We prove non local Hardy inequalities on Carnot groups and Riemannian
manifolds, relying on integral representations of fractional Sobolev norms

Russ, Emmanuel

Sire, Yannick

English

arXiv

We prove non local Hardy inequalities on Carnot groups and Riemannian manifolds, relying on integral representations of fractional Sobolev norms

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A note on precised Hardy inequalities on Carnot groupsand Riemannian manifoldsEmmanuel Russ, Yannick SireTo cite this version:Emmanuel Russ, Yannick Sire. A note on precised Hardy inequalities on Carnot groups andRiemannian manifolds. 2010. <hal-00463651>HAL Id: hal-00463651https://hal.archives-ouvertes.fr/hal-00463651Submitted on 13 Mar 2010HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.A NOTE ON PRECISED HARDY INEQUALITIES ONCARNOT GROUPS AND RIEMANNIAN MANIFOLDSEMMANUEL RUSS AND YANNICK SIREAbstract. We prove non local Hardy inequalities on Carnot groupsand Riemannian manifolds, relying on integral representations offractional Sobolev norms.AMS numbers 2000: Primary: 46E35. Secondary: 35R11, 42B35,58J35.Keywords: Hardy inequalities, Lie groups, Riemannian manifolds,fractional powers.Contents1. Introduction 11.1. Setting of the problem 21.2. The case of Lie groups 31.3. The case of Riemannian manifolds 42. Proof of Theorem 1.1 63. Proof of Theorem 1.2 7References 101. IntroductionIn the whole paper, when two quantities A(f) and B(f) depend ona function f ranging in some space V , the notationA(f) . B(f) ∀f ∈ Vmeans that there exists C > 0 such that A(f) ≤ CB(f) for all f ∈ V .Moreover, the notationA(f) ∼ B(f) ∀f ∈ Vmeans that there exists C > 0 such that C−1B(f) ≤ A(f) ≤ CB(f)for all f ∈ V .12 EMMANUEL RUSS AND YANNICK SIRE1.1. Setting of the problem. The simplest Hardy inequality on Rnasserts that, if n ≥ 3,(1.1)∫Rnu2(x)|x|2 dx .∫Rn|∇u(x)|2 dx = ‖u‖H˙1(Rn) , ∀u ∈ D(Rn).A non local version of (1.1) can be given, where the H˙1 norm in theright hand side is replaced by a H˙s norm for 0 < s < n2(see [BCG06]):(1.2)∫Rnu2(x)|x|2s dx . ‖u‖2H˙s(Rn) , ∀u ∈ D(Rn).When 0 < s < 1, it is well-known (see for instance [AF03]) that‖u‖H˙s(Rn) can be represented by means of an integral quantity involvingfirst order differences of u, and (1.2) can therefore be rewritten as(1.3)∫Rnu2(x)|x|2s dx .∫∫Rn×Rn|u(x)− u(y)|2|x− y|n+2s dxdy.These Hardy inequalities (i.e. the local and the non local version)were transposed to the framework of the Heisenberg group in [BCG06,BCX05]. More precisely, in the Heisenberg group Hd (d ≥ 1), thefollowing Hardy inequality was established in [BCX05]:(1.4)∫Hdu2(x)ρ2(x)dx . ‖∇Hu‖22 , ∀u ∈ D(Hd),where ρ(x) denotes the distance of x to the origin and ∇H stands forthe gradient associated to the vector fields Z1, ..., Z2d (see [BCX05] andthe notations therein). The non local version of (1.4), which was provenin [BCG06] (where it was derived from precised inequalities