A characterization of Hardy spaces associated with certain Schr

arXiv:1310.2262v1 [math.FA] 8 Oct 2013

A CHARACTERIZATION OF HARDY SPACES ASSOCIATED WITH ¨ CERTAIN SCHRODINGER OPERATORS ´ JACEK DZIUBANSKI AND JACEK ZIENKIEWICZ

Abstract. Let {Kt }t>0 be the semigroup of linear operators generated by a Schr¨odinger operator −L = ∆ − V (x) on Rd , d ≥ 3, where V (x) ≥ 0 satisfies ∆−1 V ∈ L∞ . We say that an L1 -function f belongs to the Hardy space HL1 if the maximal function ML f (x) = supt>0 |Kt f (x)| belongs to L1 (Rd ). We prove that the operator (−∆)1/2 L−1/2 is an isomorphism of the space HL1 with the classical Hardy space H 1 (Rd ) whose inverse is L1/2 (−∆)−1/2 . As a corollary we obtain that the space ∂ L−1/2 . HL1 is characterized by the Riesz transforms Rj = ∂x j

1. Introduction and statement of the result Let Kt (x, y) be the integral kernels of the semigroup {Kt }t>0 of linear operators on Rd , d ≥ 3, generated by a Schr¨odinger operator −L = ∆ − V (x), where V (x) is a non-negative locally integrable function which satisfies Z 1 −1 (1.1) ∆ V (x) = −cd V (y) dy ∈ L∞ (Rd ). d−2 |x − y| d R Since V (x) is non-negative, the Fenman-Kac formula implies that (1.2)

0 ≤ Kt (x, y) ≤ (4πt)−d/2 e−|x−y|

2 /4t

=: Pt (x − y).

It is known, see [14], that for V (x) ≥ 0 the condition (1.1) is equivalent to the lower Gaussian bounds for Kt (x, y), that is, there are c, C > 0 such that (1.3)

ct−d/2 e−C|x−y|

2 /t

≤ Kt (x, y).

We say that an L1 -function f belongs to the Hardy space HL1 if the maximal function ML f (x) = supt>0 |Kt f (x)| belongs to L1 (Rd ). Then we set kf kHL1 = kML f kL1 (Rd ) .

The Hardy spaces HL1 associated with Schr¨odinger operators with nonnegative potentials satisfying (1.1) were studied in [10]. It was proved that the map f (x) 7→ w(x)f (x) is an isomorphism of HL1 onto the classical Hardy space H 1 (Rd ), where Z (1.4) w(x) = lim Kt (x, y) dy, t→∞

2000 Mathematics Subject Classification. 42B30, 35J10 (primary); 42B35 (secondary). Key words and phrases. Hardy spaces, Schr¨odinger operators. The research was supported by Polish funds for sciences, grants: DEC-2012/05/B/ST1/00672 and DEC-2012/05/B/ST1/00692 from Narodowe Centrum Nauki. 1

´ JACEK DZIUBANSKI AND JACEK ZIENKIEWICZ

2

which in particular means that kf wkH 1(Rd ) ∼ kf kHL1 ,

(1.5)

see [10, Theorem 1.1]. The function w(x) is L-harmonic, that is, Kt w = w, and satisfies 0 < δ ≤ w(x) ≤ 1. Let us remark that the classical real Hardy space H 1 (Rd ) can be thought as the space 1 HL associated with the classical heat semigroup et∆ , that is, L = −∆ + V with V ≡ 0 in this case. Obviously, the constant functions are the only bounded harmonic functions for ∆. The present paper is a continuation of [10]. Our goal is to study the mappings L1/2 (−∆)−1/2

and (−∆)1/2 L−1/2

which turn out to be bounded on L1 (Rd ) (see Lemma 2.6). Our main result is the following theorem, which states another characterization of HL1 . Theorem 1.6. Assume that L = −∆ + V (x) is a Schr¨odinger operator on Rd , d ≥ 3, with a locally integrable non-negative potential V (x) satisfying (1.1). Then the mapping f 7→ (−∆)1/2 L−1/2 f is an isomorphism of HL1 onto the classical Hardy space H 1 (Rd ), that is, there is a constant C > 0 such that k(−∆)1/2 L−1/2 f kH 1 (Rd ) ≤ Ckf kHL1 ,

(1.7)

kL1/2 (−∆)−1/2 f kHL1 ≤ Ckf kH 1 (Rd ) .

