A Characterization of Hardy Spaces Associated with Certain Schrödinger Operators

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1 Potential Anal 214) 41: DOI 1.17/s A Chaacteization of Hady Spaces Associated with Cetain Schödinge Opeatos Jacek Dziubański Jacek Zienkiewicz Received: 6 June 213 / Accepted: 4 Mach 214 / Published online: 11 Apil 214 Spinge Science+Business Media Dodecht 214 Abstact et {K t } t> be the semigoup of linea opeatos geneated by a Schödinge opeato = Δ Vx) on R d, d 3, whee Vx) satisfies Δ 1 V. We say that an 1 -function f belongs to the Hady space H 1 if the maximal function M fx)= sup t> K t fx) belongs to 1 R d ). We pove that the opeato Δ) 1/2 1/2 is an isomophism of the space H 1 with the classical Hady space H 1 R d ) whose invese is 1/2 Δ) 1/2. As a coollay we obtain that the space H 1 is chaacteized by the Riesz tansfoms R j = x j 1/2. Keywods Hady spaces Schödinge opeatos Mathematics Subject Classifications 21) 42B3 35J1 42B35 1 Intoduction and Statement of the Result et = Δ + Vx) beaschödinge opeato on R d,wheevx) is a nonnegative locally integable potential and let {T t } t> be the semigoup geneated by. The action of the semigoup is given by the Feynman-Kac fomula ) T t fx)= E x e t VX s )ds fx t ), 1.1) The eseach was suppoted by Polish funds fo sciences, gants: DEC-212/5/B/ST1/672 and DEC-212/5/B/ST1/692 fom Naodowe Centum Nauki. J. Dziubański ) J. Zienkiewicz Instytut Matematyczny, Uniwesytet Wocławski, Pl. Gunwaldzki 2/4, Wocław, Poland jdziuban@math.uni.woc.pl J. Zienkiewicz zenek@math.uni.woc.pl

2 918 J. Dziubański, J. Zienkiewicz whee X t is a Bownian motion associated with the heat semigoup P t = e tδ see, e.g., Chapte V of [18]). et T t x, y) denote the integal kenel of the semigoup {T t } t>.since Vx) is non-negative, Eq. 1.1 implies that T t x, y) 4πt) d/2 e x y 2 /4t =: P t x y). 1.2) It easily follows fom Eq. 1.2 that the maximal function M fx)= sup T t fx) 1.3) t> is a bounded opeato on p R d ) fo 1 <p and of weak-type 1.1). The eal Hady space H 1 associated with is defined as H 1 ={f 1 R d ) : M f 1 R d )} 1.4) with the nom f H 1 = M f 1 R n ). 1.5) et us note that the Hady space H 1 coincides with the Hady spaces H,max,h 1 Rd ) consideed in Hofmann et al. [14, Chaptes 7 and 8], whee H,max,h 1 Rd ) is defined as the completion of the space {f 2 R d ) : M f 1 R d )} in the nom Eq We shall pesent a poof of this fact using standad aguments in the Appendix. In the pesent pape we conside the semigoup {K t } t> of linea opeatos on R d, d 3, geneated by a Schödinge opeato = Δ Vx),wheeVx)is a non-negative locally integable function which satisfies Δ 1 1 Vx)= c d R d x y d 2 Vy)dy R d ). 1.6) et K t x, y) denote the integal kenel of the semigoup {K t } t>. Clealy, the uppe Gaussian bounds 1.2) hold fo K t x, y). It is known, see [17], that fo Vx) the condition 1.6) is equivalent to the lowe Gaussian bounds fo K t x, y), thatis,theeaec, C > such that ct d/2 e C x y 2 /t K t x, y). 1.7) The Hady spaces H 1 associated with Schödinge opeatos with nonnegative potentials satisfying Eq. 1.6 wee studied in [12]. It was poved that the map fx) wx)fx) is an isomophism of H 1 onto the classical Hady space H 1 R d ),whee wx) = lim K t x, y) dy, 1.8) t which in paticula means that fw H 1 R d ) f H 1, 1.9) see [12, Theoem 1.1]. The function wx) is -hamonic, that is, K t w = w, and satisfies <δ wx) 1. et us emak that the classical eal Hady space H 1 R d ) can be thought as the space H 1 associated with the classical heat semigoup e tδ,thatis, = Δ + V with V inthis case. Obviously, the constant functions ae the only bounded hamonic functions fo Δ. The pesent pape is a continuation of [12]. Ou goal is to study the mappings 1/2 Δ) 1/2 and Δ) 1/2 1/2

