Riesz transform characterization of Hardy spaces associated with Schr¨ odinger operators

DOI: 10.1007/s11512-010-0121-5

c 2010 by Institut Mittag-Leffler. All rights reserved

Riesz transform characterization of Hardy spaces associated with Schr¨ odinger operators

with compactly supported potentials

Jacek Dziuba´nski and Marcin Preisner

Abstract. LetL=−Δ+V be a Schr¨odinger operator onR^d,d≥3. We assume thatV is a nonnegative, compactly supported potential that belongs toL^p(R^d), for somep>d/2. LetKt be the semigroup generated by−L. We say that anL¹(R^d)-functionf belongs to the Hardy space H_L¹ associated with Lif sup_t>0|Ktf|belongs to L¹(R^d). We prove thatf∈H_L¹ if and only if Rjf∈L¹(R^d) forj=1, ..., d, whereRj=(∂/∂xj)L^−1/2are the Riesz transforms associated withL.

1. Introduction

Let u(x, y)=u(x+iy) be a real-valued harmonic function in the upper half- plane {z=x+iy:y >0}. It was proved in Burkholder, Gundy, and Silverstein [2]

that for 0<p<∞ the function u is a real part of a holomorphic function F(z)=

u(z)+iv(z) which satisfies theH^p property sup

y>0

R|F(x+iy)|^pdx <∞

if and only if the maximal functionu^∗(x)=sup_|x−x|<y|u(x+iy)|belongs toL^p(R).

In Fefferman–Stein [7] the authors present other characterizations of the H^p prop- erty by means of real analysis. These characterizations lead to the notion of classical real Hardy spacesH^p(R^d). Let us recall two equivalent characterizations ofH¹(R^d) presented in [7]. AnL¹(R^d)-functionf is an element of the real Hardy spaceH¹(R^d) if and only if the maximal functionM^√₋Δf(x)=sup_t>0exp

−t√

−Δ

f(x)belongs toL¹(R^d), where exp

−t√

−Δ

denotes the Poisson semigroup. Equivalently, one

Supported by the Polish Ministry of Science and High Education—grant N N201 397137, the European Commission Marie Curie Host Fellowship for the Transfer of Knowledge “Harmonic Analysis, Nonlinear Analysis and Probability” MTKD-CT-2004-013389.

can take the maximal functionM−Δf(x) from the heat semigroup Pt in the defi- nition of the Hardy spaceH¹(R^d), that is,M−Δf(x)=sup_t>0|Ptf(x)|, where here and subsequently

P_tf(x) =f∗P_t(x) and P_t(x) = (4πt)^−d/2exp

−|x|² 4t

.

The second characterization ofH¹(R^d) is given by boundedness of certain singular integrals. Let

Rjf(x) =cdlim

ε→0

|x−y|>ε

xj−yj

|x−y|^d+1f(y)dy

= lim

ε→0

1/ε ε

R^d

∂

∂x_jPt(x−y)f(y)dydt

√t, (1.1)

j=1,2, ..., d, be the classical Riesz transforms on R^d. Clearly, for f∈L¹(R^d) the limits in (1.1) exist in the sense of distributions and defineRjf as distributions. It was proved in [7] (see also [9]) that anL¹(R^d)-function f is in the classical Hardy spaceH¹(R^d) if and only ifRjf∈L¹(R^d) forj=1, ..., d. Moreover,

(1.2) fL¹(R^d)+

d

j=1

RjfL¹(R^d),

defines a norm in the spaceH¹(R^d), which is equivalent to the normsM−ΔfL¹(R^d)

and M₋^√₋ΔfL¹(R^d). In addition, in the case of d=1, if f∈H¹(R), then u^∗∈ L¹(R), whereu(x+iy)=exp

−y√

−Δ

f(x),y >0. Conversely, every harmonic func- tionuwith the property thatu^∗∈L¹(R) is obtained in this way. In this case, the con- jugate functionv is of the formv(x+iy)=exp

−y√

−Δ

g(x) withg=−π^−1/2Rf. In this paper we consider a Schr¨odinger operator L=−Δ+V on R^d, d≥3.

We assume thatV is a nonnegative function, suppV⊆B(0,1)={x∈R:|x|<1}, and V∈L^p(R^d) for some p>d/2. Let {Kt}t>0 be the semigroup of linear operators generated by−L. SinceV≥0, by the Feynman–Kac formula, we have

(1.3) 0≤Kt(x, y)≤Pt(x−y),

whereKt(x, y) is the integral kernel of the semigroup {Kt}t>0. Let MLf(x) = sup

t>0|K_tf(x)|.

We say that anL¹(R^d)-function f belongs the Hardy spaceH_L¹ if fH_L¹=MLfL¹(R^d)<∞.

