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Lp Boundedness of Riesz transform related to Schrödinger operators on a manifold

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Lp Boundedness of Riesz transform related to Schrödinger operators on a manifold Nadine Badr? Besma Ben Ali † Université Lyon 1 Université Paris-Sud April 27, 2009 Abstract We establish various Lp estimates for the Schrödinger operator ?∆ + V on Riemannian manifolds satisfying the doubling property and a Poincaré inequal- ity, where ∆ is the Laplace-Beltrami operator and V belongs to a reverse Hölder class. At the end of this paper we apply our result on Lie groups with polynomial growth. Contents 1 Introduction 2 2 Preliminaries 6 2.1 The doubling property and Poincaré inequality . . . . . . . . . . . . . . 6 2.2 Reverse Hölder classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Homogeneous Sobolev spaces associated to a weight V . . . . . . . . . 9 3 Definition of Schrödinger operator 9 4 Principal tools 11 4.1 An improved Fefferman-Phong inequality . . . . . . . . . . . . . . . . . 11 4.2 Calderón-Zygmund decomposition . . . . . . . . . . . . . . . . . . . . . 13 4.3 Estimates for subharmonic functions . . . . . . . . . . . . . .

  • inequality valid

  • when ?1

  • proved lp maximal

  • lp boundedness

  • poincaré inequality

  • negative ricci curvature

  • riesz transform

  • negative potential

  • auscher-ben ali


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Lp
Boundedness of Riesz transform related Schrödinger operators on a manifold
Nadine Badr
Besma Ben Ali
Université Lyon 1 Université Paris-Sud
April 27, 2009
to
Abstract We establish variousLpestimates for the Schrödinger operatorΔ +Von Riemannian manifolds satisfying the doubling property and a Poincaré inequal-ity, whereΔis the Laplace-Beltrami operator andVbelongs to a reverse Hölder class. At the end of this paper we apply our result on Lie groups with polynomial growth.
Contents
1 Introduction
2 Preliminaries 2.1 The doubling property and Poincaré inequality . . . . . . . . . . . . . . 2.2 Reverse Hölder classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Homogeneous Sobolev spaces associated to a weightV. . . . . . . . .
3 Definition of Schrödinger operator
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6 6 8 9
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4 Principal tools 11 4.1 An improved Fefferman-Phong inequality . . . . . . . . . . . . . . . . . 11 4.2 Calderón-Zygmund decomposition . . . . . . . . . . . . . . . . . . . . . 13 4.3 Estimates for subharmonic functions . . . . . . . . . . . . . . . . . . . 18
5 Maximal inequalities
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6 Complex interpolation 21 Institut Camille Jordan, Université Claude Bernard, Lyon 1, UMR du CNRS 5208, 43 boulevard du 11 novembre 1918, F-69622 Villeurbanne cedex. Email: badr@math.univ-lyon1.fr  Email:besmath@yahoo.frUMR du CNRS 8628, F-91405 Orsay cedex.Université de Paris-Sud,
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7 Proof of Theorem 1.4
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Proof of point 2. of Theorem 1.3 27 8.1 Estimates for weak solutions . . . . . . . . . . . . . . . . . . . . . . . . 28 8.2 A reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 8.3 Proof of point 2. of Theorem 1.3 . . . . . . . . . . . . . . . . . . . . . 32
9 Case of Lie groups
1 Introduction