3 Hady Spaces fo Schödinge Opeatos 919 which tun out to be bounded on 1 R d ) see emma 2.6). Ou main esult is the following theoem, which states anothe chaacteization of H 1. Theoem 1.1 Assume that = Δ + Vx)is a Schödinge opeato on R d, d 3, with a locally integable non-negative potential Vx)satisfying Eq Then the mapping f Δ) 1/2 1/2 f is an isomophism of H 1 onto the classical Hady space H 1 R d ), that is, thee is a constant C> such that Δ) 1/2 1/2 f H 1 R d ) C f H 1, 1.11) 1/2 Δ) 1/2 f H 1 C f H 1 R d ). 1.12) As a coollay we immediately obtain the following Riesz tansfom chaacteization of H 1. Coollay 1.13 Unde the assumptions of Theoem 1.1 an 1 -function f belongs to the space H 1 if and only if R j f = x j 1/2 f belong to 1 R d ) fo j = 1, 2,..., d. Moeove, thee is a constant C>such that d C 1 f H 1 f 1 R d ) + R j f 1 R d ) C f H ) j=1 Example 1 It is not had to see that if fo a function Vx) defined on R d, d 3, thee is ε> such that V d/2 ε R d ) d/2+ε R d ),thenv satisfies Eq Example 2 Assume that Eq. 1.6 holds fo a function V : R d Vx 1,x 2 ) := Vx 1 ) defined on R d R n, n 1, fulfils Eq [, ), d 3. Then The eade inteested in othe esults concening Hady spaces associated with semigoups of linea opeatos, and in paticula semigoups geneated by Schödinge opeatos, is efeed to [1, 2, 6 1, 14]. 2 Boundedness on 1 We define the opeatos: Δ) 1 fx)= P t fx)dt = c d 1 fx)= K t fx)dt =: Γ x,y)fy)dy, Δ) 1/2 f = c 1 P t f dt = c d t 1/2 f = c 1 K t f dt =: t whee c 1 = Γ1/2) 1. Clealy, by Eq. 1.2, fy) dy =: x y d 2 1 fy)dy =: x y d 1 Γ x,y)fy)dy, Γ x y)fy)dy, Γ x y)fy)dy, Γ x, y) c d x y d+1, <Γx,y) c d x y d )

4 92 J. Dziubański, J. Zienkiewicz The petubation fomula assets that P t x y) = K t x, y) + = K t x, y) + t t P t s x z)v z)k s z, y) dz ds K t s x, z)v z)p s z y)dzds. 2.2) Multiplying the second inequality in Eq. 2.2 by wx) and integating with espect to dx we get t P t x y)wx)dx = wy) + wz)v z)p s z, y) ds dx, 2.3) R d since w is -hamonic. The left-hand side of Eq. 2.3 tends to a hamonic function, which is bounded fom below by δ and above by 1, as t tends to infinity. Thus thee is a constant <c w 1suchthat c w = wy) + wz)v z)γ z y)dz. 2.4) R d Similaly, integating the fist equation in Eq. 2.2 with espect to x and taking limit as t tends to infinity, we obtain 1 = wy) + V z)γ z, y) dz. 2.5) R d Fo a easonable function f the following opeatos ae well defined in the sense of distibutions: Δ) 1/2 f = c 2 P t f f) dt t 3/2,c 2 = Γ 1/2) 1, 1/2 = c 2 K t f f) dt t 3/2. emma 2.6 Thee is a constant C> such that Δ) 1/2 1/2 f 1 C f 1, 2.7) 1/2 Δ) 1/2 f 1 C f ) Poof Fom the petubation fomula 2.2)wehave Δ) 1/2 1/2 fx)= c 2 P t I) 1/2 fx) dt = c 2 P t K t ) 1/2 fx) dt t 3/2 + c 2 t = c 2 t 3/2 K t I) 1/2 fx) dt t 3/2 P t s x z)v z)k s z, y) 1/2 fy)dy dzds dt t 3/2 + fx). 2.9) Conside the integal kenel Wx,u) of the opeato t f P t s x z)v z)k s z, y) 1/2 fy)dy dzds dt t 3/2,

