x) + V(x)f(x). We assume that a>0 and V\in L^1_{loc}((0,infty), x^a dx) is a non-negative function. By definition, a function f\in L^1((0,infty), x^a dx) belongs to the Hardy space H^1(L) if sup_{t>0} |e^{-tL} f| \in L^1((0,infty), x^a dx). Under certain assumptions on V we characterize the space H^1(L) in terms of atomic decompositions of local type. In the second part we prove that this characterization can be applied to L for a \in (0,1) with no additional assumptions on the potential V.
arXiv:1603.07685v1 [math.CA] 23 Mar 2016
HARDY SPACES FOR BESSEL-SCHRÖDINGER OPERATORS
EDYTA KANIA AND MARCIN PREISNER
Abstract. Consider the Bessel operator with a potential on L2((0,), xαdx), namely
Lf(x) = f′′(x)α
xf(x) + V(x)f(x).
We assume that α > 0and VL1
loc((0,), xαdx)is a non-negative function. By definition,
a function fL1((0,), xαdx)belongs to the Hardy space H1(L)if
sup
t>0
etLf
L1((0,), xαdx).
Under certain assumptions on Vwe characterize the space H1(L)in terms of atomic decompositions
of local type. In the second part we prove that this characterization can be applied to Lfor α(0,1)
with no additional assumptions on the potential V.
1. Introduction
1.1. Background. The Schrödinger operator on Rdis given by
e
Lf=f+V · f,
where is the Laplace operator and Vis a function called a potential. If we assume that V ∈ L1
loc(Rd)
and V 0then one can find a densely defined, self-adjoint operator Lon L2(Rd), that corresponds to
e
L. It is well known that Lgenerates the semigroup of contractions Kt= exp(tL)and Ktadmits an
integral kernel Kt(x, y)such that
Ktf(x) = ZRdKt(x, y)f(y)dy,
0≤Kt(x, y)(4πt)d/2exp |xy|2
4t.
There have been wide studies on harmonic analysis related to Schrödinger operators and, more
generally, operators with Gaussian bounds. We refer the reader to [1], [2], [3], [5], [7], [8], [9], [10], [11],
[12], [14], [15], [16], [17], [18], [20], [21], [25], and references therein. In particular, the Hardy spaces
(1.1) H1(L) = (fL1(Rd) : kfkH1(L):=
sup
t>0|Ktf|
L1(Rd)
<)
related to Lwere intensively studied. At this point let us mention that the classical Hardy space
H1(∆) has many equivalent definitions, e.g. in terms of: various maximal functions, singular integrals,
square functions, etc. A particulary useful result is the atomic decomposition theorem (see [6], [22]):
a function fH1(∆) can be decomposed as f(x) = Pkλkak(x), where Pk|λk| ≃ kfkH1(∆) and
akare classical atoms, that is, there exist balls Bksuch that:
supp akBk,
kakk≤ |Bk|1,
Za(x)dx = 0.
In other words atoms satisfy some localization, size, and cancellation conditions.
2010 Mathematics Subject Classification. 42B30, 42B25, 35J10 (primary), 47D03, 43A85 (secondary).
Key words and phrases. Hardy space, Schrödinger operator, Bessel operator, maximal function, atomic decomposition.
The second author was supported by Polish funds for sciences grant DEC-2012/05/B/ST1/00672 from Narodowe
Centrum Nauki.
1
2 EDYTA KANIA AND MARCIN PREISNER
Let us mention that Hofmann et al. [21] have found general results (for Vsatisfying 0≤ V L1
loc(Rd))
saying that H1(L)given above is equal to the Hardy spaces via: square functions, atomic or molecular
decompositions. However, atoms used in [21] are given in terms of L, to be more precise: a function
aL2(Rd)is an atom if there exist a ball Band bDom(L), such that: a=Lb,supp bBand b, Lb
satisfy some size condition.
An another approach, started by Dziubański and Zienkiewicz in the 90’s, was to find atomic spaces
with simple geometric conditions that characterize H1(L). It appeared that this cannot be done in full
generality, and the properties of atoms depend strictly on the potential Vand the dimension d. For
example, if V C
c(Rd)and V 6≡ 0then atoms have modified cancellation condition Ra(x)ω(x)dx = 0,
where ωis such that 0< C1ω(x)C. For this result and generalizations see [15], [17], [24]. Other
results, see [14], lead to Hardy spaces with local atoms. It was first observed by Goldberg [19] that
if we take supremum for 0< t τ2in (1.1), then one obtains atomic space with classical atoms
complemented with the atoms of the form |B|1B(x), where the ball Bhas radius τ. In [14] the
authors assume that for 0 V L1
loc(Rd)there exists a family of cubes Q={Qk:kN}such that
kQk=Rd,|QkQj|= 0 for k6=j, d(Qk)d(Qj)if Q∗∗∗
kQ∗∗∗
j6=.
Here d(Q)is the diameter of Qand Qis a cube that has the same center as Qbut with slightly enlarged
diameter. The atomic space H1
at(Q)is built on classical atoms and atoms of the form |Qk|1Qk(x).
The main result of [14] states that under two additional assumptions on V, Q,Kt(see [14, p.41], con-
ditions: (D),(K)) we have that H1(L) = H1
at(Q), see [14, Thm. 2.2]. In other words the atoms for
H1(L)are either classical atoms or local atoms related to some Qk∈ Q.
Among examples of atoms for which one can find a family Qsuch that the assumptions of [14, Thm.
2.2] are satisfied, there are potentials Vin the Reverse Hölder class in dimension d3. For more
examples see [14]. Later, Czaja and Zienkiewicz [7, Thm. 2.4] proved that in dimension one for any
0≤ V L1
loc(R)there is a family of intervals such that [14, Thm. 2.2] gives local atomic decompositions
for H1(L).