(1.8)

As a corollary we immediately obtain the following Riesz transform characterization of HL1 . Corollary 1.9. Under the assumptions of Theorem 1.6 an L1 -function f belongs to the space HL1 if and only if Rj f = ∂x∂ j L−1/2 f belong to L1 (Rd ) for j = 1, 2, ..., d. Moreover, there is a constant C > 0 such that (1.10)

C

−1

kf kHL1 ≤ kf kL1 (Rd ) +

d X j=1

kRj f kL1 (Rd ) ≤ Ckf kHL1 .

Example 1. It is not hard to see that if for a function V (x) ≥ 0 defined on Rd , d ≥ 3, there is ε > 0 such that V ∈ Ld/2−ε (Rd ) ∩ Ld/2+ε (Rd ), then V satisfies (1.1). Example 2. Assume that (1.1) holds for a function V : Rd → [0, ∞), d ≥ 3. Then V (x1 , x2 ) := V (x1 ) defined on Rd × Rn , n ≥ 1, fulfils (1.1). The reader interested in other results concerning Hardy spaces associated with semigroups of linear operators, and in particular semigroups generated by Schr¨odinger operators, is referred to [1], [2], [4], [5], [6], [7], [8], [12].

¨ HARDY SPACES FOR SCHRODINGER OPERATORS

3

2. Boundedness on L1 We define the operators: Z ∞ Z Z f (y) −1 (−∆) f (x) = Pt f (x) dt = cd dy =: Γ0 (x − y)f (y) dy, |x − y|d−2 0 Z ∞ Z −1 L f (x) = Kt f (x) dt =: Γ(x, y)f (y) dy, 0 Z Z ∞ Z dt 1 −1/2 ′ e0 (x − y)f (y) dy, f (y) dy =: Γ (−∆) f = c1 P t f √ = cd d−1 |x − y| t Z ∞ 0 Z dt −1/2 e y)f (y) dy, L f = c1 Kt f √ =: Γ(x, t 0

where c1 = Γ(1/2)−1. Clearly, (2.1)

˜ y) ≤ c′ |x − y|−d+1 , 0 ≤ Γ(x, d

0 < Γ(x, y) ≤ cd |x − y|−d+2 .

The perturbation formula asserts that Z tZ Pt (x − y) = Kt (x, y) + Pt−s (x − z)V (z)Ks (z, y) dz ds 0 (2.2) Z tZ = Kt (x, y) + Kt−s (x, z)V (z)Ps (z − y) dz ds. 0

Multiplying the second inequality in (2.2) by w(x) and integrating with respect to dx we get Z Z Z t (2.3) Pt (x − y)w(x) dx = w(y) + w(z)V (z)Ps (z, y) ds dx, Rd

0

since w is L-harmonic. The left-hand side of (2.3) tends to a harmonic function, which is bounded from below by δ and above by 1, as t tends to infinity. Thus there is a constant 0 < cw ≤ 1 such that Z (2.4) cw = w(y) + w(z)V (z)Γ0 (z − y) dz. Rd

Similarly, integrating the first equation in (2.2) with respect to x and taking limit as t tends to infinity, we get Z (2.5) 1 = w(y) + V (z)Γ(z, y) dz. Rd

For a reasonable function f the following operators are well defined in the sense of distributions: Z ∞ dt 1/2 (−∆) f = c2 (Pt f − f ) 3/2 , c2 = Γ(−1/2)−1 , t Z ∞ 0 dt L1/2 = c2 (Kt f − f ) 3/2 . t 0

´ JACEK DZIUBANSKI AND JACEK ZIENKIEWICZ

4

Lemma 2.6. There is a constant C > 0 such that (2.7)

k(−∆)1/2 L−1/2 f kL1 ≤ Ckf kL1 ,

(2.8)

kL1/2 (−∆)−1/2 f kL1 ≤ Ckf kL1 .