5 Hady Spaces fo Schödinge Opeatos 921 that is, Wx,u) = t P t s x z)v z)k s z, y) Γy,u)dy dzds dt t 3/2. Clealy Wx,u). Integation of Wx,u)with espect to dx leads to Wx,u)dx = = 2 t 2c 1 1 = 2c 1 1 Vz)K s z, y) Γy,u)dy dzds dt t 3/2 Vz)K s z, y) Γy,u)dydz ds s Vz) Γz,y) Γy,u)dydz V z)γ z, u)dz. 2.1) Using Eq. 2.1 we see that Wx,u)dx 2c 1 1 Δ 1 V, which completes the poof of Eq The poof of Eq. 2.8 goes in the same way. We skip the details. We finish this section by poving the following two lemmas, which will be used in the sequel. emma 2.11 Assume that f 1 R d ).Then Δ) 1/2 1/2 fx)dx = fx)wx)dx. 2.12) Poof Fom Eqs. 2.9 and 2.1 we conclude that Δ) 1/2 1/2 fx)dx = c 2 Wx, u)f u) dudx + fx)dx = 2c 2 c1 1 V z)γ z, u)f u) dz du + fx)dx = wu) 1)f u) du + fx)dx, whee in the last equality we have used Eq emma 2.13 Assume that f 1 R d ).Then 1/2 Δ) 1/2 f )x)wx) dx = c w fx)dx. 2.14)

6 922 J. Dziubański, J. Zienkiewicz Poof The poof is simila to that of emma Indeed, by the petubation fomula 2.2) we have 1/2 Δ) 1/2 f )x)wx) dx = c 2 K t P t ) Δ) 1/2 )f )x) dt wx) dx t3/2 +c 2 P t I) Δ) 1/2 )f )x) dt wx) dx t3/2 t = c 2 wx)k t s x, z)v z) P s z y) Δ) 2 1 f )y) dydz ds dt dx t3/2 + wx)fx)dx t = c 2 wz)v z)p s z y) Δ) 1/2 f )y) R d R d dydzds dt t 3/2 + wx)fx)dx, whee in the last equality we have used that w is -hamonic. Integating with espect to dt and then with espect to ds yields 1/2 Δ) 1/2 f )x)wx) dx = 2c 2 wz)v z) Γ z y) Δ) 1/2 f )y) dy dz + fx)wx)dx c 1 = wz)v z)γ z u)f u) du dz + fx)wx)dx = c w fx)dx wy)fy)dy + fx)wx)dx, whee in the last equality we have used Eq Atoms and Molecules Fix 1 <q. We say that a function a is an 1,q,w)-atom if thee is a ball B R d such that supp a B, a q R d ) B q 1 1, ax)wx)dx =. The atomic nom f H 1 at,q,w is defined by f H 1 = inf λ at,q,w j, 3.1) whee the infimum is taken ove all epesentations f = j=1 λ j a j,wheeλ j C, a j ae 1,q,w)-atoms. Clealy, if w x) 1, then the 1,q,w )-atoms coincide with the classical 1,q)-atoms fo the Hady space H 1 R d ), which can be thought as H 1 Δ. As a diect consequence of Theoem 1.1 of [12]seeEq.1.9) and the esults about atomic decompositions of the classical eal Hady spaces see, e.g., [3, 15, 19]), we obtain that j=1