involvingBesov norms) says that, for 0 < s < d+ 1,(1.5)∫Hdu2(x)ρ2s(x)dx . ‖u‖2H˙s , ∀u ∈ D(Hd).When 0 < s < 1, an integral representation for the fractional Sobolevhomogeneous norm was proven in [CRTN01] (note that an analogousrepresentation holds in any connected Lie group wth polynomial vol-ume growth, and even in any unimodular Lie group if one works withthe inhomogeneous version of this norm), so that (1.5) can be rewrittenas(1.6)∫Hdu2(x)ρ2s(x)dx .∫∫Hd×Hd|u(x)− u(y)|2ρ(y−1x)2d+2+2sdxdy.PRECISED HARDY INEQUALITIES 3Hardy inequalities in local versions on more general Lie groups, namelyCarnot groups, were obtained in [Kom06]. In the present paper, we es-tablish non local versions of these Hardy inequalities on Carnot groups,in the spirit of (1.3) and (1.6). We also investigate the similar problemon Riemannian manifolds.1.2. The case of Lie groups. We now describe the general frameworkfor Lie groups. Let G be a unimodular connected Lie group endowedwith the Haar measure. By “unimodular”, we mean that the Haarmeasure is left and right-invariant. If we denote by G the Lie algebraof G, we consider a familyX = {X1, ..., Xk}of left-invariant vector fields on G satisfying the Ho¨rmander condition,i.e. G is the lie algebra generated by the X ′is. A standard metric on G ,called the Carnot-Caratheodory metric, is naturally associated with Xand is defined as follows: let ℓ : [0, 1]→ G be an absolutely continuouspath. We say that ℓ is admissible if there exist measurable functionsa1, ..., ak : [0, 1]→ C such that, for almost every t ∈ [0, 1], one hasℓ′(t) =k∑i=1ai(t)Xi(ℓ(t)).If ℓ is admissible, its length is defined by|ℓ| =∫ 10(k∑i=1|ai(t)|2 dt) 12.For all x, y ∈ G, define d(x, y) as the infimum of the lengths of alladmissible paths joining x to y (such a curve exists by the Ho¨rmandercondition). This distance is left-invariant. For short, we denote by |x|the distance between e, the neutral element of the group and x, so thatthe distance from x to y is equal to |y−1x|.For all r > 0, denote by B(x, r) the open ball in G with respect tothe Carnot-Caratheodory distance and by V (r) the Haar measure ofany ball.We denote−∆G = −k∑i=1X2ithe sub-laplacian on G and ∇G = (X1, ..., Xk) the associated gradient.The Lie group G is called a Carnot group if and only if G is simplyconnected and the Lie algebra of G admits a stratification, i.e. there4 EMMANUEL RUSS AND YANNICK SIREexist linear subspaces V1, ..., Vk of G such thatG = V1 ⊕ ...⊕ Vkwhich[V1, Vi] = Vi+1for i = 1, ..., k − 1 and [V1, Vk] = 0. By [V1, Vi] we mean the subspaceof G generated by the elements [X, Y ] where X ∈ V1 and Y ∈ Vi.Recall that the class of Carnot groups is a strict subclass of nilpotentgroups. Moreover, if G is a Carnot group, there exists n ∈ N, calledthe homogeneous dimension of G, such that, for all r > 0,(1.7) V (r) ∼ rn(see [FS82]). The Heisenberg group Hd is a Carnot group and n =2d+ 2.Let G be