Hardy inequalities on Riemannian manifolds and applications

Transcription

1 Hardy inequalities on Riemannian manifolds and alications arxiv: v2 [math.ap] 15 Ar 2013 Lorenzo D Ambrosio Diartimento di atematica, via E. Orabona, 4 I-70125, Bari, Italy Serena Diierro SISSA, Sector of athematical Analysis, via Bonomea 265 I-34136, Trieste, Italy Aril 6, 2012 Abstract We rove a simle sufficient criteria to obtain some Hardy inequalities on Riemannian ( manifolds ) related to quasilinear second-order differential oerator u := div u 2 u. Namely, if ρ is a nonnegative weight such that ρ 0, then the Hardy inequality u c ρ ρ dv g u dv g, u C0 (). holds. We show concrete examles secializing the function ρ. Our aroach allows to obtain a characterization of -hyerbolic manifolds as well as other inequalities related to Cacciooli inequalities, weighted Gagliardo- Nirenberg inequalities, uncertain rincile and first order Caffarelli-Kohn-Nirenberg interolation inequality. Keywords. Hardy inequality, Riemannian manifolds, arabolic manifolds, Cacciooli inequality, weighted Gagliardo-Nirenberg inequality, interolation inequality. SC J05, 31C12, 26D10 Address for corresondence dambros@dm.uniba.it Address for corresondence serydiierro@yahoo.it 1

2 Contents 1 Introduction 2 2 Hardy inequalities 5 3 Further inequalities 12 4 Remarks on the best constant 17 5 First order interolation inequalities 20 6 Some Alications Hardy inequality involving the distance from the boundary Hardy inequality for -hyerbolic manifold Hardy inequality on Cartan-Hadamard manifold Hardy inequalities involving the distance from the soul of a manifold Hardy-Poincaré inequality for the hyerbolic lane The Euclidean case A Aendix 33 1 Introduction An N-dimensional generalization of the classical Hardy inequality asserts that for every > 1 u c w dx u dx, u C 0 (), where R N is an oen set, and the weight w is, for instance, w(x) := x and < N, or w(x) := dist(x, ) and is convex (see for examle [BFT, B, DH, GG, P, S1, S2, i] and references therein). The reeminent role of Hardy inequalities and the knowledge of the best constants involved is a well known fact, as the reader can recognize from the wide literature that uses such atoolineuclidean orinsubellitic setting aswell asonmanifolds([bg, BGGK, BC, B, BD, DL, Ja, P] just to cite a few). On the other hand, the knowledge of the validity of a Hardy or Gagliardo-Nirenberg or Sobolev or Caffarelli-Kohn-Nirenberg inequality on a manifold and their best constants allows to obtain qualitative roerties on the manifold. For instance in [AX, CX, Xi] it was shown that if is a comlete oen Riemannian manifold with nonnegative Ricci curvature in which a Hardy or Gagliardo-Nirenberg or Caffarelli-Kohn-Nirenberg tye inequality holds, then is in some suitable sense close to the Euclidean sace. 2

3 One of our aims is to rove some Hardy inequalities on Riemannian manifolds. In 1997, Carron in [Ca2] studies weighted L 2 -Hardy inequalities on a Riemannian manifold under some geometric assumtions on the weight function ρ, obtaining, among other results, the following inequality c u 2 ρ 2 dv g u 2 dv g, u C 0 (), (1.1) where ρ is a nonnegative function such that ρ = 1, ρ γ, ρ ρ 1 {0} is a comact set of zero caacity and c = ( ) γ In [Ca2] the author alies this result to several exlicit examles of Riemannian manifolds. Under the same hyotheses on the function ρ, Kombe and Özaydin in [KO] extend Carron s result to the case 2 for functions in C0 ( \ρ 1 {0}), and the authors resent an alication to the unctured manifold B n \{0} with B n the Poincaré ball model of the hyerbolic sace and ρ the distance from the oint 0 and = 2. Li and Wang in [LW] rove that if is a hyerbolic manifold (i.e. there exists a symmetric ositive Green function G x ( ) for the Lalacian with ole at x), then 1 y G x (y) 2 u 2 (y)dv 4 G 2 g u 2 (y)dv g, u C0 ( \{x}). x(y) We also mention iklyukov and Vuorinen, which in [V] rove that the inequality ( ) 1/q ( ) 1/ α(ǫ(x))u(x) q dv g λ (β(ǫ(x)) u(x) ) dv g, u W 1, 0 (), holds for q rovided some conditions related to the isoerimetric rofile of are satisfied. In [AS], Adimurthi and Sekar use the fundamental solution of a general second-order ellitic oerator to derive Hardy-tye inequalities and then they extend their arguments to Riemannian manifolds using the fundamental solution of -Lalacian. Bozhkov and itidieri in [B1] rove the validity of (1.1) also for 2 (1 < < N), rovided there exists on a C 1 conformal Killing vector field K such that divk = µ with µ a ositive constant and ρ = K. Let > 1 and let ρ be a nonnegative function. Our rincial result is a simle criterion to establish if there holds a Hardy inequality involving the weight ρ. Namely, if ρ is - suerharmonic in, that is ρ 0, then c u ρ ρ dv g u dv g, u C 0 (), (1.2) holds (see Theorem 2.1). Such a kind of criteria is already established in [Da3] for a quite general class of second order oerators containing, among other examles, the subellitic 3

