Riesz transform characterization of Hardy spaces associated with Schrödinger operators with compactly supported potentials

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1 Ark. Ma., 48 (21), DOI: 1.17/s c 21 by Insiu Miag-Leffler. All righs reserved Riesz ransform characerizaion of Hardy spaces associaed wih Schrödinger operaors wih compacly suppored poenials Jacek Dziubański and Marcin Preisner Absrac. Le L= Δ+V be a Schrödinger operaor on, d 3. We assume ha V is a nonnegaive, compacly suppored poenial ha belongs o L p ( ), for some p>d/2. Le K be he semigroup generaed by L. We say ha an L 1 ( )-funcion f belongs o he Hardy space HL 1 associaed wih L if sup > Kf belongs o L1 ( ). We prove ha f HL 1 if and only if R j f L 1 ( )forj=1,..., d, wherer j =(/x j )L 1/2 are he Riesz ransforms associaed wih L. 1. Inroducion Le u(x, y)=u(x+iy) be a real-valued harmonic funcion in he upper halfplane {z=x+iy:y>}. I was proved in Burkholder, Gundy, and Silversein [2] ha for <p< he funcion u is a real par of a holomorphic funcion F (z)= u(z)+iv(z) which saisfies he H p propery sup y> R F (x+iy) p dx < if and only if he maximal funcion u (x)=sup x x <y u(x +iy) belongs o L p (R). In Fefferman Sein [7] he auhors presen oher characerizaions of he H p propery by means of real analysis. These characerizaions lead o he noion of classical real Hardy spaces H p ( ). Le us recall wo equivalen characerizaions of H 1 ( ) presened in [7]. An L 1 ( )-funcion f is an elemen of he real Hardy space H 1 ( ) if and only if he maximal funcion M Δ f(x)=sup ( ) > exp Δ f(x) belongs o L 1 ( ), where exp ( Δ ) denoes he Poisson semigroup. Equivalenly, one Suppored by he Polish Minisry of Science and High Educaion gran N N , he European Commission Marie Curie Hos Fellowship for he Transfer of Knowledge Harmonic Analysis, Nonlinear Analysis and Probabiliy MTKD-CT

2 32 Jacek Dziubański and Marcin Preisner can ake he maximal funcion M Δ f(x) from he hea semigroup P in he definiion of he Hardy space H 1 ( ), ha is, M Δ f(x)=sup > P f(x), where here and subsequenly ) P f(x)=f P (x) and P (x)=(4π) d/2 exp ( x 2. 4 The second characerizaion of H 1 ( ) is given by boundedness of cerain singular inegrals. Le x j y j R j f(x)=c d lim f(y) dy ε x y >ε x y d+1 (1.1) 1/ε = lim ε ε x j P (x y)f(y) dy d, j=1, 2,..., d, be he classical Riesz ransforms on. Clearly, for f L 1 ( )he limis in (1.1) exis in he sense of disribuions and define R j f as disribuions. I wasprovedin[7] (seealso[9]) ha an L 1 ( )-funcion f is in he classical Hardy space H 1 ( ) if and only if R j f L 1 ( )forj=1,..., d. Moreover, (1.2) f L1 ( )+ d R j f L1 (R ), d j=1 defines a norm in he space H 1 ( ), which is equivalen o he norms M Δ f L 1 ( ) and M Δ f L1 (R ). In addiion, in he case of d=1, if f H 1 (R), hen u d L 1 (R), where u(x+iy)=exp ( y Δ ) f(x), y>. Conversely, every harmonic funcion u wih he propery ha u L 1 (R) is obained in his way. In his case, he conjugae funcion v is of he form v(x+iy)=exp ( y Δ ) g(x) wihg= π 1/2 Rf. In his paper we consider a Schrödinger operaor L= Δ+V on, d 3. We assume ha V is a nonnegaive funcion, supp V B(, 1)={x R: x <1}, and V L p ( ) for some p>d/2. Le {K } > be he semigroup of linear operaors generaed by L. SinceV, by he Feynman Kac formula, we have (1.3) K (x, y) P (x y), where K (x, y) is he inegral kernel of he semigroup {K } >.Le M L f(x)=sup K f(x). > We say ha an L 1 ( )-funcion f belongs he Hardy space H 1 L if f H 1 L = M Lf L1 ( ) <.