7 Hady Spaces fo Schödinge Opeatos 923 the space H 1 admits atomic decomposition into 1,q,w)-atoms, that is, thee is a constant C q > suchthat Cq 1 f H 1 f at,q,w H 1 C q f H ) at,q,w et ε>, 1 <q<. We say that a function b is a 1,q,ε,w)-molecule associated with a ball B = Bx,)if and B ) 1 bx) q q 1 1 dx B q 1, bx) q q dx) 2 k B q εk 3.3) 2 k B\2 k 1 B bx)wx)dx =. 3.4) Obviously evey 1,q,w)-atom is a 1,q,ε,w)-molecule. It is also not had to see that fo fixed q > 1 and ε > thee is a constant C > such that evey1,q,ε,w)molecule b can be decomposed into a sum bx) = λ n a n, n=1 λ n C, whee λ n C, a n ae 1,q,w)-atoms. The following lemma is easy to pove. ) emma 3.5 et 1 <q<, δ,ε > be such that δ>d 1 q ε. Then thee is a constant C>such that if bx) satisfies Eq. 3.4 and bx) x y ) δ n=1 then b is a 1,q,ε,w)-molecule associated with By,). q ) 1/q dx d+d/q, 3.6) C In ode to pove Theoem 1.1 we shall use geneal esults about Hady spaces associated with Schödinge opeatos = Δ Vx) with non-negative locally integable potentials Vx) which wee poved in [11]. We say that a function a is a genealized 1,, )-atom fo the Hady space H 1 if thee is a ball B = By,) and a function b such that supp b B, b B 1, a = I T 2)b. Then we say that a is associated with the ball By,).ItwaspovedinSection6of[11] that the space H 1 admits atomic decomposition with the genealized 1,, )-atoms, that is, f H 1 f H 1, whee the nom f at,, H 1 is defined as in Eq. 3.1 with a j x) at,, eplaced by the geneal 1,, )-atoms a j x). emma3.7 Thee is a constantc > such that fo evey a being a genealized 1,, ) atom associated with By,)one has 1/2 ay) C 1 d y y ) d.

8 924 J. Dziubański, J. Zienkiewicz Poof The poof follows fom functional calculi see, e.g., [13]). Note that 1/2 a = m ) )b with m ) λ) = 2 λ) 1/2 e 2λ 1) and b such that supp b By,), b By,) 1.Fom[13] we conclude that thee is a constant C>such that fo evey >one has m ) )f x) = m ) x, y)f y) dy, R d with m ) x, y) satisfying m ) x, y) C 1 d ) x y d. 3.8) Now the lemma can be easily deduced fom Eq. 3.8 and the size and suppot popety of b. 4 Poof of Theoem 1.1 Fo eal numbes n>2, β>let ) x gx) = x ) n β, g s x) = s n/2 g. s One can easily check that t g s x) ds C x 2 n x ) 2 β ; 4.1) t g s x) ds C 2 n x ) n+2 fo >. 4.2) 2 ) Moeove, it is easily to veify that fo 1 <q<, d 1 q 1 <α d, β > one has x α d x ) d β = C α,β t α d+d/q)/2 4.3) t q R d,dx) and z y 2 d fo <γ <β<2. ) z y β y ) d+γ dy C 2 z ) d+γ +2 β 4.4) emma 4.5 Assume that Vx) satisfies the assumptions of Theoem 1.1. Then fo < γ 2 and >one has ) z y d+γ Vz) dz c R d d 1 d 2 Δ 1 V. 4.6) Poof The left-hand side of Eq. 4.6 is bounded by ) d 2 ) z y d+2 Vz) dz + Vz) dz z y z y z y > c 1 d d 2 Δ 1 V.

9 Hady Spaces fo Schödinge Opeatos 925 Poof of Theoem 1.1 We aleady have known that the opeatos Δ) 1/2 1/2 and 1/2 Δ) 1/2 ae bounded on 1 R d ). It suffices to pove Eqs and 1.12.Setγ = ) 1 1 and fix q>1andε>suchthatγ > d 1 1 q + ε. Setw x) 1.Accodingtothe atomic and molecula decompositions see Section 3) the poof of Eq will be done if we veify that Δ) 1/2 1/2 a is a multiple of a 1,q,ε,w )-molecule fo evey genealized 1,, )-atom a with a multiple constant independent of a. Identical aguments can be then applied to show that 1/2 Δ) 1/2 a is a multiple of a 1,q,ε,w)-molecule fo a being a genealized atom fo the classical Hady space H 1 R d ) = H Δ 1 with a multiple constant independent of a. et a = I K 2)b be a genealized 1,,)-atom fo H 1 associated with By,). By emma 2.11, since wx)ax) dx =, we have Δ) 1/2 1/2 ax) dx =. Set Jx) = = t 2 t P t s x z)v z)k s z, y) 1/2 a)y) dy dz ds dt... + t/2 2 + t 2 t/2 = J 1 x) + J 2 x) + J 3 x). 4.7)... t 3/2 Thanks to Eq. 2.9 and emma 3.5 it suffices to show that thee is a constant C q >, independent of ax) such that x y ) γ Jx) C q d+d/q. 4.8) q R d ) Applying emma 3.7 and Eq. 4.1 with n = d + 1, we obtain 2 t J 1 x) = P t s x z)v z)k s z, y) 1/2 a)y) dy dz ds dt t 3/2 2 t C P t s x z)v z) 1 d z y ) d dzds dt C 2 C N Consequently, J 1 x) x y C N 1 d P s x z)v z) 1 d z y x z 1 d ) γ x z d+1 x z ) d dz ds s ) N Vz) 1 d z y t 3/2 ) d dz. 4.9) x z ) N+γ Vz) z y ) d+γ dz. 4.1)