a Carnot group, denote by δ the Dirac distribution supportedat the origin and let u be a solution of−∆Gu = δ.Define N(x) = u(x)12−n for x 6= 0 and N(0) = 0. The function N isan homogeneous norm on N by [Fol75]. Kombe [Kom06] proved thefollowing Hardy inequality on G: let α > 2 − n and u ∈ D(G \ {0}),then the following holds(1.8)(n + α− 22)2 ∫Gu2(x)|∇GN(x)|2|N(x)|2 Nα(x)dx ≤∫G|∇Gu(x)|2Nα(x)dx.We prove the following non local version of (1.8):Theorem 1.1. Let G be a Carnot group with homogeneneous dimen-sion n ≥ 3. Then for all α > 2− n and all s ∈ (0, 1),(1.9)∫Gu2(x)( |∇GN ||N |)sNαdx .∫∫G×G|u(x)− u(y)|2|y−1x|n+s Nα(x) dx dy ∀u ∈ D(G\{0}).1.3. The case of Riemannian manifolds. A general principle wasdeveloped in [Car97] to derive Hardy inequalities on Riemannian man-ifolds. Let us recall here an example of such an inequality. Let M be aRiemannian manifold, denote by n its dimension, by dµ its Riemann-ian measure, by d the exterior differentiation and by ∆ the Laplace-Beltrami operator. For all x ∈ M and all r > 0, let B(x, r) be the opengeodesic ball centered at x with radius r, and V (x, r) its measure.Assume that ρ : M → [0,+∞) satisfies(1.10) |dρ| ≤ 1 on M,PRECISED HARDY INEQUALITIES 5and(1.11) ∆ρ ≤ −Cρin the distribution sense,where C > 0. Then, for all α < C − 1 and all u ∈ D(M \ ρ−1(0)),(1.12)(C − 1− α2)2 ∫M(uρ)2(x)ρα(x)dx ≤∫M|du(x)|2 ρα(x)dx.Moreover, if the codimension of ρ−1(0) is greater than 2−α, (1.12) holdsfor all function u ∈ D(M) (see [Car97], The´ore`me 1.4 and Remarque1.5) .We provide here a non local version of (1.12). To state this result,we introduce some extra assumptions on M . The first one is a Faber-Krahn inequality on M . For any bounded open subset Ω ⊂M , denoteby λD1 (Ω) the principal eigenvalue of −∆ on Ω under the Dirichletboundary condition. If p ≥ n, consider the following Faber-Krahninequality: there exists C > 0 such that(1.13) λD1 (Ω) ≥ Cµ(Ω)2p for all bounded subset Ω ⊂M.Let Λp > 0 be the greatest constant for which (1.13) is satisfied. Inother words,Λp = infλD1 (Ω)µ(Ω)2p,where the infimum is taken over all bounded subsets Ω ⊂ M . TheFaber-Krahn inequality (1.13) is satisfied in particular when an isoperi-metric inequality holds on M : namely there exists C > 0 and p ≥ nsuch that, for all bounded smooth subset Ω ⊂M ,(1.14) σ(∂Ω) ≥ Cµ(Ω)1− 1p ,where σ(∂Ω) denotes the surface measure of ∂Ω. If M has nonnegativeRicci curvature, then (1.14) with p = n and (1.13) with p = n areequivalent. More generally, if M has Ricci curvature bounded frombelow by a constant, (1.13) with p > 2n implies (1.14) with p2([Car96],Proposition 3.1, see also [Cou92] when the injectivity radius of M isfurthermore assumed to be bounded). Note that there exists a Rie-mannian manifold satisfying (1.13) for some p ≥ n but for which (1.14)does not hold for any p ≥ n ([Car96], Proposition 3.4).It is a well-known fact that (1.13) implies a lower bound for the