A question that we are concerned with is: what happens if we replace by the Bessel operator
Bf(x) = f′′(x)α/x f (x)on L2((0,), xαdx)? It is known that if α+ 1 Nthen Bcorresponds
to on radial functions on Rdwith d=α+ 1, however Bexists and generates a semigroup for all
α > 1, which can be considered as the Laplace operator on spaces with non-integer dimensions.
In this paper we prove results similar to [14] and [7] for the Bessel operator with a potential.
1.2. Definitions. For α > 0let (X, ρ, µ)be a metric-measure space, where X= (0,),ρ(x, y) =
|xy|and (x) = xαdx. Denote B(x, r) = {yX:ρ(x, y)< r}and observe that Xis a space of
homogeneous type in the sense of Coifman-Weiss [6], i.e. the doubling condition holds
µ(B(x, 2r)) (B(x, r)),
where Cdoes not depend on xXand r > 0.
The classical Bessel operator is given by
Bf(x) = f′′(x)α
xf(x).
Slightly abusing notation, we shall also write Bfor the densely defined, self-adjoint operator on L2(X, µ)
that corresponds to the differential operator above, see Subsection 2.1 for the semigroup generated by B.
In this paper we consider the Bessel-Schrödinger operator L,
(1.2) Lf=Bf+V·f,
where VL1
loc(X, µ), V 0. To be more precise, denote hf , giµ=Rfg dµ and define a quadratic
form
Q(f, g) = hf, giµ+hV f, V giµ,
with the domain
Dom(Q) = nfL2(X, µ) : f,V f L2(X , µ)o.
HARDY SPACES FOR BESSEL-SCHRÖDINGER OPERATORS 3
The quadratic form Qis positive and closed. Therefore, it defines a self-adjoint operator Lwith the
domain
Dom(L) = fDom(Q) : hL2(X, µ)gDom(Q)Q(f, g) = hh, giµ.
For f , h as above we put Lf:= h. Let Kt= exp(tL)be the semigroup generated by L. Denote by
Bsthe Bessel process on (X, µ). By using the Feynman-Kac formula,
Ktf(x) = Exexp Zt
0
V(Bs)dsf(Bt),
one gets that Kthas an integral kernel Kt(x, y)and
(1.3) 0Kt(x, y)Pt(x, y),
where Pt(x, y)is the kernel related to Pt= exp (tB), see Subsection 2.1.
We define the Hardy space H1(L)by means of the maximal operator associated with Kt, namely
(1.4) H1(L) = (fL1(X, µ) : kfkH1(L)=
sup
t>0|Ktf|
L1(X, µ)
<).
The goal of this paper is to give an atomic characterizations of local type for H1(L). Let |I|be the
diameter of I.
Definition 1.5. Let Ibe a collection of intervals that are closed with respect to the topology on (0,).
We call a family Ia proper section of Xif:
(a) for I, J ∈ I,I6=J, the intersection IJis either the empty set or a singleton,
(b) X=SI∈I I,
(c) there exists a constant C0>0such that for I, J ∈ I,IJ6=we have
C1
0|I| ≤ |J| ≤ C0|I|.
Denote τB(c, r) := B(c, τ r). For an interval I=B(x, r)(if I= (0,2A)we take I=B(A, A)), let
cI := B(x, cr). For a family Ias in Definition 1.5 we set I:= βI for some fixed β > 1, such that
I∗∗∗ J∗∗∗ 6=if and only if IJ6=.
We say that a function a:XCis an (I, µ)-atom if:
(i)there exist I∈ I and an interval JI∗∗,such that: supp(a)J, kakµ(J)1,Za dµ = 0,
or
(ii)there exists I∈ I, such that a(x) = µ(I)1I(x).
The atoms as in (ii)are called local atoms.
The atomic Hardy space H1
at(I, µ)associated with the collection Iis defined in the following way.
We say that f∈ H1
at(I, µ)if
(1.6) f(x) = X
n
λnan(x),
where λnC,anare (I, µ)-atoms, and Pn|λn|<. Set
(1.7) kfkH1
at(I):= inf X
n|λn|,
where the infimum is taken over all possible representations of fas in (1.6).
For a collection Ias above and V0,VL1
loc(X, µ)we consider the following two conditions:
- there exist constans C, ε > 0such that
(D) sup
yI∗∗ ZX
K2k|I|2(x, y)(x)Ck1εfor I∈ I, k N,
- there exist constans C, δ > 0such that
(K) Z2t
0ZX
Ps(x, y)I∗∗∗ (y)V(y)(y)ds Ct
|I|2δ
for xX, I ∈ I, t ≤ |I|2.
4 EDYTA KANIA AND MARCIN PREISNER
1.3. Statement of results. Our first main result is the following, cf. [14, Thm. 2.2]
Theorem 1.8. Assume that a proper section Iand 0VL1
loc(X, µ)are given, so that (D)and
(K)hold. Then H1(L) = H1
at(I, µ)and there exists a constant C > 0, such that
C1kfkH1
at(I)≤ kfkH1(L)CkfkH1
at(I).
In the second part we give an important application of Theorem 1.8. Let us restrict ourselves to
α(0,1). We prove that for any 0VL1
loc(X, µ)we can find a family I(V)such that the
assumptions of Theorem 1.8 hold, cf. [7, Thm. 2.4] for the case α= 0. To be more precise, let Dbe a
family of dyadic intervals on (0,), that is D={[k2n,(k+ 1)2n] : kN∪ {0}, n Z}. Consider the
family I(V)that consists of maximal dyadic closed intervals Ithat satisfy
(S) |2I|2
µ(2I)Z2I
V(y)(y)1.
In Section 4we prove that I(V)is a well defined proper section. The second main result is the
following.
Theorem 1.9. Let α(0,1) and 0VL1
loc(X, µ). Then the family I(V)satisfies the assumptions
of Theorem 1.8.