Proof. From the perturbation formula (2.2) we get Z ∞ dt 1/2 −1/2 (−∆) L f (x) = c2 (Pt − I)L−1/2 f (x) 3/2 t 0 Z ∞ Z ∞ dt dt = c2 (Pt − Kt )L−1/2 f (x) 3/2 + c2 (Kt − I)L−1/2 f (x) 3/2 (2.9) t t 0 0 Z ∞ Z t ZZ dt = c2 Pt−s (x − z)V (z)Ks (z, y)L−1/2 f (y) dy dz ds 3/2 + f (x). t 0 0 Consider the integral kernel W (x, u) of the operator Z ∞ Z t ZZ dt f 7→ Pt−s (x − z)V (z)Ks (z, y)L−1/2 f (y) dy dz ds 3/2 , t 0 0 that is, W (x, u) =

Z

∞ 0

Z t ZZ 0

˜ u) dy dz ds dt . Pt−s (x − z)V (z)Ks (z, y)Γ(y, t3/2

Clearly 0 ≤ W (x, u). Integration of W (x, u) with respect to dx leads to Z Z ∞ Z t ZZ ˜ u) dy dz ds dt W (x, u) dx = V (z)Ks (z, y)Γ(y, t3/2 0 Z ∞ 0Z Z ds e u) dy dz √ =2 V (z)Ks (z, y)Γ(y, s 0 (2.10) ZZ e y)Γ(y, e u) dy dz V (z)Γ(z, ≤ 2c−1 1 Z −1 = 2c1 V (z)Γ(z, u)dz. R −1 Using (2.1) we see that W (x, u) dx ≤ 2c−1 1 k∆ V kL∞ , which completes the proof of (2.7). The proof of (2.8) goes in the same way. We skip the details.  We finish this section by proving the following two lemmas, which will be used in the sequel. Lemma 2.11. Assume that f ∈ L1 (Rd ). Then Z Z 1/2 −1/2 (2.12) (−∆) L f (x) dx = f (x)w(x) dx.

¨ HARDY SPACES FOR SCHRODINGER OPERATORS

5

Proof. From (2.9) and (2.10) we conclude that Z

(−∆)

1/2

−1/2

L

f (x) dx = c2

Z Z

Z

W (x, u)f (u) dudx + f (x) dx Z Z −1 V (z)Γ(z, u)f (u) dz du + f (x) dx = 2c2 c1 Z Z = (w(u) − 1)f (u) du + f (x) dx,

where in the last equality we have used (2.5).



Lemma 2.13. Assume that f ∈ L1 (Rd ). Then Z

(2.14)

1/2

(L

(−∆)

−1/2

f )(x)w(x) dx = cw

Z

f (x) dx.

Proof. The proof is similar to that of Lemma 2.11. Indeed, by the perturbation formula (2.2) we have Z

(L1/2 (−∆)−1/2 f )(x)w(x) dx Z Z ∞ dt = c2 (Kt − Pt )((−∆)−1/2 )f )(x) 3/2 w(x) dx t 0 Z Z ∞ dt + c2 (Pt − I)((−∆)−1/2 )f )(x) 3/2 w(x) dx t 0 Z Z ∞ Z t ZZ = −c2 w(x)Kt−s (x, z)V (z) 0

0

1

+

Z

× Ps (z − y)((−∆)− 2 f )(y) dydz ds w(x)f (x) dx Z ∞Z tZ Z

dt dx t3/2

dt = −c2 w(z)V (z)Ps (z − y)((−∆)−1/2 f )(y) dydz ds 3/2 t 0 Rd Rd Z 0 + w(x)f (x) dx,

´ JACEK DZIUBANSKI AND JACEK ZIENKIEWICZ

6

where in the last equality we have used that w is L-harmonic. Integrating with respect to dt and then with respect to ds yields Z (L1/2 (−∆)−1/2 f )(x)w(x) dx Z Z Z 2c2 −1/2 e w(z)V (z)Γ0 (z − y)((−∆) f )(y) dy dz + f (x)w(x) dx =− c1 Z Z = w(z)V (z)Γ0 (z − u)f (u) du dz + f (x)w(x) dx Z Z Z = cw f (x) dx − w(y)f (y) dy + f (x)w(x) dx, where in the last equality we have used (2.4).