volumeof geodesic balls in M . Namely ([Car96], Proposition 2.4), if (1.13)6 EMMANUEL RUSS AND YANNICK SIREholds, then, for all x ∈M and all r > 0,(1.15) V (x, r) ≥(Λp2p+2) p2rp.We will also need another assumption on the volume growth of theballs in M . Say that M has the doubling property if and only if thereexists C > 0 such that, for all x ∈M and all r > 0,(D) V (x, 2r) ≤ CV (x, r).There is a wide class of manifolds on which (D) holds. First, it is true onLie groups with polynomial volume growth (in particular on nilpotentLie groups). Next, (D) is true if M has nonnegative Ricci curvaturethanks to the Bishop comparison theorem (see [BC64]). Recall alsothat (D) remains valid if M is quasi-isometric to a manifold with non-negative Ricci curvature, or is a cocompact covering manifold whosedeck transformation group has polynomial growth, [CSC95]. Contraryto the doubling property, the nonnegativity of the Ricci curvature isnot stable under quasi-isometry.We prove the following theorem.Theorem 1.2. Let M be a complete non compact Riemannian mani-fold. Assume that (1.13) holds and that M has the doubling property.Assume also that C > 0 and ρ : M → [0,+∞) are such that (1.10) and(1.11) hold. Then, if α < C − 1 and ρ−1(0) has codimension greaterthan 2− α, one has, for all s ∈ (0, 1),(1.16)∫Mu2(x)ρs(α−2)(x)dx .∫∫M×M|u(y)− u(x)|2d(x, y)p+sρα(x)dxdy ∀u ∈ D(M\ρ−1(0)).2. Proof of Theorem 1.1In order to prove Theorem 1.1, we need to introduce an operatorLNα on L2(G). Let L2(G,Nα) denote the L2 space on G with respectto the measure Nαdx and H1(G,Nα) the Sobolev space defined byH1(G,Nα) :={f ∈ L2(G,Nα); Xif ∈ L2(G,Nα) ∀1 ≤ i ≤ k}.Define now the operator LNα on L2(G,Nα) byLNαu := −N−αk∑i=1Xi(NαXiu).The domain of LNα is given byD (LNα) ={u ∈ L2(G,Nα); N−αXi(NαXiu) ∈ L2(G,Nα) ∀1 ≤ i ≤ k}.PRECISED HARDY INEQUALITIES 7One has, for all u ∈ D (LNα) and all v ∈ H1(G,Nα),∫GLNαu(x)v(x)Nα(x)dx =∫Gk∑i=1Xiu(x)Xiv(x)Nα(x)dx.The operator LNα is therefore clearly symmetric and nonnegative onL2(G,Nα), and the spectral theorem allows to define the usual powers(LNα)β for any β > 0 by means of spectral theory.By the definition of LNα , (1.8) means, in terms of operators inL2(G,Nα), that, for some λ > 0,(2.17) LNα ≥ λµ,where µ is the multiplication operator by |∇GN ||N | . Using a functionalcalculus argument (see [Dav80], p. 110) one deduces from (2.17) that,for any s ∈ (0, 2),(LNα)s/2 ≥ λs/2µs/2which implies, thanks to the fact (LNα)s/2 = ((LNα)s/4)2 and the sym-metry of (LNα)s/4 on L2(G,Nα), that∫G|u(x)|2 µ(x)sNα(x)dx ≤ C∫G∣∣(LNα)s/4u(x)∣∣2Nα(x)dx =C∥∥(LNα)s/4u∥∥2L2(G,Nα) .The conclusion of Theorem 1.1 follows now from the estimate∥∥(LNα)s/4u∥∥2L2(G) ≤ C ∫∫G×G|u(x)− u(y)|2|y−1x|n+2s Nα(x) dx dy,which is exactly the estimate for M = Nα provided in Lemmata 3.2and 3.3 in [RS10] (remember that (1.7) holds).3. Proof of Theorem 1.2The proof of Theorem 1.2 relies again on estimates for the powers