Corollary 1.10. Let α(0,1) and 0VL1
loc(X, µ). Then there is C > 0, such that
C1kfkH1
at(I(V))≤ kfkH1(L)CkfkH1
at(I(V)).
The paper is organized as follows. In Section 2we study the atomic Hardy spaces related to Band
its local versions. This is used in a proof of Theorem 1.8, which is provided in Section 3. Finally, in
Section 4a proof of Theorem 1.9 is given.
2. Hardy spaces for the Bessel operator
2.1. Global Hardy space for B.In this section we consider the case α > 0. Let Pt= exp(tB)be
the Bessel semigroup given by
Ptf(x) = ZX
Pt(x, y)f(y)(y),
Pt(x, y) = (2t)1exp x2+y2
4tIα1
2xy
2t(xy)α1
2,
where Iα(x) = P
m=0 1
m!Γ(m+α+1) x
22m+αis the modified Bessel function of the first kind. It is clear
that Pt(x, y) = Pt(y, x)and, since B(0,)(x) = 0, we have that
(2.1) ZX
Pt(x, y)(x) = 1.
Let us recall that
(2.2) µ(B(x, t)) t(x+t)α.
It is known that the kernel Pt(x, y)satisfies the two-side Gaussian estimates (see, e.g. [13, Lem. 4.2]),
(2.3) C1µ(B(x, t))1exp |xy|2
c1t!Pt(x, y)Cµ(B(x, t))1exp |xy|2
c2t!,
while the derivative satisfies
(2.4)
∂x Pt(x, y)Ct1/2µ(B(x, t))1exp |xy|2
ct !.
Let H1(B)be the Hardy space related to Pt, i.e. the space defined as in (1.4) with Land Ktreplaced
by Band Pt, respectively. We call a function aan µ-atom if
(iii)there exists an interval JX, such that: supp(a)J, kakµ(J)1,Za dµ = 0.
HARDY SPACES FOR BESSEL-SCHRÖDINGER OPERATORS 5
The atomic Hardy space H1
at(µ)is defined as in (1.6) and (1.7) with anbeing µ-atoms.
Theorem 2.5. [4, Thm. 1.7] Let α > 0. There is C > 0such that
C1kfkH1
at(µ)≤ kfkH1(B)CkfkH1
at(µ).
2.2. Local Hardy space for B.For τ > 0we define h1
τ(B),the local Hardy space related to B, as
the set of L1(X, µ)functions for which the norm
kfkh1
τ(B)=
sup
tτ2|Ptf|
L1(X, µ)
is finite.
Let Iτbe a proper section of Xthat consists of closed intervals of length τ.
Theorem 2.6. (a)There exists C > 0such that for τ > 0and an (Iτ, µ)-atom awe have
kakh1
τ(B)C.
(b)Let Ibe an interval such that supp(f)Iand fh1
|I|(B). Then
f=
X
n=0
λnan,
X
n=0 |λn| ≤ Ckfkh1
|I|(B),
where a0(x) = µ(I)1I(x)and anare µ-atoms supported in I∗∗ for n1.
Let us remark that another characterization of h1
τ(B), by mean of a local Riesz transform, was given
in [23, Thm. 2.11]. Theorem 2.6 will be used to prove Theorem 1.8.
Proof. (a)Obviously, if ais µ-atom, then the statement follows from Theorem 2.5. Assume then that
a(x) = µ(I)1I(x)and |I|=τ. It is well known that (2.3) implies the boundedness of the maximal
operator supt>0|Ptf|on L2(X, µ). Using this fact and the Schwarz inequality,
sup
tτ2|Pta|
L1(I∗∗)Cµ(I)1/2
sup
t>0|Pta|
L2(X,µ)(I)1/2kakL2(X,µ)C.
Denote by cIthe center of Iand notice that |xy| ≃ |xcI|when yIand x(I∗∗)c. By (2.3),
sup
tτ2|Pta|
L1((I∗∗)c)CZ(I∗∗ )c
sup
tτ2ZI
µ(B(x, t))1exp |xcI|2
c3tµ(I)1(y)(x)
CZ(I∗∗)c
sup
tτ2
t1/2exp |xcI|2
c3tdx
CZ(I∗∗)c|I|1exp |xcI|2
c4|I|2dx C.
(b)Define λ0:= Rfand g(x) = f(x)λ0µ(I)1I(x). Notice that
|λ0| ≤ kfkL1(X,µ)≤ kfkh1
|I|(B).
Therefeore, kgkL1(X,µ)2kfkL1(X,µ). Our goal is to prove that
(2.7) kgkH1(B)Ckfkh1
|I|(B).
For t≤ |I|2using (a)we obtain
sup
t≤|I|2|Ptg|
L1(X,µ)≤ kfkh1
|I|(B)+|λ0|
µ(I)1I
h1
|I|(B)Ckfkh1
|I|(B)
6 EDYTA KANIA AND MARCIN PREISNER
As for t≥ |I|2notice that Rgdµ = 0. Therefore, using (2.4),
sup
t≥|I|2|Ptg|
L1((I∗∗)c)Z(I∗∗ )c
sup
t≥|I|2ZI
(Pt(x, y)Pt(x, cI)) g(y)(y)(x)
Z(I∗∗)cZI
sup
t≥|I|2
|ycI|
tµ(B(x, t))1exp |xcI|2
c2t|g(y)|(y)(x)
CkgkL1(X,µ)|I|Z(I∗∗)c
sup
t≥|I|2
t1exp |xcI|2
c2tdx
CkfkL1(X,µ)|I|Z(I∗∗)c|xcI|2dx Ckfkh1
|I|(B).
Likewise,
sup
t≥|I|2|Ptg|
L1(I∗∗)CZI∗∗ ZI
sup
t≥|I|2
µ(B(x, t))1|g(y)|(y)(x)
CkgkL1(X,µ)ZI∗∗
sup
t≥|I|2
t1/2dx Ckfkh1
|I|(B).