4 oerators on Carnot grous. For this goal we shall mainly use a technique introduced by itidieri in [i] and develoed in [Da1, Da2, Da3] and in [B1, B2]. The roof is based on the divergence theorem and on the careful choice of a vector field. Let us oint out some interesting outcomes of our aroach. A first issue is that, since it is quite general, our aroach includes Hardy inequalities already studied in [AS, B1, Ca2, KO, LW] in the case = 2 as well as their generalization for > 1. Indeed, in all these cited aers, the authors assume extra conditions on the function ρ or on the manifold. Furthermore, in concrete cases, our result yields an exlicit value of the constant c. oreover, in several cases, this value is also the best constant (see [Da3]). To this regard, we discuss if the best constant is achieved or not and, in the latter case, we study the ossibility to add a remainder term. Another asect of our technique is that it allows to characterize the -hyerbolic manifolds. We remind that a manifold is called -hyerbolic if there exists a symmetric ositive Green function G x ( ) for the -Lalacian with ole at x. We rove that is -hyerbolic if and only if there exists a nonnegative non trivial function f L 1 loc () such that f u dv g u dv g, u C0 (). Notice that one of the imlications of this characterization for = 2 is the result roved in [LW]. During the review rocess of this work, we have received the aer of Devyver, Fraas and Pinchover [DFP]. In [DFP] a general linear second order differential oerator P in the Euclidean framework is studied. The authors find a rofound relation between the existence of ositive suersolutions of P u = 0, Hardy tye inequalities involving P and a weight W and the characterization of the sectrum of the weighted oerator. We refer the interested reader to [DFP, DFP2]. We also obtain a generalization of (1.2). Namely, for a nonnegative function ρ, the inequality ( ) 1 α u ρ ρα ρ dv g u ρ α dv g, u C0 (), (1.3) holds, rovided ( 1 α) ρ 0 (see Theorem 3.1). The above inequality contains, as secial case, the Cacciooli inequality. Indeed, if ρ is a -subharmonic function, that is ρ 0, then (1.3) holds for α > 1 and, in articular, for α = we have 1 u ρ dv g u ρ dv g, u C 0 (). This is the so called Cacciooli inequality (see for instance [PRS] and the references therein for the version = 2 on manifolds). Another advantage of our aroach is that it allows to obtain also other new and known results, like wighted Gagliardo-Nirenberg inequalities and the uncertain rincile. 4

5 Finally we show that if (1.3) and a Sobolev tye inequality (that is c u L u L ) hold on, then we obtain an interolation inequality involving, as weights, ρ and its gradient. As articular case, our results contain inequalities on manifolds related to the celebrated Caffarelli-Kohn-Nirenberg inequality. The aer is organized as follows. We resent the roof of (1.2) in Section 2, where imortant consequences and observations are derived. In Section 3 we show natural extensions of (1.2), obtaining also Hardy inequality with weights, Cacciooli-tye inequalities, weighted Gagliardo-Nirenberg inequalities and the uncertain rincile. Some remarks on the best constant and if it is attained are discussed in Section 4. In Section 5 we resent a first order interolation inequality. Finally Section 6 is devoted to resent some concrete examles of Hardy-tye inequalities on manifolds. Notation In what follows (, g) is a comlete Riemannian N-dimensional manifold, is an oen set, dv g is the volume form associated to the metric g, u and divh stand resectively for the gradient of a function u and the divergence of a vector field h with resect to the metric g (see [Au] for further details). Throughout this aer > 1. 2 Hardy inequalities In order to state Hardy inequalities involving a weight ρ, the basic assumtion we made on ρ is that ρ is -suerharmonic in weak sense. Namely, we assume that ρ L 1 loc (), ρ L 1 loc (), and ρ 0 on in weak sense, that is for every nonnegative ϕ C 1 0 (), we have The main result on Hardy inequalities is the following: ρ 2 ( ρ ϕ)dv g 0. (2.4) Theorem 2.1. Let ρ W 1, loc () be a nonnegative function on such that Then ρ ρ ρ 0 on in weak sense. L 1 loc (), and the following inequality holds: ( ) 1 u ρ ρ dv g u dv g, u C0 (). (2.5) Before roving Theorem 2.1, we shall resent some immediate consequences and extensions of the main result. 5

6 Definition 2.2. Let be an oen set. We denote by D 1, () the comletion of C 0 () with resect to the norm ( u D 1, = u dv g ) 1/. It is ossible to extend the validity of (2.5) to function u C0 (). This extension is based on the inclusion D 1, () D 1, (). (2.6) Theaboveinclusionissatisfied, forinstance,when\isacomactsetofzero-caacity (see Aendix A). Corollary 2.3. Let ρ L 1 loc () be a function satisfying the assumtions of Theorem 2.1. If (2.6) holds, then ρ L 1 ρ loc (), and the following inequality holds: ( ) 1 u ρ ρ dv g u dv g, u C0 (). (2.7) Proof. The inequality (2.5) holds for every u C 0 (), then it holds for every u D 1, (). Since C0 () D1, (), by using (2.6) we conclude the roof. In order to illustrate further consequences of Theorem 2.1 we give the following: Definition 2.4. A manifold is said -hyerbolic 1 if there exists a symmetric ositive Green function G x ( ) for the -Lalacian with ole at x 2, if it is not the case we call it -arabolic. Several equivalent definitions of -arabolic manifolds can be given. For instance in [Tr1] there is the following (see also the literature therein and [Ho2]) Proosition 2.5. Let > 1. The following statements are equivalent a) is -arabolic; b) there exists a comact set K with non emty interior such that ca (K,) = 0; c) there is no non constant ositive -suerharmonic function on ; d) there exists a sequence of functions u j C 0 () such that 0 u j 1, u j 1 uniformly on every comact subset of and u j dv g 0. 1 any authors call these manifolds non -arabolic. 2 That is G x = x where x is the Dirac measure concentrated at oint x. 6