ofa suitable operator. Namely, defineLραu = ρ−αdiv(ρα∇u),where ∇ is the gradient induced by the Riemannian metric and divis the divergence operator on M . As before, Lρα is a nonnegativesymmetric operator on L2(M, ραdx) and Lβρα is defined by spectraltheory for all β > 0. If µ denotes the multiplication operator by ρ−2,8 EMMANUEL RUSS AND YANNICK SIRE(1.12) means that Lρα ≥ cµ in L2(M, ραdx). Spectral theory thenyields Ls/2ρα ≥ cµs/2, which means that∫Mu2(x)ρs(α−2)(x)dx .∥∥∥Ls/4ρα u∥∥∥2L2(M,ραdx),and we are therefore left with the task of checking∥∥∥Ls/4ρα u∥∥∥2L2(M,ραdx).∫∫M×M|u(y)− u(x)|2d(x, y)p+sρα(x)dxdy.To that purpose, one first notices that∥∥∥Ls/4ρα u∥∥∥2L2(M,ραdx).∫ +∞0t−1−s/2∥∥t Lρα (I + t Lρα)−1u∥∥2L2(M,ραdx) dtand it is therefore enough to show that∫ +∞0t−1−s/2∥∥t Lρα (I + t Lρα)−1u∥∥2L2(M,ραdx) dt.∫∫M×M|u(y)− u(x)|2d(x, y)p+sρα(x)dxdy.The proof follows the same lines as the one of Lemma 3.3 in [RS10]and we will therefore be sketchy, only indicating the main differences.Using (D), one can pick up a countable family xtj , j ∈ N, such that theballs B(xtj ,√t)are pairwise disjoint and(3.18) M =⋃j∈NB(xtj , 2√t).By (D), there exist constants C˜ > 0 and κ > 0 such that for allθ > 1 and all x ∈ G, there are at most C˜ θ2κ indexes j such that|x−1xtj | ≤ θ√t.For fixed j, one hast Lρα (I + t Lρα)−1u = t Lρα (I + t Lρα)−1 gj,twhere, for all x ∈M ,gj,t(x) := u(x)−mj,tand mj,t is defined bymj,t :=1V(xtj , 2√t) ∫B(xtj ,2√t)u(y)dy.PRECISED HARDY INEQUALITIES 9Note that, here, the mean value of u is computed with respect to theRiemannian measure on M . Since (3.18) holds, one clearly has∥∥t Lρα (I + t Lρα)−1u∥∥2L2(M,ραdx) ≤ ∑j∈N∥∥t Lρα (I + t Lρα)−1u∥∥2L2(B(xtj ,2√t),ραdx)=∑j∈N∥∥t Lρα (I + t Lρα)−1gj,t∥∥2L2(B(xtj ,2√t),ραdx) ,and it is therefore enough to ckeck∑j∈N∥∥t Lρα (I + t Lρα)−1gj,t∥∥2L2(B(xtj ,2√t),ραdx)(3.19).∫∫M×M|u(y)− u(x)|2d(x, y)p+sρα(x)dxdy.As in [RS10], this is a consequence of L2 off-diagonal estimates for Lραand upper estimates for the functions gj,t. Let us recall the off-diagonalestimates for Lρα for completeness:Lemma 3.1. There exists C with the following property: for all closeddisjoint subsets E, F ⊂ M with d(E, F ) =: d > 0, all function f ∈L2(M, ραdx) supported in E and all t > 0,∥∥(I+ t Lρα)−1f∥∥L2(F,ραdx) + ∥∥t Lρα(I+ t Lρα)−1f∥∥L2(F,ραdx) ≤8 e−C d√t ‖f‖L2(E,ραdx) .The proof of Lemma 3.1 is analogous to the one of Lemma 3.1 in[RS10].As far as estimates for gj,t are concerned, set, for all k ≥ 1,Cj,t0 = B(xtj , 4√t)and Cj,tk = B(xtj , 2k+2√t)\B(xtj , 2k+1√t),and gj,tk := gj,t 1Cj,tk, k ≥ 0, where, for any subset A ⊂ M , 1A is theusual characteristic function of A. We then have:Lemma 3.2. There exists C¯ > 0 such that, for all t > 0 and all j ∈ N:A.