From (2.7) we have that gH1(B), so using Theorem 2.5 we obtain λkand µ-atoms aksuch that
g=P
k=1 λkak. Consequently,
f=
X
k=0
λkak,
X
k=0 |λk| ≤ Ckfkh1
|I|(B),
where a0(x) = µ(I)1I(x).
The only problem we have to deal with is that akare not necessarily supported in I∗∗ . Let ψbe
a function such that ψ1on I,ψ0on (I∗∗ )c, and kψkC|I|1. To complete the proof we
will show, that for every µ-atom akthere exist a sequence aj
k, such that each aj
kis either µ-atom or
aj
k=µ(I)1I. Moreover, ak=Pjλj
kaj
kand Pj|λj
k|< C, with Cthat not depend on k. Fix a=ak
and an interval JXsuch that: supp(a)J,kakµ(J)1. Obviously supp(ψa)I∗∗ J,
and if JI, then ψa =ais an µ-atom. Furthermore, if J(I∗∗ )c, then ψa = 0, so it suffices
to consider the case that J(I)cI∗∗ 6=. Observe that |J| ≤ |I|and let NNbe such that
1/2N+1|I| ≤ |J| ≤ 1/2N|I|.
Define λ:= Rψa dµ and notice that
|λ|=ZJ
ψ(x)a(x)(x)=ZJ
a(x)(ψ(x)ψ(cJ)) (x)
µ(J)1ZJ|xcJ||ψ(ξ)|(x)
C|J||I|1C2N.
Let us choose intervals Ij, such that J=: I0I1⊂ ··· ⊂ INI∗∗, where |Ij+1 |/|Ij|= 2 and
|IN| ≃ |I|. Then
ψa =ψa λµ(I0)1I0+
N
X
j=1
λµ(Ij1)1Ij1µ(Ij)1Ij
+λ(µ(IN)1INµ(I)1I) + λµ(I)1I=
N+2
X
j=0
bj.
Observe that:
(1) for j= 0, ..., N we have: supp(bj)Ij,Rbj= 0, and
kbjkC|λ|µ(Ij)1C2Nµ(Ij)1,
(2) supp(bN+1)I∗∗ ,kbN+1kC|λ|µ(I)1and RbN+1 = 0.
HARDY SPACES FOR BESSEL-SCHRÖDINGER OPERATORS 7
We conclude that bjare multiples of (I|I|, µ)-atoms and
N+2
X
j=0
bj
H1
at(I|I|)
C
N+2
X
j=0
2NC.
Corollary 2.8. There exists a constant C > 0such that for τ > 0we have
C1kfkH1
at(Iτ)≤ kfkh1
τ(B)CkfkH1
at(Iτ).
The right inequality in Corolarry 2.8 follows easily from Theorem 2.6(a). For the left inequality one
uses Theorem 2.6(b)with a suitable partition of unity and methods as in Lemmas 3.2 and 3.5 below.
We omit the details.
3. Proof of Theorem 1.8
3.1. Auxiliary estimates. For a proper section Ilet {φI}I∈I be a partition of unity associated with
I, that is a family of Cfunctions on X, such that supp(φI)I,0φI1,kφ
Ik≤ |I|1and
PI∈I φI(x) = 1 for all xX.
The perturbation formula states that
(3.1) Pt(x, y)Kt(x, y) = Zt
0ZX
Pts(x, z)V(z)Ks(z, y)(z)ds.
To prove Theorem 1.8 we closely follow the proof of [14, Thm. 2.2]. However, in our weighted space,
some technical difficulties appear. Therefore we present all the details for convenience of the reader.
Lemma 3.2. For I∈ I and fL1(X, µ),
sup
t4|I|2|Pt(φIf)|
L1((I∗∗)c, µ)CkφIfkL1(X, µ).
Proof. Denote by cIthe center of the interval I∈ I. For x(I∗∗)cand yIwe have |xy| ≃
|xcI|. Notice that (2.2) implies
(3.3) µ(B(x, t))1xαt1/2.
Using (2.3) the left-hand side is bounded by
Z(I∗∗)c
sup
t4|I|2ZI
µ(B(x, t))1e|xy|2
c2t|φI(y)f(y)|(y)(x)
CkφIfkL1(X,µ)Z(I∗∗)c
sup
t4|I|2
xαt1
2e|xcI|2
c3t(x)
CkφIfkL1(X,µ)Z(I∗∗)c|I|1e|xcI|2
c4|I|2dx
CkφIfkL1(X, µ).
Corollary 3.4. For I∈ I and fL1(X, µ),
kφIfkh1
|I|(B)
sup
t≤|I|2|Pt(φIf)|
L1(I∗∗, µ)
.
Denote
e
fI:= X
J:IJ6=
φJf, ¯
fI:= fe
fI=X
J:IJ=
φJf.
8 EDYTA KANIA AND MARCIN PREISNER
Lemma 3.5. For I∈ I and fL1(X, µ),
sup
t≤|I|2KtφIe
fIφIKte
fI
L1(I∗∗, µ)CX
J:IJ6=kφJfkL1(X, µ)
Proof. Denote e
I=J:IJ6=J. Note that for xI∗∗ and ye
I, there is |xy| ≤ C|I|and
(3.6) sup
t≤|I|2
t1
2e|xy|2
c2tC|xy|1.
Using (1.3), (2.3), (3.3), and (3.6),
sup
t≤|I|2KtφIe
fIφIKte
fI
L1(I∗∗, µ)ZI∗∗
sup
t≤|I|2Ze
I|φI(y)φI(x)|Kt(x, y)e
fI(y)(y)(x)
CZI∗∗ Ze
I|xy|kφ
Iksup
t≤|I|2
t1
2e|xy|2
c2te
fI(y)(y)dx
CZI∗∗ |I|1dx
e
fI
L1(X,µ).
Lemma 3.7. Assume that Vand Iare given, so that (D)holds. Then
X
I∈I
sup
t>0Kt¯
fI
L1(I∗∗∗, µ)CkfkL1(X, µ).