Forj=1, ..., dlet us define theRiesz transformsRj associated withLby setting

(1.4) Rjf=√

π ∂

∂xj

L^−1/2f= lim

ε→0

1/ε ε

∂

∂xj

Ktf dt

√t,

where the limit is understood in the sense of distributions. The fact that for any f∈L¹(R^d) the operators

(1.5) R^ε_jf=

1/ε ε

∂

∂x_jK_tf dt

√t

are well defined and the limits lim_ε→0R^ε_jf exist in the sense of distributions will be discussed below.

The main result of this paper is the following theorem.

Theorem 1.1. Assume that f∈L¹(R^d). Thenf is in the Hardy spaceH_L¹ if and only ifRjf∈L¹(R^d)for everyj=1, ..., d. Moreover,there existsC >0such that

(1.6) 1

CfH_L¹≤ fL¹(R^d)+ d j=1

RjfL¹(R^d)≤CfH_L¹.

The Hardy spaces H_L¹ associated with the Schr¨odinger operators Lwith com- pactly supported potentials were studied in [6]. It was proved there that the el- ements of the space H_L¹ admit special atomic decompositions. Furthermore, the spaceH_L¹ is isomorphic to the classical Hardy spaceH¹(R^d). To be more precise, let

Γ(x, y) = _∞

0

Kt(x, y)dt and Γ0(x, y) =− _∞

0

Pt(x, y)dt, and set

L⁻¹f(x) =

R^d

Γ(x, y)f(y)dy and Δ⁻¹f(x) =

R^d

Γ0(x, y)f(y)dy.

The operatorsI−VΔ⁻¹and I−V L⁻¹ are bounded and invertible onL¹(R^d), and I= (I−VΔ⁻¹)(I−V L⁻¹) = (I−V L⁻¹)(I−VΔ⁻¹).

Moreover,

(I−V L⁻¹) :H_L¹−→H¹(R^d) is an isomorphism (whose inverse isI−VΔ⁻¹) and (1.7) (I−V L⁻¹)fH¹(R^d) fH_L¹

for f∈H_L¹ (see [6, Corollary 3.17]). Therefore, in order to prove Theorem 1.1, it suffices to show that Rj(I−V L⁻¹)−Rj, j=1,2, ..., d, are bounded operators onL¹(R^d).

The proof of the relevant atomic decompositions presented in [6] was based on the following identity

(1.8) Pt(I−V L⁻¹) =Kt− t

0

(Pt−Pt−s)V Ksds− _∞

t

PtV Ksds=Kt−Wt−Qt, which comes from the perturbation formula

Pt=Kt+ t

0

Pt−sV Ksds.

Formula (1.8) will be used also here in the analysis of the integral (1.5) for larget, while for smallt we shall use its slightly different version, namely

(1.9) Pt(I−V L⁻¹) =Kt+ t

0

Pt−sV Ksds−PtV L⁻¹=Kt+Wt−Qt.

Boundedness of Riesz transforms for Schr¨odinger operators onL^p-spaces at- tracted attention of many authors. We refer the reader to [1], [4], and [8] for background and references. Under very relaxed assumptions on V ≥0, the weak type (1,1) inequality

(1.10) ∇f1,∞+V^1/2f1,∞≤C(−Δ+V)^1/2fL¹(R^d), f∈C_c^∞(R^d), is an unpublished result of Ouhabaz (see also [8] for a proof based on finite speed propagation of the wave equation).

In the case of Schr¨odinger operators with potentials V ≥0, V ≡0, satisfying the reverse H¨older inequality with the exponent d/2 (which clearly implies that suppV=R^d), Riesz transform characterizations of the relevant Hardy spacesH₋¹_Δ+V were obtained in [5].

2. Auxiliary estimates In this section we will use notationft(x)=t^−d/2f

x/√ t

.

Forf∈L¹(R^d) and 0<ε<1 we definetruncated Riesz transforms by setting R^ε_jf=

1/ε ε

∂

∂xj

K_tf dt

√t and R^εjf= 1/ε

ε

∂

∂xj

P_tf dt

√t.

Let

G(x, y) = _∞

0

Kt(x, y)dt

√t and G0(x, y) = _∞

0

Pt(x−y)dt

√t.

Then

G(x, y)≤G0(x, y) =c|x−y|^−d+1 and, consequently, forϕfrom the Schwartz classS(R^d) we have

limε→0R^ε_jf, ϕ=−

R^2d

G(x, y)f(y) ∂

∂x_jϕ(x)dy dx.

HenceRjf is a well-defined distribution and

|Rjf, ϕ| ≤CfL¹(R^d)

∂

∂xj

ϕ L¹(R^d)

+ ∂

∂xj

ϕ L^∞(R^d)

.