3 Riesz ransform characerizaion of Hardy spaces 33 For j=1,..., d le us define he Riesz ransforms R j associaed wih L by seing (1.4) R j f = π 1/ε L 1/2 f = lim K f d, x j ε ε x j where he limi is undersood in he sense of disribuions. The fac ha for any f L 1 ( ) he operaors 1/ε (1.5) Rjf ε = K f d ε x j are well defined and he limis lim ε Rj ε f exis in he sense of disribuions will be discussed below. The main resul of his paper is he following heorem. Theorem 1.1. Assume ha f L 1 ( ). Then f is in he Hardy space H 1 L if and only if R j f L 1 ( ) for every j=1,..., d. Moreover, here exiss C> such ha (1.6) 1 C f H 1 L f L 1 ( )+ d j=1 R j f L1 ( ) C f H 1 L. The Hardy spaces HL 1 associaed wih he Schrödinger operaors L wih compacly suppored poenials were sudied in [6]. I was proved here ha he elemens of he space HL 1 admi special aomic decomposiions. Furhermore, he space HL 1 is isomorphic o he classical Hardy space H1 ( ). To be more precise, le Γ(x, y)= K (x, y) d and Γ (x, y)= P (x, y) d, and se L 1 f(x)= Γ(x, y)f(y) dy and Δ 1 f(x)= Γ (x, y)f(y) dy. The operaors I V Δ 1 and I VL 1 are bounded and inverible on L 1 ( ), and Moreover, I =(I V Δ 1 )(I VL 1 )=(I VL 1 )(I V Δ 1 ). (I VL 1 ):H 1 L H 1 ( ) is an isomorphism (whose inverse is I V Δ 1 )and (1.7) (I VL 1 )f H1 ( ) f H 1 L

4 34 Jacek Dziubański and Marcin Preisner for f HL 1 (see [6, Corollary 3.17]). Therefore, in order o prove Theorem 1.1, i suffices o show ha R j (I VL 1 ) R j, j=1, 2,..., d, are bounded operaors on L 1 ( ). The proof of he relevan aomic decomposiions presened in [6] wasbasedon he following ideniy (1.8) P (I VL 1 )=K (P P s )VK s ds which comes from he perurbaion formula P = K + P s VK s ds. P VK s ds = K W Q, Formula (1.8) will be used also here in he analysis of he inegral (1.5) for large, while for small we shall use is slighly differen version, namely (1.9) P (I VL 1 )=K + P s VK s ds P VL 1 = K + W Q. Boundedness of Riesz ransforms for Schrödinger operaors on L p -spaces araced aenion of many auhors. We refer he reader o [1], [4], and [8] for background and references. Under very relaxed assumpions on V, he weak ype (1,1) inequaliy (1.1) f 1, + V 1/2 f 1, C ( Δ+V) 1/2 f L1 ( ), f C c ( ), is an unpublished resul of Ouhabaz (see also [8] for a proof based on finie speed propagaion of he wave equaion). In he case of Schrödinger operaors wih poenials V, V, saisfying he reverse Hölder inequaliy wih he exponen d/2 (which clearly implies ha supp V = ), Riesz ransform characerizaions of he relevan Hardy spaces H 1 Δ+V were obained in [5]. 2. Auxiliary esimaes In his secion we will use noaion f (x)= d/2 f ( x/ ). For f L 1 ( )and<ε<1 we define runcaed Riesz ransforms by seing 1/ε Rjf ε = ε K f d 1/ε and R ε x jf = j ε x j P f d.

5 Riesz ransform characerizaion of Hardy spaces 35 Le G(x, y)= K (x, y) d and G (x, y)= P (x y) d. Then G(x, y) G (x, y)=c x y d+1 and, consequenly, for ϕ from he Schwarz class S( )wehave lim ε Rε jf,ϕ = G(x, y)f(y) ϕ(x) dy dx. R x 2d j Hence R j f is a well-defined disribuion and ( R j f,ϕ C f L 1 ( ) ϕ x j ϕ L 1 (R )+ x d j Using (1.8) and(1.9) wewrie L ( ) (2.1) R ε jf = R ε j(i VL 1 )f W ε jf + Q ε jf +W ε j f +Q ε jf, ). where Q ε j, Qε j, W ε j,and W ε j are operaors wih he inegral kernels 1/ε Q ε j(x, y)= 1 1/ε Wj ε (x, y)= 1 Q (x, y) d 1 and Qε x j (x, y)= j ε x j W (x, y) d and Wε j (x, y)= 1 ε x j Q (x, y) d, x j W (x, y) d, respecively. We shall prove ha Q ε j, Wε j,and W ε j converge in he norm-operaor opology on L 1 ( ), while Q ε j converges srongly on L1 ( )asεends o. Lemma 2.1. The operaors Q ε j L 1 ( ) as ε. converge in he norm-operaor opology on Proof. There exiss φ S( ), φ, such ha (2.2) P (x z) x j 1/2 φ (x z).