Proof. Denote sm= 2m|I|2and let m2. By the semigroup property, (1.3), (2.3),
sup
smtsm+1
Kt(|φJf|)
L1(X, µ)
ZX
sup
smtsm+1 ZXZX
Ptsm1(x, z)Ksm1(z, y)|φJ(y)f(y)|(z)(y)(x)
CZX|φJ(y)f(y)|ZX
Ksm1(z, y)ZX
s1/2
mexp |xz|2
c3smdx dµ(z)(y)
C(m1)1ε· kφJfkL1(X, µ).
In the last inequality we have applied (D). Using the above estimate and Lemma 3.2,
X
I∈I
sup
t>0Kt X
J:IJ=
φJf!
L1(I∗∗∗, µ)
X
J∈I X
I:JI=
sup
t>0
Kt(|φJf|)
L1(I∗∗∗, µ)CX
J∈I
sup
t>0
Kt(|φJf|)
L1((J∗∗)c, µ)
CX
J∈I
sup
t4|J|2
Kt(|φJf|)
L1((J∗∗)c, µ)
+CX
J∈I
X
m=2
sup
smtsm+1
Kt(|φJf|)
L1((J∗∗)c, µ)
CX
J∈I kφJfkL1(X, µ)+CX
J∈I
X
m=2
m1ε· kφJfkL1(X, µ)CkfkL1(X, µ).
Lemma 3.8. Z
0ZX
V(z)Ks(|f|)(z)(z)ds CkfkL1(X, µ)
HARDY SPACES FOR BESSEL-SCHRÖDINGER OPERATORS 9
Proof. By integrating (3.1) and using (2.1) we obtain
Zt
0ZX
V(z)Ks(z, y)(z)ds C.
The proof is finished by setting t→ ∞.
Lemma 3.9. Assume that Vand Iare given, so that (K)holds. Then for I∈ I,
sup
t≤|I|2|(PtKt)(φIf)|
L1(X, µ)CkφIfkL1(X, µ).
Proof. By (1.3) and Lemma 3.2,
sup
t≤|I|2|(PtKt)(φIf)|
L1((I∗∗)c, µ)CkφIfkL1(X, µ).
Consider the integral on I∗∗ . From (3.1),
(PtKt)(φIf)(x) = Zt
0
PtsV′′Ks(φIf)(x)ds +Zt
0
PtsVKs(φIf)(x)ds
where V=I∗∗∗ V+(I∗∗∗)cV=V+V′′ . Repeating the argument from the proof of Lemma 3.2 and
using Lemma 3.8 we obtain
sup
t≤|I|2Zt
0
PtsV′′Ks(φIf)ds
L1(I∗∗
,µ)CkφIfkL1(X, µ).
To estimate the integral that contains Vwrite
Zt
0
PtsVKs(φjf))(x)dsZt/2
0
PtsVPs(|φjf|)(x)ds +Zt
t/2
PtsVPs(|φjf|)(x)ds
=A1,t(x) + A2,t (x).
Let tm= 2m|I|2. Similarly as in proof of the Lemma 3.7 we obtain
sup
t≤|I|2
A1,t
L1(X, µ)
X
m=0 ZX
sup
tm+1ttmZt/2
0ZX
Pts(x, y)V(y)Ps(|φIf|)(y)(y)ds dµ(x)
C
X
m=0 ZXZtm
0
V(y)Ps(|φIf|)(y)ZX
t1
2
me|xy|2
c3tmdx ds dµ(y)
C
X
m=0 Ztm
0ZX
I∗∗∗ (y)V(y)Ps(|φIf|)(y)ds dµ(y)
C
X
m=0 2m|I|2
|I|2δ
kφIfkL1(X, µ).
In the last inequality we have used (K). To estimate A2,t we proceed similarly noticing that for
t[tm+1, tm]and s[t/2, t]we have s[tm+2, tm]. The details are left to the reader.
3.2. Proof of Theorem 1.8. First inequality. Observe that φIf=φIe
fIand
Kt(φIf) = KtφIe
fIφIKte
fIφIKt¯
fI+φIKt(f).
From Lemmas 3.5 and 3.7 we deduce that
10 EDYTA KANIA AND MARCIN PREISNER
X
I∈I
sup
t≤|I|2|Kt(φIf)|
L1(I∗∗, µ)X
I∈I
sup
t≤|I|2KtφIe
fIφIKte
fI
L1(I∗∗, µ)
+X
I∈I
sup
t≤|I|2φIKt¯
fI
L1(I∗∗, µ)
+X
I∈I
sup
t≤|I|2|φIKtf|
L1(I∗∗, µ)
CkfkL1(X, µ)+kfkH1(L)CkfkH1(L).
The above estimate, together with Corollary 3.4 and Lemma 3.9, lead to
X
I∈I kφIfkh1
|I|(B)CX
I∈I
sup
t≤|I|2|(PtKt)(φIf)|
L1(I∗∗, µ)
+CX
I∈I
sup
t≤|I|2|Kt(φIf)|
L1(I∗∗, µ)
CkfkH1(L).
Now we use Theorem 2.6(b)for each φIfgetting (λI
n)nand (I, µ)-atoms (aI
n)n, so that
φIf(x) = X
n
λI
naI
n(x),and X
nλI
nCkφIfkh1
|I|(B).
Summing up for all I∈ I we finish the first part of the proof.
Second inequality. Let abe an (I, µ)-atom, such that supp(a)I∗∗ . There exists an integer
m0, independent of I, such that
inf |I|2:JI6=2m|I|2.
Denote tn= 2n|I|2. Observe that
sup
ttm|Kt(a)|
L1(X, µ)
sup
ttm|(KtPt)(a)|
L1(X, µ)
+
sup
ttm|Pt(a)|
L1(X, µ)≤ kakL1(X, µ).