10 926 J. Dziubański, J. Zienkiewicz Theefoe, using the Minkowski integal inequality togethe with Eqs. 4.3 and 4.6,weget J 1x) x y ) γ C d+d/q. 4.11) q dx) In ode to estimate J 2 x) we use emma 3.7 and Eq. 4.1 with n = d to obtain J 2 x) x y ) γ C 2 x y ) γ t/2 K s z, y) 1 d y y C 2 z y 2 d t d/2 e c x z 2 /t Vz) ) d dy dz ds dt t 3/2 t 2γ d 3)/2 e c x z 2 /t Vz) ) z y N+γ 1 d 2γ t y y ) d+γ dy dzdt. 4.12) Setting N = β + γ with <γ <β<2 and applying the Minkowski integal inequality togethe with Eqs. 4.4 and 4.6 we conclude that J 2x) x y ) γ q dx) C t d+3 2γ d/q)/2 Vz) 2 ) z y 2 d z y β 1 d 2γ y y ) d+γ dy dzdt t C t d+3 2γ d/q)/2 Vz) 2 z y 2 d C C d+d/q. 2d+2+d/q Vz) z y ) β ) β t 1 d 2γ y y ) d+2+γ β dz z y ) d+γ dy dzdt 4.13)

11 Hady Spaces fo Schödinge Opeatos 927 By emma 3.7 and Eq. 4.1 with n = d,wehave t J 3 x) C Hence, J 3 x) 2 t 2 y y C N 2 t d 2 e c z y 2 /t x y ) γ C 2 P t s x z)v z)t d 2 e c z y 2 t ) d 1 d dy dzds dt x z 2 d t d 2 e c z y 2 /t t 3 2 x z t ) N Vz) y y ) d 1 d dy dz dt. 4.14) t3/2 x z 2 d ) x z N+γ t γ Vz) t 1+ y y ) d+γ 1 d 2γ dy dz dt t 3/2. By Minkowski s integal inequality combined with Eq. 4.3 we aive to J 3x) x y ) γ q dx) t d+2+d/q)/2+γ 3/2 Vz) 2 t d/2 e c z y 2 /t y y ) d+γ 1 d 2γ dy dzdt. Application of Eq. 4.2 with n = 2d + 1 d q 2γ andtheneq.4.6 yields J 3x) x y ) γ q dx) ) C 2 3d+d/q z y 2d+1+d/q+2γ Vz) y y ) d+γ dy dz 2 2d+d/q Vz) z y ) 2d+1+d/q+3γ dz C d+d/q. The above inequality togethe with Eqs and 4.13 gives desied Eq. 4.8 and, consequently, the poof of Eq is complete. et us note that in the poof Eq we use only emmas 2.11, 3.7, and the uppe Gaussian bounds fo the kenels. The poof of Eq goes identically to that of Eq by eplacing emma 2.11 by emma Poof of the Riesz Tansfom Chaacteization of H 1 Poof Poof of Coollay 1.13 Assume that f H 1. Then, thanks to Theoem 1.1, thee is g H 1 R d ) such that f = 1/2 Δ) 1/2 g. By the chaacteization of the classical Hady