6 36 Jacek Dziubański and Marcin Preisner On he oher hand, by (1.3), K s (z,y) Cs d/2. Hence for <ε 2 <ε 1 <1wehave (x, y) Qε2 j (x, y) dx Q ε1 j C 1/ε2 1/2 φ (x z)v (z)s d/2 dz ds d dx (2.3) 1/ε2 C d/2 d V L 1 (R ), d which ends o zero uniformly wih respec o y as ε 1,ε 2. Lemma 2.2. The operaors Wj ε L 1 ( ) as ε. converge in he norm-operaor opology on Proof. The proof borrows ideas from [6]. Le <ε 2 <ε 1 < 1 2.Then W ε1 j (x, y) Wε2(x, y) dx + j 1/ε2 8/9 1/ε2 8/9 = W (y)+w (y). (P (x z) P s (x z)) x j V (z)k s(z,y) dz ds d dx (P (x z) P s (x z)) x j V (z)k s(z,y) dz ds d dx Observe ha here exiss φ S(R), φ, such ha for <s< 8/9, (P (x z) P s (x z)) x j s 3/2 φ (x z). Since sk s (z,y) Cs (2 d)/2 exp( c z y 2 /s) C z y 2 d,wehaveha (2.4) 1/ε2 8/9 W (y) s 2 φ (x z)v (z)k s (z,y) dz ds d dx 1/ε2 1/9 d V (z) z y 2 d dz Cε 1/9 1 uniformly in y. The las inequaliy is a simple consequence of he Hölder inequaliy and he assumpion p>d/2.

7 Riesz ransform characerizaion of Hardy spaces 37 For 8/9 <s< we have K s (z,y) Cs d/2 C 4d/9.Using(2.2) we ge ha 1/ε2 W (y) C ( 1/2 φ (x z)+( s) 1/2 φ s (x z)) 8/9 V (z) 4d/9 dz ds d dx C V L 1 ( ) + C V L1 ( ) 1/ε2 4d/9 d 1/ε2 4d/9 1/2 ( s) 1/2 ds d (2.5) Cε 4d/9 1 1 uniformly in y. Now he lemma follows from (2.4) and(2.5). Lemma 2.3. There exiss a limi of he operaors W ε j opology on L 1 ( ) as ε. in he norm-operaor (2.6) Proof. Le <ε 1 <ε 2 <1. Applying (2.2) we obain ha ε2 (x, y) W j (x, y) dx W ε2 j ε2 ( s) 1/2 φ s (x z)v (z)k s (z,y) dz ds d dx ε 1 ε2 /2 ε2 = ε 1 ε 1 /2 = W (y)+ W (y). If <s</2, hen, of course, ( s) 1/2 C 1/2. Noe ha ) K s (z,y) ds C z y 2 d exp ( z y 2 ψ (z y) 8 for some ψ L p ( ), ψ (p denoes he Hölder conjugae exponen o p). Hence, ε2 /2 ε2 W (y) C 1 V (z)k s (z,y) dz ds d C ε 1 ε2 ε2 ε 1 V (z)ψ (z y) dz d (2.7) C V p ψ p d C ε 1 ε 1 d/2p ψ 1 p d Cε 1 d/2p 2 uniformly in y.