In the last inequality we applied Lemma 3.9 and Theorem 2.6(b), since ais also an (I|I|, µ)-atom.
It suffices to estimate
supt>tm|Kta|
L1(X, µ). This is done by using (1.3) and (K). Indeed, using
similar methods as in the proof of Lemma 3.2,
sup
t>tm|Kta|
L1(X, µ)X
nm
sup
tnttn1|Kta|
L1(X, µ)
X
nm
sup
tn+1t3tn+1
KtKtn+1 a
L1(X, µ)
CX
nm+1 ZXZX
K2n|I|2(x, y)|a(y)|(y)(x)
CkakL1(X, µ)C,
since the operator suptn+1 t3tn+1 Ktis bounded on L1(X, µ), see (2.3).
4. Proof of Theorem 1.9.
In the whole section we assume that α(0,1). Recall that for 0a < b,
(4.1) µ((a, b)) = b1+αa1+α
1 + α(bα+1 2ab
(ba)aα2ab.
HARDY SPACES FOR BESSEL-SCHRÖDINGER OPERATORS 11
By (4.1) and the Mean-Value Theorem we easily get
(4.2) |I|2
µ(I)b1αa1α.
Lemma 4.3. For α(0,1) and IJ(0,),
|I|2
µ(I)ZI
V(y)(y)|J|2
µ(J)ZJ
V(y)(y).
Proof. Obviously, since V0, it is enough to prove that
(4.4) |I|2
µ(I)|J|2
µ(J).
Let abcdand I= (b, c)(a, d) = J. Denote Γ(x, y) = (yx)2/(yα+1 xα+1 ). Now, (4.4) is
equivalent to Γ(b, c)Γ(a, d). This is done in two steps.
Step 1: Γ(b, c)Γ(a, c). Denote a=sc and b=tc, where 0< s t < 1. It is enough to prove
(1 t)2
1tα+1 (1 s)2
1sα+1 .
By a simple calculus argument, the function F1(x) = (1 x)2/(1 xα+1 )is monotonically decreasing
for x(0,1).
Step 2: Γ(a, c)Γ(a, d). Similarly, let c=ta,d=sa,1< ts. The function F2(x) =
(x1)2/(xα+1 1) is monotonically increasing in (1,), thus
(t1)2
(t)α+1 1(s1)2
(s)α+1 1.
Proposition 4.5. Let VL1
loc(X, µ),V0. Then the family I(V)of maximal dyadic intervals
satisfying (S)is a well-defined proper section (see Definition 1.5).
Proof. For a closed dyadic interval Iconsider F(I) = |2I|2µ(2I)1R2IV dµ and denote by Idthe
smallest dyadic interval containing I. Notice that 2I2Idand, by Lemma 4.3, we have F(I)F(Id).
Also, for an increasing sequence of dyadic intervals InIn+1 we have limn→∞ F(In) = , see (4.2).
This justifies the choice of I(V)as maximal dyadic intervals such that (S) holds. What is left to
prove is that I(V)is a proper section, namely we need to show that for I , J ∈ I(V),IJ6=we have
|I| ≃ |J|.
By contradiction, suppose that there exist Ik,Jksuch that IkJk6=and |Ik|/|Jk| → ∞. We can
assume that 2Jd
k2Ikfor all k. Denote, ak=|Ik|2µ(Ik)1|Jk|2µ(Jk). By the choice of I,
1|2Ik|2
µ(2Ik)Z2Ik
V(y)(y)|2Ik|2µ(2Jd
k)
µ(2Ik)2Jd
k22Jd
k2
µ(2Jd
k)Z2Jd
k
V(y)(y)C1ak.
The proof will be finished when we show that ak→ ∞. This follows from (4.1) by considering several
cases. Let a, b, c be such that 0a < b < c.
Case 1: Jk= [a, b],Ik= [b, c].
Subcase 1: 4a2bc. Using (4.1) we have
ak(c/b)2(b/c)1+α= (c/b)1α(|Ik|/|Jk|)1α.
Subcase 2: 4a2bc. This subcase can hold only for finite k.
Subcase 3: 4a2bc. Using (4.1) we have
akc2
(ba)2
(ba)aα
cα+1 =aαc
cα(ba)c1α
(ba)1α(|Ik|/|Jk|)1α.
12 EDYTA KANIA AND MARCIN PREISNER
Subcase 4: 4a2bc. Using (4.1) we have
akcb
ba2(ba)aα
(cb)bα≃ |Ik|/|Jk|.
Case 2: Jk= [b, c],Ik= [a, b]. Then
akC|Ik|
|Jk|2|Jk|bα
|Ik|bα≃ |Ik|/|Jk|.
Recall that hf, g iµ=RXfg dµ, so that h−Bφ, ψ iµ=hφ, Bψiµfor appropriate ψ , φ. For y > 0the
distributional equation
Bφ=δy
has the solution given by φy(x) = 1
2(1α)x1αy1α. We shall use φyto construct superharmonic
functions that will be crucial in the proof of (D).
Lemma 4.6. Let α(0,1),0VL1
loc(X, µ), and I(V)is as in Proposition 4.5. Then
ZX
K2n|I|2(x, y)(x)C21α
2n
for yI∗∗,I∈ I(V), and n0.
Proof. Let Ibe a dyadic interval such that
|2I|2
µ(2I)Z2I
V dµ 1,|2Id|2
µ(2Id)Z2Id
V dµ > 1.
By continuity argument there exists Jsuch that I∗∗ 2IJ2Idand |J|2
µ(J)RJV dµ = 1. Let
J= (a, b)and observe that |J| ≃ |I|. Define
φI(x) = 1 + 1
2(1 α)ZJ
V(y)x1αy1α(y).
Fix zI∗∗. By (4.2) and the doubling condition,
(4.7) φI(z)1 + Csup
y,yJ|y1α(y)1α|ZJ
V dµ C.