12 928 J. Dziubański, J. Zienkiewicz space H 1 R d ) by the Riesz tansfoms we have x j 1/2 f = x j 1/2 1/2 Δ) 1/2 g = x j Δ) 1/2 g 1 R d ). 5.1) Convesely, assume that fo f 1 R d ) we have x j 1/2 f 1 R d ) fo j = 1, 2,..., d. Set g = Δ) 1/2 1/2 f. Then by emma 2.6, g 1 R d ) and Δ) 1/2 g = Δ) 1/2 Δ) 1/2 1/2 f = 1/2 f 1 R d ), 5.2) x j x j x j which implies that g H 1 R d ). Consequently, by Theoem 1.1, f H 1. Finally Eq can be deduced fom Eqs. 5.1, 5.2, and Theoem 1.1. The authos want to thank the efeee fo he/his comments which impoved pesenta- Acknowledgments tion of the pape. Appendix In the appendix we shall pove that the spaces H 1 and H,max,h 1 Rd ) defined in the intoduction coincide. The poof goes by standad aguments and we do not use atomic decompositions. et us note that the poof woks in moe geneal settings, e.g. fo semigoups satisfying Gaussian bounds on spaces of homogeneous type in the sense of Coifman-Weiss [4]. Since {T t } t> is a stongly continuous semigoup in e.g. 2 R d ), one can easily deduce fom Eq. 1.2 and the semigoup popety that {T t } t> is a pointwise appoximate of the identity, that is, lim T tfx)= fx) a.e. fo f 1 R d ) + R d ). 6.1) t Consequently, fx) M fx) a.e. fo f 1 R d ) + R d ). 6.2) et f n 2 R d ) be a Cauchy sequence in the nom Eq. 1.5 and let q = lim n f n H 1. By vitute of Eq. 6.2 we have f n f m 1 R d ) f n f m H 1. Hence the sequence f n conveges to a unique function f in 1 R d ). We shall pove that M f 1 R d ) and the convegence of f n to f is also in the H 1 -nom. To see this take a subsequence n k such that f nk f nj H 1 3 k fo j k 1 6.3) and wite f = f nk + f nj+1 f nj )convegence in the 1 nom). 6.4) j=k Obviously, M f 1 R d ) f nk H j=k f nj+1 f nj H 1, which gives M f 1 R d ) q. Futhe, by Eqs. 6.4 and 6.3,wehave M f f nk ) 1 R d ) f nj+1 f nj H 1 ask. j=k

13 Hady Spaces fo Schödinge Opeatos 929 Thus we have poved that H,max,h 1 Rd ) H 1. Assume now that f H 1,thatis,f 1 R d ) and M f 1 R d ).ByEq.1.2 fo t>wehavet t f 2 R d ),andsot t f H,max,h 1 Rd ). We shall pove that lim M T t f f) t + 1 R d ) =. 6.5) It is well-known that thee ae constants C,c > such that t T t x, y) Ct 1 t d/2 exp c x y 2 /t) 6.6) see e.g., [5], [16, Theoem 6.17], [7] and efeences theein). We claim that thee exists a constant C 1 > such that fo evey t>anda>1one has sup T t+s fx) T s fx) C 1 A 1 f 1 R d ). 6.7) s>at 1 R d ) To pove the claim, we note that fo s>at, thanks to Eq. 6.6,wehave t T t+s x, y) T s x, y) = u T s+u x, y) du Hence, C C t t sup T t+s x, y) T s x, y) s>at t + s) 1 d/2 exp c x y 2 /t + s))du s 1 d/2 exp c x y 2 /s) du Cts 1 d/2 exp c x y 2 /s). { CtAt) 1 d/2 if x y At, Ct x y 2 d if x y > At, and, consequently, ) sup T t+s x, y) T s x, y) dx C 1 A 1, R n s>at which implies Eq We ae now in a position to complete the poof of Eq. 6.5.UsingEq.6.7 we obtain T t f f H 1 sup T t+s f T s f + s>at 1 R d sup T t+s f T s f ) s At 1 R d ) C 1 A 1 f C sup T t+s f f +C sup T s f f s At C 1 A 1 f 1 R d ) + 2C sup s A+1)t 1 R d ) T s f f s At 1 R d ) 1 R d ). 6.8) Fix ε>andthen take A = ε 1. Clealy, sup s A+1)t T s fx) fx) 2M fx) 1 R d ). 6.9) Since lim t sup s<a+1)t T s fx) fx) =a.e., we get Eq. 6.5 fom Eqs. 6.8 and 6.9 by applying the ebesgue dominated convegence theoem.