7 Other characterizations of -arabolic manifolds are based on several roerties, for instance on the volume growth, on the isoerimetric rofile of the manifold, on some roerties of some cohomology, on the recurrence of the Brownian motion. See [Gr1, Gr2, Gr3, Gr4, LT, Tr1] and the references therein. From Theorem 2.1 we deduce the following characterization of -hyerbolicity. Theorem 2.6. A manifold is -hyerbolic if and only if there exists a nonnegative non trivial function f L 1 loc () such that f u dv g u dv g, u C0 (). (2.8) Proof. If is -hyerbolic then inequality (2.8) holdswith f = ( 1 ) Gx G x. Indeed G x is nonnegative and satisfies the hyotheses of Theorem 2.1 (see Theorem 6.4 for further details). Conversely, assume that is -arabolic and that (2.8) is valid for a function f 0. Then from d) of Proosition 2.5 there exists a sequence of functions u j C 1 0 () such that 0 u j 1, u j 1 uniformly on every comact subset of and u j dv g 0, (as j + ). It imlies that fdv D g = 0 for every comact subset D of and then f 0. This concludes the roof. Remark 2.7. Since the -hyerbolicity of is equivalent to the existence of a non constant ositive -suerharmonic ( function ) ρ on, then by Theorem 2.1 we obtain that 1 inequality (2.8) holds with f = ρ. ρ Remark 2.8. Our Theorem 2.6 imlies that if the manifold admits a C 1 conformal Killing vector field K (see i.e. [B1] for the definition) such that divk = µ 0 with µ constant and K L 1 loc (), then is -hyerbolic. This follows combining Theorem 2.6 and Theorem 4 of [B1] (see also Remark 2.11 ii) below). In order to rove Theorem 2.1 we fix some notation. Let h L 1 loc () be a vector field. We remind that the distribution divh is defined as ϕ divh dv g = ( ϕ h)dv g, (2.9) for every ϕ C 1 0 (). 7

8 Let h L 1 loc () be a vector field and let A L1 loc () be a function. In what follows we write A divh meaning that the inequality holds in distributional sense, that is for every ϕ C0 1 () such that ϕ 0, we have ϕ A dv g ϕ divh dv g = ( ϕ h)dv g. (2.10) Remark 2.9. Let f C 1 (R) be a real function such that f (0) = 0. Taking ϕ = f (u) with u C 1 0 () in (2.9), we have f (u)divh dv g = f (u)( u h)dv g. In articular, choosing f (u) = u, with > 1, we get u divh dv g = u 2 u( u h)dv g, u C0 1 (). (2.11) Lemma Let h L 1 loc () be a vector field and let A h L 1 loc () be a nonnegative function such that i) A h divh, ii) h A 1 h L 1 loc (). Then for every u C 1 0 () we have u A h dv g h A 1 h u dv g. (2.12) Proof. We note that the right hand side of (2.12) is finite since u C 1 0 (). Using the identity (2.11) and Hölder inequality we obtain u A h dv g u divh dv g u 1 h u dv g = u 1 A ( 1)/ h h u dv g This comletes the roof. ( A ( 1)/ h u A h dv g ) ( 1)/ ( h A 1 h u dv g ) 1/. Secializing the vector field h and the function A h, we shall deduce from (2.12) Hardytye inequalities on Riemannian manifolds. 8

9 Remark Letting us to oint out a strategy to get Hardy inequalities at least in some secial cases. Under the hyotheses of Lemma 2.10, if A h = divh, then (2.12) reads as u divh dv g h divh 1 u dv g. (2.13) i) Let V be a function in L 1 loc () such that its weak artial derivatives of order u to two are in L 1 loc (). If V 0, choosing h = V, we obtain A h = divh = div ( V) = V 0. Then from (2.13) u V dv g V V 1 u dv g. (2.14) This kind of inequalities for the Euclidean setting = R N are already found by Davies e Hinz in [DH]. At this oint, in order to deduce from (2.14) an inequality like c u ρ dv g u dv g, (2.15) we have to choose a suitable function V. Let us to consider the case when ρ is the distance from a oint o and ρ = 1. A suitable choice for V is V = ρ 2 if 1 < < 2, V = lnρ if = 2 and V = ρ 2 if 2 < < N. Following [Ca2], if we require that ρ γ ρ the above choices yield the inequality (2.15) with c = ( γ +1 The success of this strategy is deely linked to the hyothesis ρ = 1. Indeed, it seems that such a strategy does not work even in the subellitic setting, where the analogous of the hyothesis ρ = 1 does not hold. Furthermore, the fact that the hyothesis ρ = 1 is sometimes restrictive even in the Euclidean case, can be seen in the following examle. In the Euclidean unit ball B 1 R N the inequality c B 1 ). u x ln x dx u dx (2.16) B 1 holds for 1 < N (see Section 6.3, Section 6.6 and [Da3]). If we wish to deduce (2.16) from (2.15) we are forced to choose ρ = x ln x. However ρ 1. ii) Let < N. Assume that there exists a C 1 conformal Killing vector field K (see i.e. [B1] for the definition) such that divk = N K µ > 0. Choosing h = 2 K, we have A h := divh = N µ (see Lemma 3 in [B1]) and the inequality (2.13) reads as 2 K ( ) N divk N K u dv g (divk) 1 u dv g. (2.17) Therefore, by Lemma 2.10, (2.17) holds for every u C0 1 () rovided K L 1 loc (). This last fact was obtained in [B1, Theorem 4]. 9