Also, we claim that for xX,
(4.8) φI(x)1 + µ(J)x1αz1α
|J|2.
Indeed, if xis such that |x1αz1α| ≤ C|J|2µ(J)1, with Clarge enough, this follows exactly as in
(4.7). In the opposite case |x1αz1α| ≥ C|J|2µ(J)1, we have |x1αz1α| ≃ |x1αy1α|for
yJand the claim follows.
Now we proceed to a crucial argument that uses superharmonicity. Observe that formal calculation
gives
LφI(x) = BφI(x)V(x)φI(x) = V(x)( J(x)φI(x)) 0
and, consequently,
∂t KtφI(z) = ZX
Kt(z, x)(LφI(x)) (x)0.
This leads to
(4.9) KtφI(z)φI(z), t > 0.
However, φIis not in Dom(L)(or even in L2(X, µ)), thus we provide a detailed proof of (4.9) in
Appendix.
Denote
θ(t) = ZX
Kt(z, x)(x).
HARDY SPACES FOR BESSEL-SCHRÖDINGER OPERATORS 13
Our goal is prove that
(4.10) θ(2n|I|2)c021α
2n.
This will follow by induction argument. By (1.3) and (2.3),
(4.11) K2t(z, x) = ZX
Kt(z, y)Kt(y, x)(y)Cθ(t)µ(B(x, t))1
and
θ(2t) = Z|x1αz1α|<R
K2t(z, x)(x) + Z|x1αz1α|>R
K2t(z, x)(x) = A1+A2.
By (4.8), (4.9), and (4.7) we have
A2|J|2
(J)Z|x1αz1α|>R
K2t(z, x)µ(J)x1αz1α
|J|2(x)
CR1|J|2µ(J)1ZK2t(z, x)φI(x)(x)
CR1|J|2µ(J)1.
To estmate A1we use (4.11) and (2.2),
A1(t)Z|x1αz1α|<R
µ(B(x, t))1(x)
(t)t1/2Z|x1αz1α|<R
dx.
Case A: z1α2R, then
Z|x1αz1α|<R
dx =1
1α(z1α+R)1
1α(z1αR)1
1αCRzα.
In this case
(4.12) θ(2t)c1θ(t)t1/2Rzα+R1|J|2µ(J)1.
Case B: z1α<2R,
Z|x1αz1α|<R
dx C(z1α+R)1
1αCR 1
1α.
In this case
(4.13) θ(2t)c2θ(t)t1/2R1
1α+R1|J|2µ(J)1.
Now we proceed to the proof of (4.10). The first step, θ(|I|2)C, follows simply by (1.3). Assume
that (4.10) holds for some n. Consider the following cases:
Case 1: ρ(0, I )2|I|. In this case µ(I)≃ |I|zα.
Subcase 1.1: θ(2n|I|2)2n/2|I|2z2. Observe that
R1:= 21θ(2n|I|2)1/2(2n|I|2)1/4|I|1/2zα< z1α/2.
Putting R=R1and t= 2n|I|2into (4.12) and using the induction hypothesis,
θ(2n+1|I|2)c3θ(2n|I|2)1/2(2n|I|2)1/4|I|1/2
c3c1/2
02n1α
42n
4c02(n+1) 1α
2.
The last inequality holds if we choose c0such that c0c2
321α.
Subcase 1.2: θ(2n|I|2)2n/2|I|2z2. One easily checks that
R2:= 21θ(2n|I|2)1(2n|I|2)1/2|I|zα1α
2α> z1α/2.
14 EDYTA KANIA AND MARCIN PREISNER
Putting R=R2and t= 2n|I|2into (4.13) and using the induction hypothesis,
θ(2n+1|I|2)c4θ(2n|I|2)(2n|I|2)1/2|I|1
1αzα
1α1α
2α
c4θ(2n|I|2)(2n|I|2)1/2|I|1α
2α
c4c0(2n)1α
2+1
21α
2αc02(n+1) 1α
2.
In the last inequality we choose c0such that c0c2α
42(1α)(2α)/2. Notice that we have used z≥ |I|,
which follows from ρ(0, I)2|I|.
Case 2: ρ(0, I )2|I|. In this case µ(I)≃ |I|α+1 . Notice that z4|I|.
Subcase 2.1 θ(2n|I|2)2n/2|I|2αzα2. Observe that
R3:= 21θ(2n|I|2)1/2(2n|I|2)1/4|I|1α
2zα/2< z1α/2.
Putting R=R3and t= 2n|I|2into (4.12) and using the induction hypothesis,
θ(2n+1|I|2)c5θ(2n|I|2)1/22n/4zα/2|I|α/2
c5c1/2
02n1α
42n
4zα/2|I|α/2
c5c1/2
02α2n1α
42n
4
c02(n+1) 1α
2.
The last inequality holds if we choose c0such that c0c2
521+α.
Subcase 2.2 θ(2n|I|2)<2n/2|I|2αzα2. One easily checks that
R4:= 21θ(2n|I|2)1(2n|I|2)1/2|I|1α1α
2α> z1α/2.
Putting R4and t= 2n|I|2into (4.13) we obtain
θ(2n+1|I|2)c6θ(2n|I|2)(2n|I|2)1/2|I|1α
2α
c6c0(2n)1α
2+1
21α
2αc02(n+1) 1α
2,
similarly as in subcase 1.2.
Lemma 4.14. Let α(0,1),0VL1
loc(X, µ), and I(V)is as in Proposition 4.5. Then the pair
(V, I(V)) satisfies (K).
Proof. Case 1: ρ(0, I )2|I|. In this case µ(I)≃ |I|1+α. Using (2.3) and (2.2),
Z2t
0ZI∗∗∗
Ps(x, y)V(y)(y)ds CZ2t
0
s1+α
2ds ·ZI∗∗∗
V dµ C t 1α
2µ(I)
|I|2Ct
|I|21α
2
.