3. Atoms and molecules Fix 1 < q ≤ ∞. We say that a function a is an (1, q, w)-atom if there is a ball R 1 B ⊂ Rd such that supp a ⊂ B, kakLq (Rd ) ≤ |B| q −1 , a(x)w(x) dx = 0. The atomic norm kf kH 1 at,q,w is defined by 1 = inf kf kHat,q,w

(3.1)

∞ nX j=1

o |λj | ,

P where the infimum is taken over all representations f = ∞ j=1 λj aj , where λj ∈ C, aj are (1, q, w)-atoms. Clearly, if w0 (x) ≡ 1, then the (1, q, w0)-atoms coincide with the classical (1, q)-atoms 1 for the Hardy space H 1 (Rd ), which can be thought as H−∆ . As a direct consequence of Theorem 1.1 of [10] (see (1.5)) and the results about atomic decompositions of the classical real Hardy spaces (see, e.g., [3], [13], [15]), we obtain that the space HL1 admits atomic decomposition into (1, q, w)-atoms, that is, there is a constant Cq > 0 such that 1 1 Cq−1 kf kHat,q,w ≤ kf kHL1 ≤ Cq kf kHat,q,w .

(3.2)

Let ε > 0, 1 < q < ∞. We say that a function b is a (1, q, ε, w)-molecule associated with a ball B = B(x0 , r) if Z  q1 Z  q1 1 1 −1 q q q |b(x)| dx ≤ |2k B| q −1 2−εk (3.3) |b(x)| dx ≤ |B| , 2k B\2k−1 B

B

and (3.4)

Z

b(x)w(x) dx = 0.

Obviously every (1, q, w)-atom is a (1, q, ε, w)-molecule. It is also not hard to see that for fixed q > 1 and ε > 0 there is a constant C > 0 such that every (1, q, ε, w) molecule

¨ HARDY SPACES FOR SCHRODINGER OPERATORS

7

b can be decomposed into a sum b(x) =

∞ X

λn an ,

∞ X n=1

n=1

|λn | ≤ C,

where λn ∈ C, an are (1, q, w)-atoms. The following lemma is easy to prove. Lemma 3.5. Let 1 < q < ∞, δ, ε > 0 be such that δ > d(1 − 1q ) + ε. Then there is a constant C > 0 such that if b(x) satisfies (3.4) and Z  |x − y0 | δ q 1/q r −d+d/q (3.6) b(x) 1 + dx , ≤ r C then b is a (1, q, ε, w)-molecule associated with B(y0 , r). In order to prove Theorem 1.6 we shall use general results about Hardy spaces associated with Schr¨odinger operators with non-negative potentials which were proved in [9]. Let {Tt }t>0 be a semigroup of linear operators generated by a Schr¨odinger operator −L = ∆ − V(x) on Rd , where V(x) is a non-negative locally integrable potential. The Hardy space HL1 is define by means of the maximal function, that is, HL1 = {f ∈ L1 (Rd ) : kf kHL1 := k sup |Tt f (x)|kL1 (Rd ) < ∞}. t>0

We say that a function a is a generalized (1, ∞, L)-atom for the Hardy space HL1 if there is a ball B = B(y0 , r) and a function b such that supp b ⊂ B,

kbkL∞ ≤ |B|−1 ,

a = (I − Tr2 )b.

Then we say that a is associated with the ball B(y0 , r). It was proved in Section 6 of [9] that the space HL1 admits atomic decomposition with the generalized (1, ∞, L)-atoms, 1 1 is defined as in (3.1) with aj (x) , where the norm kf kHat,∞,L that is, kf kHL1 ∼ kf kHat,∞,L replaced by the general (1, ∞, L)-atoms aj (x). Lemma 3.7. There is a constant C > 0 such that for every a being a generalized (1, ∞, L) atom associated with B(y0 , r) one has  |y − y0 | −d −1/2 1−d 1+ |L a(y)| ≤ Cr . r Proof. The proof follows from functional calculi (see, e.g., [11]). Note that L−1/2 a = 2 m(r) (L)b with m(r) (λ) = r(r 2 λ)−1/2 (e−r λ − 1) and b such that supp b ⊂ B(y0 , r), kbkL∞ ≤ |B(y0, r)|−1 . From [11] we conclude that there is a constant C > 0 such that for every r > 0 one has Z m(r) (L)f (x) = m(r) (x, y)f (y) dy, Rd

with m(r) (x, y) satisfying (3.8)