10 Proof of Theorem 2.1. Let 0 < < 1, and ρ := ρ+. In order to aly Lemma 2.10, we define h and A h as h := ρ 2 ρ ρ 1 and A h := ( 1) ρ. (2.18) ρ Since 1 ρ 1, the fact that ρ W1, loc () imlies that h L1 loc () and A h L 1 loc (). oreover, by comutation we have h A 1 h = ρ ( 1) ρ ( 1) ( 1) ρ 1 ρ ( 1) ( 1) = 1 ( 1) ( 1) 1 L1 loc(), that is ii) of Lemma 2.10 is fulfilled. The hyothesis i) of Lemma 2.10 is satisfied rovided ρ ( 1) ρ ϕ dv g ( ρ 2 ρ ρ 1 ϕ ) dv g (2.19) holds for every nonnegative function ϕ C0 1 (). Then, for a fixed ϕ C1 0 () nonnegative, we have to rove (2.19). Let K = suϕ and let U be a neighborhood of K with comact closure in. We note that both integrals in (2.19) are finite since 1 ρ 1 and ρ W 1, loc (). Since lnρ = ρ ρ ρ L loc (), (2.20) and lnρ L loc (), we have that lnρ W 1, (U). Thus, for every n N there exists φ n C (U) such that φ n lnρ W 1, < 1/n, φ n lnρ ointwise a.e. and ln φ 3 n. Setting ψ n = e φn we have that ψ n C (U), ψ n, ψ n ρ a.e. and lnψ n lnρ dv g 0, ψ n ρ ψ n dv g 0, (as n 0+ ). (2.21) K 3 Reminding that the Sobolev sace W 1, () is the comletion of the set with resect to the norm K ρ { u C () : u dv g < and ( u W 1, = u dv g + } u dv g < u dv g ) 1/, the aroximation result follows by slight modification of classical arguments that the reader can find, for instance, in [Au]. 10

11 For every n N, the function ϕ n defined as ϕ n := ϕ belongs to C 1 ψn 1 0 () and it is nonnegative since ϕ C0 1 () is nonnegative and ψ n > 0. Using ϕ n as test function in (2.4) we have 0 ρ 2 ( ρ ϕ n )dv g = ( which, since by comutation ϕ ψ 1 n ) = ϕ ψ 1 n ρ 2 ρ ψ n ( 1) ψn ϕ dv g ( )) ϕ ρ ( ρ 2 ψn 1 dv g, (2.22) ( 1) ψn ψn ϕ, imlies ( ) ρ 2 ρ ψn 1 ϕ Now, letting n + we obtain by dominated convergence: ( ) ρ 2 ( ) ρ ρ 2 ρ ψn 1 ϕ dv g ρ 1 ϕ dv g, because ρ 2 ρ ϕ ψn 1 Indeed, C ρ 1 1 ρ 2 ρ ψ n ψ n and, since ρ 2 ρ ψn 1 have that ρ 2 ρ ψ 1 n ρ 1 1 L 1 (U). Now we claim that ρ 2 ρ ϕdv g = ψn 1 ρ 2 ρ ψ 1 n ρ 2 ρ ρ 1 ψ n ψ n ϕdv g ointwise a. e. dv g. (2.23) ρ ϕdv g. L (U), by Lebesgue dominated convergence theorem we ρ 2 ρ ρ 1 in L (U). From this and the fact that ψ n ρ in L (U) ψ n ρ we get the claim. Therefore, letting n + in (2.23), we have ρ ( ) ρ 2 ρ ( 1) ρ ϕ dv g ρ 1 ϕ dv g, which is exactly (2.19), since ρ = ρ. An alication of Lemma 2.10 gives ( ) 1 u ρ ρ dv g u dv g. (2.24) Finally, letting 0 in (2.24) and using Fatou s Lemma, we conclude the roof. 11 ρ

12 3 Further inequalities In this section we shall resent some slight but natural extensions of Theorem 2.1 and Lemma As byroducts of these generalizations we shall obtain Hardy inequalities with a weight in the right hand side, Cacciooli-tye inequalities, weighted Gagliardo- Nirenberg inequalities and the uncertain rincile. A first examle of a ossible generalization of Theorem 2.1 is the following: Theorem 3.1. Let α R, and let ρ W 1, loc () be a nonnegative function satisfying the following roerties: i) ( 1 α) ρ 0 on in weak sense, ii) ρ,ρ α L 1 ρ α loc (). Then the following Hardy inequality holds ( ) 1 α ρ α u ρ ρ dv g ρ α u dv g, u C 0 (). (3.25) The roof of the above theorem is similar to the one of Theorem 2.1 and it is based on a careful choice of the vector field h and of the function A h in Lemma Proof. Let 0 < < 1, and ρ := ρ +. In order to aly Lemma 2.10 we choose the vector field h and the function A h as h := ( 1 α) ρ 2 ρ ρ 1 α Arguing as in the roof of Theorem 2.1, we have to show that ( 1 α) 2 ρ ρ α ϕ dv g ( 1 α), A h := ( 1 α) 2 ρ. (3.26) ( ρ 2 ρ ρ 1 α ϕ ρ α ) dv g, (3.27) for every nonnegative function ϕ C 1 0 (). Let K := suϕ and let U be a neighborhood of K. Let k >, and define ρ k := inf{ρ,k}. Arguing as in the roof of Theorem 2.1, we have that there exists a sequence {ψ n } such that ψ n k, and lnψ n lnρ k dv g 0, ψ n ρ k ψ n dv g 0, (as n + ). (3.28) K Then we use ϕ n := ϕ ψ 1 α n ( 1 α) 2 K ρ k as test function in the hyothesis i), obtaining ρ 2 ρ ψ n ψ α n ( ρ 2 ρ ϕ dv g ( 1 α) ψn 1 α 12 ϕ ) dv g. (3.29)