8 38 Jacek Dziubański and Marcin Preisner If /2 s, hen here exiss ϕ S( ), ϕ, such ha K s (z,y) ϕ (z y). Therefore ε2 W (y) ( s) 1/2 V (z)k s (z y) dz ds d ε 1 /2 ε2 /2 C (s) 1/2 V (z)ϕ (z y) ds dz d (2.8) ε 1 ε2 C V p ϕ p d ε 1 Cε 1 d/2p 2 uniformly in y. Now he lemma is a consequence (2.7) (2.8). Lemma 2.4. Assume ha f L 1 ( ). Then he limi F =lim ε Qε j f exiss in he L 1 ( )-norm. Moreover, F L 1 ( ) C f L 1 ( ) wih C independen of f. Proof. Of course, for any fixed y, he funcion z U(z,y)=V (z)γ(z,y) is suppored in he uni ball and U(z,y) Lr (dz) C r for fixed r [1,dp/(dp+d 2p)) wih C r independen of y. The las saemen follows from (1.3) andhehölder inequaliy. Le 1 Hj ε (x, z)= P (x z) d, ε x j Hj ε g(x)= Hj ε (x, z)g(z) dz, Hj g(x)= sup Hj ε g(x). <ε<1 I follows from he heory of singular inegral convoluion operaors (see, e.g., [3, Chaper 4]) ha for 1<r< here exiss C r such ha (2.9) H j g Lr ( ) C r g Lr ( ) for g L r ( ) and lim ε Hj εg(x)=h jg(x) a.e. and in he L r ( )-norm. Noe ha Q ε j (x, y)=hε j U(,y)(x). Hence, here exiss a funcion Q j (x, y) such ha lim ε Qε j (x, y)= Q j (x, y) a.e.and (2.1) sup y sup Q ε j(x, y) r dx C r for 1 <r< <ε<1 dp dp+d 2p.

9 Riesz ransform characerizaion of Hardy spaces 39 Fix N>d.Since Hj ε(x, z) C N x z N for x z >1, (2.11) Q ε j(x, y) = Hj ε (x, z)u(z,y) dz C N x N for x > 2andy. z 1 The Hölder inequaliy combined wih (2.1) and(2.11) implies ha (2.12) sup sup Q ε j(x, y) dx C y <ε<1 and (2.13) lim Q ε ε j(x, y) Q j (x, y) dx = for every y. Now he lemma can be easily concluded from (2.12), (2.13), and Lebesgue s dominaed convergence heorem. 3. Proof of he main heorem Recall ha I VL 1 is an isomorphism in L 1 ( ). Consider f L 1 ( ). Using (2.1) and Lemmas we ge ha R j f belongs o L 1 ( ) if and only if j=1 R j (I VL 1 )f L 1 ( ) and d d f L 1 (R )+ j f L 1 ( ) (I VL 1 )f L 1 (R )+ j (I VL 1 )f L 1 (R ). d Applying he characerizaion of he classical Hardy space H 1 ( ) by means of he Riesz ransforms R j (see (1.2)) and (1.7) we obain Theorem 1.1. j=1 Acknowledgemen. commens. The auhors would like o hank he referee for valuable References 1. Auscher, P.andBen Ali, B., Maximal inequaliies and Riesz ransform esimaes on L p spaces for Schrödinger operaors wih nonnegaive poenials, Ann. Ins. Fourier (Grenoble) 57 (27), Burkholder, D.L.,Gundy, R.F.andSilversein, M. L., A maximal funcion characerizaion of he class H p, Trans. Amer. Mah. Soc. 157 (1971), Duoandikoexea, J.,Fourier Analysis, Graduae Sudies in Mahemaics 29, American Mahemaical Sociey, Providence, RI, 21.

10 31 Jacek Dziubański and Marcin Preisner: Riesz ransform characerizaion of Hardy spaces 4. Duong, X.T.,Ouhabaz, E.M.andYan, L., Endpoin esimaes for Riesz ransforms of magneic Schrödinger operaors, Ark. Ma. 44 (26), Dziubański, J.andZienkiewicz, J., Hardy space H 1 associaed o Schrödinger operaor wih poenial saisfying reverse Hölder inequaliy, Rev. Ma. Iberoamericana 15 (1999), Dziubański, J.andZienkiewicz, J., Hardy space H 1 for Schrödinger operaors wih compacly suppored poenials, Ann. Ma. Pura Appl. 184 (25), Fefferman, C.andSein, E.M.,H p spaces of several variables, Aca Mah. 129 (1972), Sikora, A., Riesz ransform, Gaussian bounds and he mehod of wave equaion, Mah. Z. 247 (24), Sein, E.M.,Harmonic Analysis: Real-Variable Mehods, Orhogonaliy, and Oscillaory Inegrals, Princeon Universiy Press, Princeon, NJ, Jacek Dziubański Insyu Maemayczny Uniwersye Wroc lawski Pl. Grunwaldzki 2/4 PL Wroc law Poland jdziuban@mah.uni.wroc.pl Marcin Preisner Insyu Maemayczny Uniwersye Wroc lawski Pl. Grunwaldzki 2/4 PL Wroc law Poland preisner@mah.uni.wroc.pl Received February 19, 29 in revised form January 2, 21 published online March 26, 21