Using (1.8) and (1.9) we write

(2.1) R^ε_jf=R^εj(I−V L⁻¹)f−W^εjf+Q^εjf+Wj^εf+Q^εjf, whereQ^εj,Q^εj,Wj^ε, andW^εj are operators with the integral kernels

Q^εj(x, y) = 1/ε

1

∂

∂x_jQt(x, y) dt

√t and Q^εj(x, y) = 1

ε

∂

∂x_jQt(x, y) dt

√t,

Wj^ε(x, y) = 1/ε

1

∂

∂xj

Wt(x, y) dt

√t and W^εj(x, y) = 1

ε

∂

∂xj

Wt(x, y) dt

√t,

respectively. We shall prove thatQ^εj,Wj^ε, andW^εj converge in the norm-operator topology onL¹(R^d), while Q^εj converges strongly onL¹(R^d) asεtends to 0.

Lemma 2.1. The operators Q^εj converge in the norm-operator topology on L¹(R^d)as ε→0.

Proof. There exists φ∈S(R^d),φ≥0, such that

(2.2)

∂

∂xj

Pt(x−z)

≤t^−1/2φt(x−z).

On the other hand, by (1.3),Ks(z, y)≤Cs^−d/2. Hence for 0<ε2<ε1<1 we have

R^d|Q^εj¹(x, y)−Q^εj²(x, y)|dx

≤C

R^d

1/ε₂ 1/ε₁

_∞

t

R^d

t^−1/2φt(x−z)V(z)s^−d/2dz dsdt

√tdx

≤C 1/ε₂

1/ε₁

t^−d/2dtVL¹(R^d), (2.3)

which tends to zero uniformly with respect toy as ε1, ε2→0.

Lemma 2.2. The operators Wj^ε converge in the norm-operator topology on L¹(R^d) asε→0.

Proof. The proof borrows ideas from [6]. Let 0<ε2<ε1<¹₂. Then

R^d|Wj^ε1(x, y)−Wj^ε2(x, y)|dx

≤

R^d

1/ε2

1/ε₁

t^8/9 0

R^d

∂

∂xj

(P_t(x−z)−P_t−s(x−z))

V(z)K_s(z, y)dz ds dt

√tdx

+

R^d

1/ε2

1/ε₁

t t^8/9

R^d

∂

∂xj

(P_t(x−z)−P_t−s(x−z))

V(z)K_s(z, y)dz ds dt

√tdx

=W(y)+W(y).

Observe that there existsφ∈S(R),φ≥0, such that for 0<s<t^8/9, ∂

∂x_j(Pt(x−z)−Pt−s(x−z))

≤st^−3/2φt(x−z).

SincesK_s(z, y)≤Cs^(2−d)/2exp(−c|z−y|²/s)≤C|z−y|^2−d, we have that W(y)≤

R^d

1/ε2

1/ε1

t^8/9 0

R^d

st⁻²φt(x−z)V(z)Ks(z, y)dz ds dt dx

≤ 1/ε2

1/ε1

t^−10/9dt

R^d

V(z)|z−y|^2−ddz≤Cε^1/9₁ (2.4)

uniformly iny. The last inequality is a simple consequence of the H¨older inequality and the assumptionp>d/2.

Fort^8/9<s<t we haveKs(z, y)≤C s^−d/2≤C t^−4d/9. Using (2.2) we get that W(y)≤C

R^d

1/ε₂ 1/ε₁

t t^8/9

R^d

(t^−1/2φt(x−z)+(t−s)^−1/2φt−s(x−z))

×V(z)t^−4d/9dz dsdt

√tdx

≤CVL¹(R^d)

1/ε₂ 1/ε₁

t^−4d/9dt

+CVL¹(R^d)

1/ε₂ 1/ε1

t^{−4d/9−1/2} t

0

(t−s)^−1/2ds dt

≤Cε^4d/9₁ ⁻¹ (2.5)

uniformly iny. Now the lemma follows from (2.4) and (2.5).

Lemma 2.3. There exists a limit of the operators W^εj in the norm-operator topology on L¹(R^d)asε→0.

Proof. Let 0<ε₁<ε₂<1. Applying (2.2) we obtain that

R^d|W^εj²(x, y)−W^εj²(x, y)|dx

≤

R^d

ε2

ε₁

t 0

R^d

(t−s)^−1/2φ_t−s(x−z)V(z)K_s(z, y)dz dsdt

√tdx

=

R^d

ε2

ε₁

t/2 0

R^d

...+

R^d

ε2

ε₁

t t/2

R^d

...

=W(y)+W(y).