13 In the case α < 1 we obtain (3.27) from (3.29) by slight modifications of the roof of Theorem 2.1, so we will omit the roof. Let α > 1. We claim that, letting n + in (3.29), and eventually taking a subsequence, we get ( 1 α) 2 ρ 2 ( ) ρ ρ k ρ 2 ρ ρ α ϕ dv g ( 1 α) k ρ 1 α ϕ dv g. k (3.30) In fact, for the right hand side the limit follows by dominated convergence, since ρ 2 ρ ψ 1 α n ϕ = ρ 2 ρ ϕ ψn α +1 C ρ 1 k α +1 L 1 (U). Dealing with the left hand side of (3.29), we set ρ 2 ρ ψ n ψ α n As in the roof of Theorem 2.1, we have = ρ 2 ρ ψ 1 n f n ρ 2 ρ ρ 1 k while from the relations g n ψ n ψ kα ρ k n ρ k ψ n ψ n ψ α n =: f n g n. in L (U), (3.31) kα in L (U) we obtain that the sequence g n is bounded in L (U). Therefore, u to a subsequence, g n is weakly convergent in L (U). Since g n ρ k ρ k ρ α k ointwise a. e., we have that the convergence is in the weak sense. This fact with (3.31) concludes the claim. Next ste is letting k + in (3.30). Let us rewrite the integrand in the right hand side as ρ 2 ρ ρ 1 α k ρ ϕ = 2 ρ(ρ k ) α (ρ k ) α +1 ϕ C ρ 1 (ρ ) α (ρ ) α, which is in L 1 (U), since C ρ 1 (ρ ) α L (U) and (ρ ) α L (U) by hyothesis ii). Thus we can use the dominated convergence to obtain the limit for the right hand side of (3.30). 13

14 In order to ass to the limit for k + in the left hand side of (3.30), we rewrite the integrand as ρ 2 ρ ρ k ρ α k ϕ = ρ 2 ρ ρ ρ α k χ {ρ k}ϕ = ρ ρ α k χ {ρ k}ϕ, (3.32) where we have used the fact that ρ = ρ. Now, if α we aly the dominated convergence, since the term in (3.32) is dominated by the function C ρ L 1 (U). α Whereas, if α > we use the monotone convergence, since ρ ρ α k χ { ρ k}ϕ is an increasing sequence of nonnegative functions. Thus, letting k + in (3.30), we get ( 1 α) 2 ρ ρ α ϕ dv g ( 1 α) ( ) ρ 2 ρ ρ 1 α ϕ dv g, (3.33) which is exactly (3.27), since ρ = ρ. As in Theorem 2.1, an alication of lemma 2.10 gives ( ) α +1 ρ α u ρ ρ dv g ρ α u dv g. (3.34) Finally, letting 0 in (3.34), we conclude the roof. Indeed we can use the dominated convergence for the right hand side, since ρ α u C(ρ+) α C(ρ+1) α L 1 loc (U), and aly the Fatou s Lemma for the left hand side. Remark 3.2. If α, then the hyothesis ii) in the above Theorem 3.1, can be avoided. Indeed, since ρ W 1, loc () we get that ρα L 1 loc () and from the roof it follows that ρ L 1 ρ α loc (). As a consequence of Theorem 3.1 (it suffices to take α = +q), we obtain the following Cacciooli-tye inequality for -subharmonic functions, which is worth of mention: Corollary 3.3. (L -Cacciooli-tye inequality) Let ρ L 1 loc () and q > 1. Assume that ρ is nonnegative on an oen set, and ρ W 1, loc (), ρq ρ,ρ +q L 1 loc (). If ρ 0 on in weak sense, then we have ( ) q +1 ρ q ρ u dv g ρ +q u dv g, u C0 (). (3.35) Notice that for q = 0 and = 2 the above theorem is a version of the classical Cacciooli inequality on manifolds. See also [PRS] for a version of Cacciooli inequality related to subharmonic functions on manifolds. Now we resent a ossible generalization of Lemma 2.10, and some of its consequences, like the weighted Gagliardo-Nirenberg inequality and the uncertain rincile on manifolds. 14

15 Lemma 3.4. Let h L 1 loc () be a vector field and let A h L 1 loc () be a nonnegative function such that i) A h divh, ii) h A 1 h L 1 loc (). Then for every u C 1 0 (), q R, s > 0 and a > 1 we have u s h q dv g /a ( rovided h q L 1 loc (). h A 1 h u dv g ) 1/a ( h qa A a 1 h In articular, setting w := h Ah, we have 1. ( ) 1/s ( ) b/ ( u s h q dv g q( 1)/s w u dv g where t, > 0 and 1 u as a 1 dvg ) 1/a, (3.36) w t u dv g ) (1 b)/, (3.37) 1 s = b + 1 b, 1 q = t, b = t( 1) 1+t( 1). 2. u s dv g /a ( where s > 0 and a > 1. w u dv g ) 1/a ( 1 A a 1 h u as a 1 dvg ) 1/a, (3.38) Proof. By Hölder inequality with exonent a we have u s h q dv g = u /a A 1/a h h q A 1/a h u s /a dv g ( ) 1/a ( u A h dv g h qa A a /a h u as a 1 dvg ) 1/a, which by using (2.12), imlies (3.36). From (3.36) we get (3.37) by choosing a = 1+, and (3.38) by choosing q = 0. t SecializinghandA h weobtainfrom(3.37)and(3.38)aweightedgagliardo-nirenberg inequality and an uncertain rincile resectively. In articular, choosing h and A h as in (2.18), we have the following 15

16 Theorem 3.5. Let ρ W 1, loc () be nonnegative. Assume that ρ is -suerharmonic function on and satisfies the hyotheses of Theorem 2.1. Let > 0 and 0 b 1. Then for every u C0 1 (), we have ( where u s ρ q( 1) ρ q( 1) dv g ) 1/s 1 s = b + 1 b, ( ) q( 1)/s ( ) b/ ( ) (1 b)/ u dv g u dv g, 1 (3.39) 1 q( 1) = b b. In articular, if ρ = d α for some α 0 with d = 1, then we have u s ( d dv 1 g α ( 1) where s = 1+. ) ( 1) ( ) 1/ ( ) 1/ u dv g u dv g, (3.40) Notice that for s = = 2 the inequality (3.40) is the weighted Gagliardo-Nirenberg inequality on manifold. Its counterart in Euclidean setting is largely studied by many authors, see for instance [DV]. Further examles of manifolds and functions ρ satisfying the hyotheses of the above theorem are given in Section 6. Theorem 3.6. Let ρ W 1, loc () be nonnegative. Assume that ρ is -suerharmonic function on and satisfies the hyotheses of Theorem 2.1. Let s > 0 and a > 1. Then for every u C 1 0 (), we have u s dv g ( ) /a ( ) ( 1/a u dv g u as a 1 1 ρ (a 1) ρ (a 1) dv g In articular, if ρ = d α for some α 0 with d = 1, then we have ( ) /a ( u s dv g α ( 1) u dv g ) 1/a ( ) 1/a. (3.41) u as a 1 d (a 1) dv g ) 1/a. (3.42) Notice that if a = s = = 2 the inequality (3.42) in the Euclidean setting coincides with the celebrated uncertain rincile with d = x, the Euclidean norm. Remark 3.7. Different choices of the vector field h and of the function A h in Lemma 3.4, roduce inequalities different than (3.39) (3.42). For instance, one can define h and A h as in (3.26), obtaining a version of (3.39) (3.42) with further weights. 16