 |x − y| −d . |m(r) (x, y)| ≤ Cr 1−d 1 + r

8

´ JACEK DZIUBANSKI AND JACEK ZIENKIEWICZ

Now the lemma can be easily deduced from (3.8) and the size and support property of b.  4. Proof of Theorem 1.6 For real numbers n > 2, β > 0 let g(x) = (1 + |x|)−n−β ,

x
gs (x) = s−n/2 g √ . s

One can easily check that Z t  |x| −2−β (4.1) gs (x) ds ≤ C|x|2−n 1 + √ ; t 0 Z ∞  |x| −n+2 (4.2) for r > 0. gs (x) ds ≤ Cr 2−n 1 + r r2 Moreover, it is easily to verify that for 1 < q < ∞, d(1 − 1q ) < α ≤ d, β > 0 one has

 |x| −d−β

α−d 1+ √ = Cα,β t(α−d+d/q)/2 (4.3)

|x|

q d L (R , dx) t and Z   |y| −d+γ |z| −d+γ+2−β |z − y| −β  2 2−d 1+ dy ≤ Cr 1 + (4.4) |z − y| 1+ r r r for 0 < γ < β < 2 . Lemma 4.5. Assume that V (x) satisfies the assumptions of Theorem 1.6. Then for 0 < γ ≤ 2 and r > 0 one has Z  |z − y| −d+γ d−2 (4.6) V (z) 1 + dz ≤ c−1 k∆−1 V kL∞ . d r r d R Proof. The left-hand side of (4.6) is bounded by Z Z  |z − y| −d+2  r d−2 dz + V (z) dz V (z) |z − y| r |z−y|>r |z−y|≤r d−2 ≤ c−1 k∆−1 V kL∞ . d r



Proof of Theorem 1.6. We already have known that the operators (−∆)1/2 L−1/2 and 1 L1/2 (−∆)−1/2 are bounded on L1 (Rd ). It suffices to prove (1.7) and (1.8). Set γ = 10 1 and fix q > 1 and ε > 0 such that γ > d(1 − q ) + ε. Set w0 (x) ≡ 1. According to the atomic and molecular decompositions (see Section 3) the proof of (1.7) will be done if we verify that (−∆)1/2 L−1/2 a is a multiple of a (1, q, ε, w0)-molecule for every generalized (1, ∞, L)-atom a with a multiple constant independent of a. Identical arguments can be then applied to show that L1/2 (−∆)−1/2 a is a (1, q, ε, w)-molecule for a being a 1 generalized atom for the classical Hardy space H 1 (Rd ) = H−∆ with a multiple constant independent of a.

¨ HARDY SPACES FOR SCHRODINGER OPERATORS

9

1 Let a = (I − Kr2 )b be R a generalized (1, ∞, L)-atom for HL associated with B(y0 , r). By Lemma 2.11, since w(x)a(x) dx = 0, we have that Z (−∆)1/2 L−1/2 a(x) dx = 0.

Set J(x) = (4.7)

=

Z

Z

∞ 0

Z t ZZ 0

r2 0

Z t ZZ

Pt−s (x − z)V (z)Ks (z, y)(L−1/2 a)(y) dy dz ds ... +

0

Z

∞

r2

Z

0

t/2

ZZ

+

Z

∞

r2

Z

t

t/2

ZZ

dt t3/2

...

= J1 (x) + J2 (x) + J3 (x). Thanks to (2.9) and Lemma 3.5 it suffices to show that there is a constant Cq > 0, independent of a(x) such that

 |x − y0 | γ

J(x) 1 +

q d ≤ Cq r −d+d/q .

r L (R )

(4.8)

Applying Lemma 3.7 and (4.1) with n = d + 1, we obtain Z |J1 (x)| =

r2

0

Z

Z t ZZ 0

Z tZ

Pt−s (x − z)V (z)Ks (z, y)(L−1/2 a)(y) dy dz ds

dt t3/2

 dt |z − y0 | −d dz ds 3/2 Pt−s (x − z)V (z)r 1−d 1 + r t 0 0 Z r2 Z   |z − y0 | −d ds ≤C Ps (x − z)V (z)r 1−d 1 + dz √ r s 0 Z    −N |x − z| |z − y0 | −d ≤ CN |x − z|1−d 1 + V (z)r 1−d 1 + dz. r r

≤C

(4.9)

r2

Consequently,

(4.10)

 |x − y0 | γ |J1 (x)| 1 + Z r   |z − y0 | −d+γ |x − z| −N +γ V (z) 1 + dz. ≤ CN r 1−d |x − z|−d+1 1 + r r

Therefore, using the Minkowski integral inequality together with (4.3) and (4.6), we get (4.11)

 |x − y0 | γ

≤ Cr −d+d/q .