14 93 J. Dziubański, J. Zienkiewicz et us finally emak that thanks to the subodination fomula and esults of [14, Chapte 8] we can adapt the above aguments to pove the equivalence of the definitions of Hady spaces given by means of the Poisson semigoup e t. Refeences 1. Ausche, P., Duong, X.T., McIntosh, A.: Boundedness of Banach space valued singula integal opeatos and Hady spaces, Unpublished pepint 25) 2. Benicot, F., Zhao, J.: New abstact Hady spaces. J. Funct. Anal. 255, ) 3. Coifman, R.: A eal vaiable chaacteization of H p. Stud. Math. 51, ) 4. Coifman, R., Weiss, R.G.: Extensions of Hady spaces and thei use in analysis. Bull. Am. Math. Soc. 83, ) 5. Coulhon, T., Duong, X.T.: Maximal egulaity and kenel bounds: obsevations on a theoem by Hiebe and Püss. Adv. Diffe. Equ. 5, ) 6. Czaja, W., Zienkiewicz, J.: Atomic chaacteization of the Hady space H 1 R) of one-dimensional Schödinge opeatos with nonnegative potentials. Poc. Am. Math. Soc. 1361), ) 7. Duong, X.T., Yan,.X.: Duality of Hady and BMO spaces associated with opeatos with heat kenel bounds. J. Am. Math. Soc. 18, ) 8. Dziubański, J., Gaigós, G., Matínez, T., Toea, J.., Zienkiewicz, J.: BMO spaces elated to Schödinge opeatos with potentials satisfying a evese Hölde inequality. Math. Z. 249, ) 9. Dziubański, J., Zienkiewicz, J.: Hady space H 1 associated to Schödinge opeato satisfying evese Hölde inequality. Rev. Mat. Ibeoam. 15, ) 1. Dziubański, J., Zienkiewicz, J.: Hady spaces H 1 fo Schödinge opeatos with cetain potentials. Stud. Math. 164, ) 11. Dziubański, J., Zienkiewicz, J.: On Hady spaces associated with cetain Schödinge opeatos in dimension 2. Rev. Mat. Ibeoam. 284), ) 12. Dziubański,J.,Zienkiewicz,J.:Onisomophismsof hadyspacesassociatedwithschödinge opeatos. J. Fouie Anal. Appl. 19, ) 13. Hebisch, W.: A multiplie theoem fo Schödinge opeatos. Colloq. Math. 6/61, ) 14. Hofmann, S., u, G.Z., Mitea, D., Mitea, M., Yan,.X.: Hady spaces associated with non-negative self-adjoint opeatos satisfying Davies-Gafney estimates. Mem. Am. Math. Soc ) 211) 15. atte, R.H.: A decomposition of H p R n ) in tems of atoms. Stud. Math. 621), ) 16. Ouhabaz, E.M.: Analysis of Heat Equations on Domains, ondon Math. Soc. Mono-gaphs, vol. 31, Pinceton Univ Pess 25) 17. Semenov, Yu.A.: Stability of p -spectum of genealized Schödinge opeatos and equivalence of Geen s functions. Int. Math. Res. Not. 12, ) 18. Simon, B.: Functional Integation and Quantum Physics, 2nd edn. AMS Chelsea Publishing, Povidence 25) 19. Stein, E.: Hamonic Analysis: Real-Vaiable Methods, Othogonality, and Oscillatoy Integals. Pinceton Univesity Pess, Pinceton 1993)

Boundedness for Marcinkiewicz integrals associated with Schrödinger operators

Boundedness for Marcinkiewicz integrals associated with Schrödinger operators Poc. Indian Acad. Sci. (Math. Sci. Vol. 24, No. 2, May 24, pp. 93 23. c Indian Academy of Sciences oundedness fo Macinkiewicz integals associated with Schödinge opeatos WENHUA GAO and LIN TANG 2 School