17 To end this section, we want to oint out that it is ossible to extend all the results of this aer considering vector fields of the tye µ u := µ( u), where µ is a (1,1)-tensor (say C 1 ). In this case, relacing with µ, a Hardy-tye inequality like (2.5) holds rovided µ( µ u 2 µ u ) 0, where µ stands for the adjoint of µ. We leave the details to the interested reader. Notice that the study of Hardy inequalities for the vector field µ was already studied in [Da3], when the suort of the manifold is R N. 4 Remarks on the best constant Theorems 2.1 and 3.1 affirm the validity of some Hardy inequalities ( ) with an exlicit value ( ) 1 of the constants involved. In many cases these constants, and 1 α, result to ( ) 1 α be shar. For examle in [Da3] the author roves the sharness of the constant involved in the inequality of Theorem 3.1 in several cases. oreover the question of the existence of functions that realize the best constant is analyzed in many aers (for the Euclidean case see for instance [B, DH, P, S1, S2]). On the other hand, the knowledge of the best constants for the inequalities lays a crucial role in [AX, CX, Xi]. For the sake of simlicity, we shall focus our attention on the inequality (2.5). We denote by c() the best constant in (2.5), namely c() := inf u dv g. (4.43) u D 1, (),u 0 u ρ dv ρ g Then, we have the following: Theorem 4.1. Under the same hyotheses of Theorem 2.1 we have: ( ) 1. If ρ 1 D 1, 1 1 (), then c() = and ρ is a minimizer. ( ), 2. If ρ 1 / D 1, (), 2 and c() = then the best constant c() is not achieved. ( ) 1. 1 Proof. 1) From (2.5) we have c() oreover, if ρ D 1, (), by comutation ( ) ρ 1 1 ( dv g = ) ρ 1/ ρ dv g ( ) 1 ρ = dv g ρ ( ) 1 ρ 1 = ρ dv ρ g. 17 1

18 Thus, taking u = ρ 1, we obtain the infimum in (4.43). 2) Let u C 0 (). We define the functional I as I(u) := ( ) 1 u dv g u ρ ρ dv g. We note that the functional I is nonnegative, since (2.5) holds, and the best constant will be achieved if and only if I(u) = 0 for some u D 1, (). Let v be the new variable v := u with γ := 1. By comutation we have ρ γ u 2 = (vρ γ ) 2 = γ 2 v 2 ρ 2γ 2 ρ 2 +ρ 2γ v 2 +2γvρ 2γ 1 ρ v. (4.44) (If ρ is not smooth enough, we can consider ψ n as in the roof of Theorem 2.1 and after the comutation take the limit as n + ). We remind that the inequality (ξ η) s ξ s sηξ s 1 (4.45) holds for every ξ,η,s R, with ξ > 0, ξ > η and s 1 (see [GG]). Alying (4.45) and (4.44), with s = /2, ξ = γ 2 v 2 ρ 2γ 2 ρ 2 and η = 2γvρ 2γ 1 ρ v ρ 2γ v 2, we have u γ v ρ γ ρ + γ 2 γ v 2 v ρ 2 ρ v Then, taking into account that v = u ρ γ, we have I(u) = u dv g γ + 2 γ 2 v 2 ρ ρ 2 v 2. u ρ ρ dv g (4.46) γ 2 γ v 2 v ρ 2 ( ρ v)dv g + 2 γ 2 v 2 ρ ρ 2 v 2 dv g =: I 1 (v)+i 2 (v). (4.47) Re-arranging the exression in I 1 (v) and integrating by arts we obtain I 1 (v) = = ( ) 1 1 ( ) ( v ) ρ 2 ρ dv g v ( ρ)dv g 0,

19 where we have used the hyothesis ρ 0. On the other hand we can rewrite I 2 (v) as I 2 (v) = 2 γ 2 ρ ρ 2 /2 2 v dvg. (4.48) Thus, we conclude that for every u D 1, () I (u) 2 γ 2 ρ ρ 2 2 /2 v dvg > 0, and this inequality imlies the non existence of minimizers in D 1, (). We end this section by showing a further result that arises from the fact that the best constant, in some cases, is not achieved. Indeed, if the best constant involved in an inequality is not achieved, it is natural to ask if a reminder term can be added. The next result shows that in the inequality (2.5) one can add a reminder term. Theorem 4.2. Let = 2 and let ρ be as Theorem 2.1. We define Λ 1 := inf ρ u 2 dv g. u C0 1() ρ u 2 dv g Assume that Λ 1 > 0. Then u 2 dv g 1 4 u 2 ρ 2 ρ 2 dv g +Λ 1 u 2 dv g, u C 1 0 (). (4.49) Proof. We shall give a sketch of the roof since it is similar to the roof of Theorem 4.1. By using the same notation of the roof of Theorem 4.1, from (4.44), we deduce that I(u) ρ v 2 dv g Λ 1 ρ v 2 dv g = Λ 1 u 2 dv g, where we have used the fact that ρ 0, the definition of Λ 1 and v = u/ρ 1/2. This concludes the roof. An examle of manifold where Theorem 4.2 alies is the following. Let ρ be a nonnegative suerharmonic function on R N and let R N a bounded oen set. Then ρ belongs to the uckenhout class A 1 and this imlies that Λ 1 > 0 (indeed it suffices to combine Theorems 3.59 and of [HK]). In articular, with the choice ρ := x 2 N, N > 2 and R N a bounded oen set, (4.49) reads as u 2 dx (N 2)2 4 u 2 x 2 dx+λ 1 u 2 dx, u C 1 0 (), which is the celebrated inequality roved in [BV]. See also [Da2, GG] for related results in Euclidean and subellitic setting for > 1 and for further references. 19