J1 (x) 1 +

q L (dx) r

In order to estimate J2 (x) we use Lemma 3.7 and (4.1) with n = d to obtain

´ JACEK DZIUBANSKI AND JACEK ZIENKIEWICZ

10

(4.12)

 |x − y0 | γ |J2 (x)| 1 + r Z ZZ Z ∞ |x − y0 | γ t/2 2 t−d/2 e−c|x−z| /t V (z) ≤C 1+ r 0 r2  |y − y0 | −d dt × Ks (z, y)r 1−d 1 + dy dz ds 3/2 r t Z ZZ ∞

2

t(2γ−d−3)/2 e−c|x−z| /t V (z)

≤C

r2

× |z − y|

2−d

 |y − y0 | −d+γ |z − y| −N +γ 1−d−2γ  1+ dy dz dt. r 1+ √ r t

Setting N = β + γ with 0 < γ < β < 2 and applying the Minkowski integral inequality together with (4.4) and (4.6) we conclude that

 |x − y0 | γ

J2 (x) 1 +

q L (dx) r Z ∞ ZZ ≤C t−(d+3−2γ−d/q)/2 V (z) r2

(4.13)

 |z − y| −β 1−d−2γ  |y − y0 | −d+γ dy dz dt 1+ × |z − y|2−d 1 + √ r r t Z ∞ ZZ ≤C t−(d+3−2γ−d/q)/2 V (z) 2 r −β  √t β   |y − y0 | −d+γ |z − y| 1−d−2γ 2−d r 1+ dy dz dt × |z − y| 1+ r r r Z  |z − y0 | −d+2+γ−β ≤ C r −2d+2+d/q V (z) 1 + dz r ≤ Cr −d+d/q .

By Lemma 3.7 and (4.1) with n = d, we have |J3 (x)| Z ≤C

∞ r2

Z t ZZ t 2

d

Pt−s (x − z)V (z)t− 2 e−

c|z−y|2 t

 |y − y0 | −d 1−d dt × 1+ r dy dz ds 3 r t2 Z ∞ ZZ   −N |x − z| ≤ CN V (z) |x − z|2−d 1 + √ t r2  d |y − y0 | −d 1−d dt 2 × t− 2 e−c|z−y| /t 1 + r dy dz ds 3/2 . r t

¨ HARDY SPACES FOR SCHRODINGER OPERATORS

11

Hence,  |x − y0 | γ |J3 (x)| 1 + Z ∞ ZZ r  |x − z| −N +γ γ ≤C |x − z|2−d 1 + √ t V (z) t r2  d |y − y0 | −d+γ 1−d−2γ dt ′ 2 × t− 2 e−c |z−y| /t 1 + r dy dz ds 3/2 . r t By Minkowski’s integral inequality combined with (4.3) we arrive to

 |x − y0 | γ

q

J3 (x) 1 + r L (dx) Z ∞ ZZ ≤ t(−d+2+d/q)/2+γ−3/2 V (z) r2

′

× t−d/2 e−c |z−y|

2 /t



|y − y0 | −d+γ 1−d−2γ r dy dz dt. r − 2γ and then (4.6) yields

1+

Application of (4.2) with n = 2d + 1 − dq

 |x − y0 | γ

q

J3 (x) 1 + L (dx) r ZZ  |z − y| −2d+1+d/q+2γ  |y − y0 | −d+γ ≤C r 2−3d+d/q V (z) 1 + 1+ dy dz r r Z  |z − y0 | −2d+1+d/q+3γ 2−2d+d/q ≤ r V (z) 1 + dz r ≤ Cr −d+d/q .