Case 2: ρ(0, I)>2|I|. In this case µ(I)≃ |I|cα
I, where cIdenotes the center of I. Using (2.3) and
the doubling condition,
Z2t
0ZI∗∗∗
Ps(x, y)V(y)(y)ds CZ2t
0ZI∗∗∗
µ(B(y, s))1V(y)(y)ds
CZ2t
0
s1
2cα
Ids ·ZI∗∗∗
V(y)(y)
Ct 1
2µ(I)
|I|2cα
ICt
|I|21
2
.
Combining Lemmas 4.6 and 4.14 we obtain Theorem 1.9.
HARDY SPACES FOR BESSEL-SCHRÖDINGER OPERATORS 15
Appendix
The goal of this Appendix is to give a precise proof of the formula (4.9). Recall that J= (a, b). By
the definition of φIwe have
(4.15)
φI(x) = 1 + 1
2(1 α)
RJV(y)y1α(y)x1αRJV(y)(y), x < a
Rx
aV(y)(x1αy1α)(y)Rb
xV(y)(y1αx1α)(y), x (a, b)
x1αRJV(y)(y)RJV(y)y1α(y), x > b.
Recall that RJV=µ(J)|J|2. The formula (4.15) easily implies that
(4.16) |φ
I(x)| ≤ (J)|J|2xα
for all xX.
Lemma 4.17. Let ψDom(Q),ψ0,ηC
c(X, µ),η1on supp(ψ). Then φIηDom(Q)and
Q(ψ, φIη) = ZX
ψ(x)V(x)(φI(x)J(x)) (x)0.
Proof. Observe that using (4.15) we obtain that
φ′′
I(x)α
xφ
I(x) = J(x)V(x).
Since supp(η)is compact, using facts (4.8), (4.16) we deduce that φIηDom(Q). Since η1on
supp (ψ),ZX
ψ(x)(φI(x)η(x))(x) = ZX
ψ(x)φ
I(x)(x)
=ZX
ψ(x)φ′′
I(x)α
xφ
I(x)(x)
=ZX
ψ(x)J(x)V(x)(x).
The lemma follows, since φI1.
Recall that zJis fixed and denote ϑ(u) = KuφI(z).Our goal is to prove that ϑ(t+s)ϑ(t)for
t, s > 0.
Denote k(x) := Kt(x, z) = Kt/2(Kt/2(·, z ))(x). Since the semigroup Ktis analytic and Kt/2(·, z )
L2(X, µ)(see (1.3) and (2.3)), we have kDom(L)Dom(Q). Let ηnX,ηnC
c(X, µ),
supp(ηn)(0, n + 1],ηn1on (0, n], and kη
nk+kη′′
nkC.
First, observe that
(Ku(k)ηn)(φIηn+1)= (Ku(k)ηn)φ
I=Ku(k)φ
Iηn+Ku(k)φ
Iη
n.
Using this,
Q(Ku(k), φIηn) = ZKu(k)(φIηn)+ZV Ku(k)φIηn
=ZKu(k)φ
Iηn+ZKu(k)φIη
n+ZV Ku(k)φIηn
=Z(Ku(k)ηn)(φIηn+1)ZKu(k)φ
Iη
n+ZKu(k)φIη
n
+ZV(Ku(k)ηn)(φIηn+1)
=Q(Ku(k)ηn, φIηn+1)2ZKu(k)φ
Iη
nZKu(k)φIη′′
nZKu(k)φIη
n
α
x
=Q(Ku(k)ηn, φIηn+1)2B1B2B3.
(4.18)
16 EDYTA KANIA AND MARCIN PREISNER
Proposition 4.19. The function ϑis non-increasing.
Proof. Using (4.18),
ϑ(t+s)ϑ(t) = ZX
(Ks(k)k) (x)φI(x)(x)
= lim
n→∞ ZX
(Ks(k)k)(x)φI(x)ηn(x)(x)
= lim
n→∞ ZXZs
0
(L)Ku(k)(x)duφI(x)ηn(x)(x)
=lim
n→∞ Zs
0ZX
LKu(k)(x)φI(x)ηn(x)(x)du
=lim
n→∞ Zs
0
Q(Ku(k), φIηn)du
=lim
n→∞ Zs
0
(Q(Ku(k)ηn, φIηn+1)2B1B2B3)du.
Having in mind Lemma 4.17 it is enough to show that Rs
0Bidu 0as n→ ∞ for i= 1,2,3.
This follows from Lebesgue’s Dominated Convergence Theorem and the estimates we have already
established. For example, for B2observe that |φI(x)η′′
n(x)| ≤ Cµ(J)|J|2|x|1α[n,n+1](x)for nN
with Nlarge enough. Then the majorant is
Zs
0ZX
sup
nN|Ku(k)(x)φI(x)η′′
n(x)|(x)du CZs
0ZX
|x|1α
µ(B(x, t+u)) exp |xz|2
c2(t+u)xαdx du
CZs
0ZX
|x|1α
(t+u)1/2x
x+t+uα
exp |xz|2
c2(t+u)dx du
Ct1/2sZX|x|1αexp |xz|2
c2(t+s)dx
C(I, t, s).
The integrals with B3and B1goes similarly. For the latter one we use (4.16).
Acknowledgments: The authors would like to thank Jacek Dziubański for his helpful remarks.
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Edyta Kania
Instytut Matematyczny, Uniwersytet Wrocławski
pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
E-mail address :edyta.kania@math.uni.wroc.pl
Marcin Preisner
Instytut Matematyczny, Uniwersytet Wrocławski
pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
E-mail address :marcin.preisner@math.uni.wroc.pl
... We are interested in proving the lower Gaussian estimates, but this can be done only for some potentials V . For other potentials Hardy spaces may have a local character, (see e.g., [17]). Let ...
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... This can be done by a standard procedure, for details see e.g. [24,Thm. 2.2(b)]. ...
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