20 5 First order interolation inequalities In this section we shall study some inequalities of Hardy-Sobolev tye. As already said above, interolation inequalities as well as the knowledge of an estimate of the best constant have an imortant role in several areas of mathematical science. Thus we shall address some efforts to kee track of exlicit values of the involved constants. We shall assume that the Sobolev inequality ( ) 1/ ( ) 1/ S() u dv g u dv g, u C0 () holds for some > 0, and the Hardy inequality H(α,) ρ α u ρ ρ dv g ρ α u dv g, u C 0 () (H α) holds for an exonent α R. In some cases, the validity of (S) imlies that (H α ) holds as well. Indeed, let N > 2 and let be a N-dimensional comlete and connected Riemannian manifold with infinite volume, if (S) holds with = 2 and = 2N/(N 2) then is hyerbolic (see [Ca1]). In this case, from Theorem 2.6 we have that a Hardy inequality holds. Therefore, there exists a nonnegative non constant suerharmonic function ρ ad hence (H α ) holds with = 2 and α < 1 (see Theorem 3.1 and Remark 3.2). In order to state our main result of this section, we need the following reliminary theorem: Theorem 5.1. Assume that (S) holds on. Let θ R and ρ 0 be a function such that (H α ) holds with α = θ. Then there exists C 2 > 0 such that oreover C 2 ( ) 1/ ( ρ θ u dv g (S) ρ θ u dv g ) 1/, u C 0 (). (5.50) H(θ,) 1/ C 2 = S() θ +H(θ,) 1/. In articular, if ρ L 1 loc () is a nonnegative function satisfying the hyotheses of Theorem 3.1 with α = θ and (S) holds, then we obtain (5.50) with 1 θ C 2 = S() θ + 1 θ 20

21 Proof. The case θ = 0 corresonds to the Sobolev inequality. Let θ 0. Let u C 0 () and define v as v := ρ θ u. By comutation we have v = ( ρ θ u ) = ρ θ u+θρ θ 1 u ρ ( ρ θ u + θ ρ θ 1 u ρ ) ( = ρ θ u + θ ) ρ θ 1 u ρ (5.51) H 1/H1/ where, for sake of brevity, H = H(θ,) and S = S(). By using the inequality (a+ 1 ǫ b) ǫ 1 a + 1 ǫ b (0 < ǫ < 1, a,b > 0) ǫ ǫ with ǫ := H1/ H 1/ + θ, a := ρθ u and b := H 1/ ρ θ 1 u ρ, we have ( v ǫ 1 ρ θ u + 1 ǫ ǫ Hρ θ u ρ. (5.52) Then, by (S) and using (H α ) with α = θ, we obtain ) / ρ θ u dv g ( ) / = v dv g S v (5.53) [ S ǫ 1 ρ θ u dv g + 1 ǫ ] H ρ θ u ǫ ρ ρ dv g [ S ǫ ǫ ] ρ θ u dv ǫ g = (Sǫ) ρ θ u dv g, which concludes the roof. Theorem 5.2. Assume that (S) holds on with >. Let θ R and ρ 0 be a function such that (H α ) holds with α = θ. Let r > 0, 0 a 1, γ, ǫ,σ and be real numbers satisfying the following relations 1 1 r 1 a + a, (5.54) and γ + (r ) r( ) = (1 θ)a+(1 a) (5.55) ǫ = θa+σ(1 a). (5.56) 21

22 Then there exists C 3 > 0 such that C 3 ( u r ) 1/r ρ γr ρ (γ+ǫ)r dv g ( ) ρ θ u dv g)a/( u ρ (+σ) (1 a)/ dv ρ g, u C 0 (), (5.57) that is rovided ρ σ L 1 ρ loc (). oreover C 3 u ρ γ+ǫ ρ γ L r ρ θ u a L u ρ +σ ρ (r ) r( C 3 = C ) 2 H(θ,) a (r ) r( ). 1 a, (5.58) L In articular, if ρ L 1 loc () is a nonnegative function satisfying the hyotheses of Theorem 3.1 with α = θ and (S) holds, then we obtain (5.57) with (r ) r( C 3 = C ) 2 ( 1 θ ) a (r ) r( ). Proof. From condition (5.54) it follows that r. We shall distinguish tree cases. Case: r =. From (5.54) necessarily we have a = 1 and hence from (5.55) and (5.56) ǫ = θ = γ. The inequality to rove is actually the thesis of Theorem 5.1. Case: r =. If a = 0 there is nothing to rove. If a = 1, then the thesis is the the inequality (H α ). Let 0 < a < 1. By using (5.55) and (5.56) we have u r u a u (1 a) ρ γr ρ (γ+ǫ)r dv g = ρ (1 θ)a ρ a ρ (1 a) ρ (+σ) dv g. Now the claim follows alying Hölder inequality with exonent 1/a and then Hardy inequality (H α ). Case: > r >. Let q R be a arameter that we shall fix later. Using Hölder inequality with exonent s > 1 we obtain ( u r ρ γr ρ (γ+ǫ)r dv g = ) ( 1/s u (r q)s ρ θ dv g u r q ρ θ/s u q ρ γr+ θ/s ρ (γ+ǫ)r dv g u qs ρ (γr+ θ/s)s ρ (γ+ǫ)rs dv g ) 1/s. (5.59) 22