∥∥gj,t0 ∥∥2L2(Cj,t0,ραdx)≤ C¯tp/2∫B(xtj ,4√t)∫B(xtj ,4√t)|u(x)− u(y)|2 ρα(x)dx dy.B. For all k ≥ 1, ∥∥gj,tk ∥∥2L2(Cj,tk,ραdx)≤C¯(2k√t)p∫x∈B(xtj ,2k+2√t)∫y∈B(xtj ,2k+2√t)|u(x)− u(y)|2 ρα(x)dx dy.10 EMMANUEL RUSS AND YANNICK SIREThe proof of Lemma 3.2 is analogous to the one of Lemma 3.4 in[RS10], the only extra ingredient being the lower bound (1.15) appliedwith the balls B(xtj , 2k+2√t). We then conclude the proof of (3.19)in the same way as for the conclusion of the proof of Lemma 3.3 in[RS10].References[AF03] R.A. Adams and J.J.F. Fournier. Sobolev spaces, volume 140 of Pureand Applied Mathematics (Amsterdam). Elsevier/Academic Press, Am-sterdam, second edition, 2003.[BC64] R. Bishop and R. Crittenden. Geometry of manifolds. Academic Press,New York, 1964.[BCG06] H. Bahouri, J.-Y. Chemin, and I. Gallagher. Precised Hardy inequalities.Ann. Scuol. Norm. Sup. Pisa, 5:375–391, 2006.[BCX05] H. Bahouri, J.Y. Chemin, and C.J. Xu. Trace and trace lifting theoremsin weighted Sobolev spaces. Journal of the Institute of Mathematics ofJussieu, 4(04):509–552, 2005.[Car96] G. Carron. Ine´galite´s isope´rime´triques de Faber-Krahn et conse´quences.In Actes de la table ronde de ge´ome´trie diffe´rentielle (Luminy, 1992),Collection SMF Se´minaires et Congre`s, volume 1, pages 205–232, 1996.[Car97] G. Carron. Ine´galite´s de Hardy sur les varie´te´s riemanniennes non com-pactes. J. Math. Pures Appl., 76:883–891, 1997.[Cou92] T. Coulhon. Sobolev inequalities on graphs and on manifolds, pages 207–214. Plenum Press, New York, 1992.[CRTN01] T. Coulhon, E. Russ, and V. Tardivel-Nachef. Sobolev algebras on Liegroups and Riemannian manifolds. Amer. J. Math., 123:283–342, 2001.[CSC95] T. Coulhon and L. Saloff-Coste. Varie´te´s riemanniennes isome´triques a`l’infini. Rev. Mat. Iberoam., 11:687–726, 1995.[Dav80] E.B. Davies. One-parameter semigroups, volume 15 of London Mathe-matical Society Monographs. Academic Press, Inc., London-New York,second edition, 1980.[Fol75] G. Folland. Subelliptic estimates and function spaces on nilpotent Liegroups. Ark. Mat., 13:161–207, 1975.[FS82] G. Folland and E. M. Stein. Hardy spaces on homogeneous groups.Princeton Univ. Press, 1982.[Kom06] I. Kombe. Hardy, Rellich and Uncertainty principle inequalities onCarnot Groups. http://arxiv.org/abs/math/0611850, 2006.[RS10] E. Russ and Y. Sire. Non local poincare´ inequalities on lie groups withpolynomial volume growth. http://jp.arxiv.org/abs/1001.4075, 2010.Emmanuel Russ– Universite´ Paul Ce´zanne, LATP,Faculte´ des Sciences et Techniques, Case cour AAvenue Escadrille Normandie-Niemen, F-13397 Marseille, Cedex 20,France etCNRS, LATP, CMI, 39 rue F. Joliot-Curie, F-13453 Marseille Cedex13, FrancePRECISED HARDY INEQUALITIES 11Yannick Sire– Universite´ Paul Ce´zanne, LATP,Faculte´ des Sciences et Techniques, Case cour AAvenue Escadrille Normandie-Niemen, F-13397 Marseille, Cedex 20,France etCNRS, LATP, CMI, 39 rue F. Joliot-Curie, F-13453 Marseille Cedex13, France.

A note on precised Hardy inequalities on Carnot groups and Riemannian manifolds

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