The above inequality together with (4.11) and (4.13) gives desired (4.8) and, consequently, the proof of (1.7) is complete. Let us note that in the proof (1.7) we use only Lemmas 2.11, 3.7, and the upper Gaussian bounds for the kernels. The proof of (1.8) goes identically to that of (1.7) by replacing Lemma 2.11 by Lemma 2.13.  5. Proof of the Riesz transform characterization of HL1

Proof of Corollary 1.9. Assume that f ∈ HL1 . Then, thanks to Theorem 1.6, there is g ∈ H 1 (Rd ) such that f = L1/2 (−∆)−1/2 g. By the characterization of the classical Hardy space H 1 (Rd ) by the Riesz transforms we have ∂ −1/2 ∂ −1/2 1/2 ∂ (5.1) L f= L L (−∆)−1/2 g = (−∆)−1/2 g ∈ L1 (Rd ). ∂xj ∂xj ∂xj Conversely, assume that for f ∈ L1 (Rd ) we have ∂x∂ j L−1/2 f ∈ L1 (Rd ) for j = 1, 2, ..., d. Set g = (−∆)1/2 L−1/2 f . Then by Lemma 2.6, g ∈ L1 (Rd ) and ∂ ∂ −1/2 ∂ (−∆)−1/2 g = (−∆)−1/2 (−∆)1/2 L−1/2 f = L f ∈ L1 (Rd ), (5.2) ∂xj ∂xj ∂xj

12

´ JACEK DZIUBANSKI AND JACEK ZIENKIEWICZ

which implies that g ∈ H 1 (Rd ). Consequently, by Theorem 1.6, f ∈ HL1 . Finally (1.10) can be deduced from (5.1), (5.2), and Theorem 1.6. 

References [1] Auscher, P., Duong, X.T., McIntosh, A.:

Boundedness of Banach space valued singular integral

operators and Hardy spaces, Unpublished preprint (2005) [2] Bernicot, F., Zhao, J.: New abstract Hardy spaces, J. Funct. Anal., 255, 1761–1796 (2008) [3] Coifman, R.: A real variable characterization of H p , Studia Math., 51, 269–274 (1974) [4] Czaja, W., Zienkiewicz, J.: Atomic characterization of the Hardy space HL1 (R) of one-dimensional Schr¨odinger operators with nonnegative potentials, Proc. Amer. Math. Soc. 136, no. 1, 89–94 (2008) [5] Duong, X.T., Yan, L.X.: Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Amer. Math. Soc., 18, 943–973 (2005) [6] Dziuba´ nski, J., Garrig´os, G., Mart´ınez, T., Torrea, J.L., Zienkiewicz, J.: BMO spaces related to Schr¨odinger operators with potentials satisfying a reverse H¨ older inequality, Math. Z., 249, 329–356 (2005). [7] Dziuba´ nski, J., Zienkiewicz, J.: Hardy space H 1 associated to Schr¨odinger operator satisfying reverse H¨ older inequality, Rev. Mat. Iberoamericana, 15, 279–296 (1999) [8] Dziuba´ nski, J., Zienkiewicz, J.: Hardy spaces H 1 for Schr¨odinger operators with certain potentials, Studia Math., 164, 39–53 (2004) [9] Dziuba´ nski, J., Zienkiewicz, J.: On Hardy spaces associated with certain Schr¨odinger operators in dimension 2, Rev. Mat. Iberoam., 28, no. 4, 1035–1060 (2012) [10] Dziuba´ nski, J., Zienkiewicz, J.: On Isomorphisms of Hardy Spaces Associated with Schr¨odinger Operators, J. Fourier Anal. Appl., 19, 447–456 (2013) [11] Hebisch, W.:

A multiplier theorem for Schr¨odinger operators, Colloq. Math., 60/61, 659–664

(1990) [12] Hofmann, S., Lu, G.Z., Mitrea, D., Mitrea, M., Yan, L.X.:

Hardy spaces associated with non-

negative self-adjoint operators satisfying Davies-Gafney estimates, Mem. Amer. Math. Soc., 214, no. 1007 (2011) [13] Latter, R.H.:

A decomposition of H p (Rn ) in terms of atoms, Studia Math., 62 no. 1, 93–101

(1978) [14] Semenov, Yu.A.: Stability of Lp -spectrum of generalized Schr¨odinger operators and equivalence of Green’s functions, Internat. Math. Res. Notices, 12, 573–593 (1997). [15] Stein, E.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, (1993)

¨ HARDY SPACES FOR SCHRODINGER OPERATORS

13

Instytut Matematyczny, Uniwersytet Wroclawski, pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland E-mail address: [email protected], [email protected]