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The main goal of this paper is to establish theLpboundedness for the Riesz trans-formsr(Δ +V)12,V12(Δ +V)21and related inequalities on certain classes of Riemannian manifolds. Here,Vis a non-negative, locally integrable function onM. For the Euclidian case, this subject was studied by many authors under different conditions onV mention the . Weworks of Helffer-Nourrigat [35], Guibourg [31], Shen [51], Sikora [52], Ouhabaz [47] and others. Recently, Auscher-Ben Ali [3] provedLpmaximal inequalities for these operators under less restrictive assumptions. They assumed thatVbelongs to some reverse Hölder classRHq natural step further is to extend(for a definition, see section 2). A the above results to the case of Riemannian manifolds. For Riemannian manifolds, theLpboundedness of the Riesz transform ofΔ +V was discussed by many authors. We mention Meyer [45], Bakry [9] and Yosida [59]. The most general answer was given by Sikora [52]. LetMsatisfying the doubling property(D)and assume that the heat kernel verifieskpt(x, .)k2µ(B(Cx,t))for all xMandt >0 these hypotheses, Sikora proved that if. UnderVLl1oc(M),V0, then the Riesz transforms ofΔ +VareLpbounded for1< p2and of weak type (1,1). Li [41] obtained boundedness results on Nilpotent Lie groups under the restriction VRHqandq2D,Dbeing the dimension at infinity ofG(see [23]). Following the method of [3], we obtain new results forp >2on complete Rie-mannian manifolds satisfying the doubling property(D), a Poincaré inequality(P2) and takingVin someRHq manifolds of polynomial type we obtain additional. For results. This includes Nilpotent Lie groups. Let us summarize the content of this paper. LetMbe a complete Riemannian manifold satisfying the doubling property(D)and admitting a Poincaré inequality (P2). First we obtain the range ofpfor the following maximal inequality valid for uC0(M): kΔukp+kV ukp.k(Δ +V)ukp.(1) Here and after, we useu.vto say that there exists a constantCsuch thatuCv. The starting step is the followingL1inequality foruC0(M), kΔuk1+kV uk13k(Δ +V)uk1(2) which holds for any non-negative potentialVLl1oc(M). This allows us to define Δ +Vas an operator onL1(M)with domainD1(Δ)∩ D1(V).
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For larger range ofp, we assume thatVLopcl(M)andΔ +Vis a priori defined onC0. The(1) can be obtained if one imposes for the potential validity of Vto be more regular:
Theorem 1.1.LetMbe a complete Riemannian manifold satisfying(D)and(P2). ConsiderVRHqfor some1< q≤ ∞. Then there is >0depending only onV such that(1)holds for1< p < q+.
This new result for Riemannian manifolds is an extension of the one of Li [41] on Nilpotent Lie groups settings obtained under the restrictionq2D. The second purpose of our work is to establish someLpestimates for the square root ofΔ +V. Notice that we always have the identity k |ru| k22+kV12uk22=k(Δ +V)12uk22, uC0(M).(3) The weak type(1,1)inequality proved by Sikora [52] is satisfied under our hypotheses: k |ru| k1,+kV12uk1,.k(Δ +V)21uk1.(4) Interpolating (3) and (4), we obtain 1 k |ru| kp+kV2ukp.k(Δ +V)12ukp(5) when1< p <2anduC0(M). Here,k kp,is the norm in the Lorentz spaceLp,. It remains to find good assumptions onVandMto obtain (5) for some/all2< p <. Recall before the following result
Proposition 1.2.([4]) LetMbe a complete Riemannian manifold satisfying(D)and (P2) there exists. Thenp0>2such that the Riesz transformT=r(Δ)12isLp bounded for1< p < p0. We now letp0= supnp]2,[;r(Δ)12isLpboundedo obtain the fol-. We lowing theorem.
Theorem 1.3.LetMbe a complete Riemannian manifold. LetVRHqfor some q >1and >0such thatVRHq+. 1. Assume thatMsatisfies(D)and(P2). Then for alluC0(M), k |ru| kp.k(Δ +V)12ukpfor1< p <inf(p0,2(q+));(6) kV12ukp.k(Δ +V)12ukpfor1< p <2(q+).(7) 2. Assume thatMis of polynomial type and admits(P2). Suppose thatD < p0, whereDis the dimension at infinity and thatD2q <p20. a. Ifq < D, then (6) holds for1< p <inf(qD+, p0), (qD=DDqq). b. IfqD, then (6) holds for1< p < p0.
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