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

Nadine Badr

Description

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

Sujets

• Jordan
• Bernard
• Université de Lyon
• Poincaré inequality
• Riesz transform
• Membrane potential

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 17
• Langue: English

Extrait

L p

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 variousL p estimates for the Schrödinger operator−Δ +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: [e-mail] †  Email:[e-mail] 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 theL p boundedness for the Riesz trans- formsr(−Δ +V) − 1 2 ,V 1 2 (−Δ +V) − 21 and 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] provedL p maximal inequalities for these operators under less restrictive assumptions. They assumed thatVbelongs to some reverse Hölder classRH q  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, theL p boundedness of the Riesz transform of−Δ +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 verifieskp t (x, .)k 2 ≤ µ(B( C x, √ t)) for all x∈Mandt >0 these hypotheses, Sikora proved that if. UnderV∈L l 1 oc (M),V≥0, then the Riesz transforms of−Δ +VareL p bounded for1< p≤2and of weak type (1,1). Li [41] obtained boundedness results on Nilpotent Lie groups under the restriction V∈RH q andq≥ 2 D ,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(P 2 ) and takingVin someRH q  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 (P 2 ). First we obtain the range ofpfor the following maximal inequality valid for u∈C 0 ∞ (M): kΔuk p +kV uk p .k(−Δ +V)uk p .(1) Here and after, we useu.vto say that there exists a constantCsuch thatu≤Cv. The starting step is the followingL 1 inequality foru∈C 0 ∞ (M), kΔuk 1 +kV uk 1 ≤3k(−Δ +V)uk 1 (2) which holds for any non-negative potentialV∈L l 1 oc (M). This allows us to define −Δ +Vas an operator onL 1 (M)with domainD 1 (Δ)∩ D 1 (V).

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For larger range ofp, we assume thatV∈L o p cl (M)and−Δ +Vis a priori defined onC 0 ∞ . 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(P 2 ). ConsiderV∈RH q for 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 restrictionq≥ 2 D . The second purpose of our work is to establish someL p estimates for the square root of−Δ +V. Notice that we always have the identity k |ru| k 2 2 +kV 1 2 uk 2 2 =k(−Δ +V) 1 2 uk 2 2 , u∈C 0 ∞ (M).(3) The weak type(1,1)inequality proved by Sikora [52] is satisfied under our hypotheses: k |ru| k 1,∞ +kV 1 2 uk 1,∞ .k(−Δ +V) 21 uk 1 .(4) Interpolating (3) and (4), we obtain 1 k |ru| k p +kV 2 uk p .k(−Δ +V) 1 2 uk p (5) when1< p <2andu∈C 0 ∞ (M). Here,k k p,∞ is the norm in the Lorentz spaceL p,∞ . 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 (P 2 ) there exists. Thenp 0 >2such that the Riesz transformT=r(−Δ) − 1 2 isL p bounded for1< p < p 0 . We now letp 0 = sup n p∈]2,∞[;r(−Δ) − 1 2 isL p bounded o  obtain the fol-. We lowing theorem.

Theorem 1.3.LetMbe a complete Riemannian manifold. LetV∈RH q for some q >1and >0such thatV∈RH q+ . 1. Assume thatMsatisfies(D)and(P 2 ). Then for allu∈C 0 ∞ (M), k |ru| k p .k(−Δ +V) 1 2 uk p for1< p <inf(p 0 ,2(q+));(6) kV 1 2 uk p .k(−Δ +V) 1 2 uk p for1< p <2(q+).(7) 2. Assume thatMis of polynomial type and admits(P 2 ). Suppose thatD < p 0 , whereDis the dimension at infinity and that D 2 ≤q < p 2 0 . a. Ifq < D, then (6) holds for1< p <inf(q ∗ D +, p 0 ), (q ∗ D = D D −q q ). b. Ifq≥D, then (6) holds for1< p < p 0 .

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