(2.6)

If 0<s<t/2, then, of course, (t−s)^−1/2≤Ct^−1/2. Note that t

0

K_s(z, y)ds≤C|z−y|^2−dexp

−|z−y|² 8t

≤tψ_t(z−y)

for someψ∈L^p(R^d),ψ≥0 (p denotes the H¨older conjugate exponent top). Hence, W(y)≤C

ε2

ε₁

t/2 0

R^d

t⁻¹V(z)K_s(z, y)dz ds dt≤C ε2

ε₁

R^d

V(z)ψ_t(z−y)dz dt

≤C ε2

ε1

Vpψ_tpdt≤C ε2

ε1

t^−d/2pψ₁pdt≤Cε¹₂^−d/2p (2.7)

uniformly iny.

If t/2≤s≤t, then there exists ϕ∈S(R^d), ϕ≥0, such that Ks(z, y)≤ϕt(z−y).

Therefore

W(y)≤ ε2

ε1

t t/2

R^d

(t−s)^−1/2V(z)Ks(z−y)dz dsdt

√t

≤C ε2

ε₁

R^d

t/2 0

(st)^−1/2V(z)ϕ_t(z−y)ds dz dt

≤C ε₂

ε1

Vpϕtpdt

≤Cε¹₂^−d/2p (2.8)

uniformly iny. Now the lemma is a consequence (2.7)–(2.8).

Lemma 2.4. Assume that f∈L¹(R^d). Then the limit F=limε→0Q^εjf exists in theL¹(R^d)-norm. Moreover, FL¹(R^d)≤CfL¹(R^d) withC independent of f.

Proof. Of course, for any fixedy∈R^d, the functionz→U(z, y)=V(z)Γ(z, y) is supported in the unit ball andU(z, y)L^r(dz)≤Cr for fixed r∈[1, dp/(dp+d−2p)) with Cr independent of y. The last statement follows from (1.3) and the H¨older inequality. Let

H_j^ε(x, z) = 1

ε

∂

∂xj

Pt(x−z)dt

√t,

H_j^εg(x) =

R^d

H_j^ε(x, z)g(z)dz, H_j^∗g(x) = sup

0<ε<1|H_j^εg(x)|.

It follows from the theory of singular integral convolution operators (see, e.g., [3, Chapter 4]) that for 1<r<∞there exists Cr such that

(2.9) H_j^∗gL^r(R^d)≤CrgL^r(R^d) forg∈L^r(R^d) and limε→0H_j^εg(x)=Hjg(x) a.e. and in theL^r(R^d)-norm.

Note thatQ^εj(x, y)=H_j^εU(·, y)(x). Hence, there exists a functionQj(x, y) such that limε→0Q^εj(x, y)=Qj(x, y) a.e. and

(2.10) sup

y

R^d

sup

0<ε<1|Q^εj(x, y)|^rdx≤C_r for 1< r < dp dp+d−2p.

FixN >d. Since|H_j^ε(x, z)|≤CN|x−z|^−N for|x−z|>1, (2.11) |Q^εj(x, y)|=

|z|≤1

H_j^ε(x, z)U(z, y)dz

≤C_N|x|^−N for|x|>2 andy∈R^d. The H¨older inequality combined with (2.10) and (2.11) implies that

(2.12) sup

y

R^d

sup

0<ε<1|Q^εj(x, y)|dx≤C and

(2.13) lim

ε→0

R^d|Q^εj(x, y)−Qj(x, y)|dx= 0 for every y.

Now the lemma can be easily concluded from (2.12), (2.13), and Lebesgue’s domi- nated convergence theorem.

3. Proof of the main theorem

Recall thatI−V L⁻¹is an isomorphism inL¹(R^d). Considerf∈L¹(R^d). Using (2.1) and Lemmas2.1–2.4we get that Rjf belongs to L¹(R^d) if and only if

Rj(I−V L⁻¹)f∈L¹(R^d) and

fL¹(R^d)+ d j=1

RjfL¹(R^d)∼ (I−V L⁻¹)fL¹(R^d)+ d j=1

Rj(I−V L⁻¹)fL¹(R^d).

Applying the characterization of the classical Hardy spaceH¹(R^d) by means of the Riesz transformsRj (see (1.2)) and (1.7) we obtain Theorem 1.1.

Acknowledgement. The authors would like to thank the referee for valuable comments.

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Jacek Dziuba´nski Instytut Matematyczny Uniwersytet Wroclawski Pl. Grunwaldzki 2/4 PL-50-384 Wroclaw Poland

[email protected]

Marcin Preisner Instytut Matematyczny Uniwersytet Wroclawski Pl. Grunwaldzki 2/4 PL-50-384 Wroclaw Poland

[email protected]

Received February 19, 2009 in revised form January 20, 2010 published online March 26, 2010