23 Now we aly Hölder inequality with exonent t > 1 to the second term of (5.59) and obtain u qs ρ (γ+ǫ)rs dv ρ (γr+ θ/s)s g = ( u qs /t ρ qs /t u qs ρ qs ρ qs ρ θ dv g)1/t( Now, requiring that the following conditions are satisfied we get /t ρ qs /t ρ (γ+ǫ)rs qs /t θ/t u qs ρ dv g ρ (γr+ θ/s)s qs /t+θ/t ) 1/t u qs ρ (γ+ǫ)rs t qs t /t ρ (γr+ θ/s)s t qs t /t+θt /t dv g.(5.60) qs =, (r q)s =, (5.61) s = r since (5.54) holds. Using (5.61), by (5.59) and (5.60) we have ( > 1, (5.62) u r ( ) 1/s ρ γr ρ (γ+ǫ)r dv g u ρ θ dv g u ) ( 1/s t ) ρ ρ ρ θ u ρ (γ+ǫ)rs t t /t 1/s t dv g ρ dv (γr+ θ/s)s t t /t+θt /t g C /s 2 H(θ,) 1/s t ( ( ρ θ u dv g ) /s+1/s t ) u ρ (γ+ǫ)rs t t /t 1/s t ρ dv (γr+ θ/s)s t t /t+θt /t g, (5.63) where, in the last inequality, we have used (5.50) and and (H α ) with α = θ. To conclude we have to choose t > 1 such that and (γ +ǫ)rs t t /t = ( +σ), (γr + θ/s)s t t /t+θt /t =, (5.64) /s+1/s t = ar/, 1/s t = (1 a)r/. (5.65) s s (asr ) Firstofall,notethatin(5.60)wecanmakethechoicet = > 1,since (5.54)holds. Using the exressions of t and s, equalities(5.65) follow by simle comutations. oreover, from (5.65) we obtain that s t = t and = asr. Using this two exressions and r(1 a) t sr(1 a) the conditions (5.55) and (5.56) we get also (5.64). This concludes the roof. 23

24 Remark 5.3. The condition (5.56) takes into account the resence of the ρ in the weights aearing in(5.57) and it is also a necessary condition. Indeed, to see the necessity of (5.56) we argue as follows. Assume that Theorem 5.2 were true. If (S) and (H α ) hold with a function ρ, then those inequalities still hold with the function λρ for every λ > 0, and hence the conclusion of Theorem 5.2 holds relacing ρ with λρ. By homogeneous consideration one derives the necessity of (5.56). Remark 5.4. Since the condition (5.56) is a requirement on the arameters ǫ and σ, if ρ = 1, these arameters do not aear in the inequality (5.57). Therefore, condition (5.56) is always fulfilled (i.e. choosing ǫ = aθ and σ = 0). The next corollary deals with a generalization of this case. Corollary 5.5. Assume that (S) holds on with >. Let θ R and ρ 0 be a function such that (H α ) holds with α = θ and ρ = d β with β R and d = 1. Let r > 0, 0 a 1 and γ, be real numbers satisfying (5.54) and γ + (r ) r( ) Then there exists C 3 > 0 such that C 3 ( that is = (1 βθ)a+(1 a). (5.66) u r ) 1/r ( ) a/ ( d dv γr g d βθ u u ) (1 a)/ dv g d dv g, u C 0 (),(5.67) C 3 u ρ γ L r rovided 1 ρ L 1 loc (). In articular, C 3 = C 3β γ+βθa (1 a). ρ βθ u a 1 a u L ρ, (5.68) L Remark 5.6. Notice that, if in the revious corollary we take = N, condition (5.66) N becomes 1 r γ N = 1 + a N (βθ 1) (1 a), (5.69) N and (5.67) is a articular case on manifold of a result obtained by Caffarelli, Kohn and Nirenberg in [CKN] in Euclidean setting. 6 Some Alications In what follows we aly the results roved in the above sections to concrete cases, obtaining Hardy inequalities which in some cases are new. For sake of brevity we shall limit ourselves to show some alications of Theorems 2.1 and 3.1 by secializing the 24

25 function ρ. With the same technique it is ossible to obtain alications of the other theorems resented in the revious sections (Cacciooli inequality, uncertain rincile, Gagliardo-Nirenberg inequality, first order interolation inequalities, and so on). We leave the details to the interested reader. 6.1 Hardy inequality involving the distance from the boundary In order to rove a Hardy inequality involving the distance from the boundary, we need the following result, which is an immediate consequence of Theorems 2.1 and 3.1. Theorem 6.1. Let be a comact Riemannian manifold with boundary of class C, let ϕ 1 be the first eigenfunction related to the first eigenvalue of the -Lalacian 4, and α < 1. If ϕ 1 > 0, then the following inequality holds on ( ) 1 α ϕ α 1 In articular we have ( ) 1 u ϕ 1 u ϕ 1 ϕ 1 dv g ϕ 1 dv g The main theorem of this section is the following: ϕ α 1 u dv g, u C 0 u dv g, u C 0 (). (6.71) (). (6.72) Theorem 6.2. Let be a comact Riemannian manifold with boundary of class C, let ϕ 1 be the first eigenfunction related to the first eigenvalue of the -Lalacian. Assume that ϕ 1 C 1( ), ϕ 1 > 0 on and ϕ 1 0 on. Denoted by d(x) := dist(x, ), there exists a constant c > 0 such that c u d dv g u dv g, u C 0 (). (6.73) Theroofoftheabovetheoremreliesonthefollowingresult,whichisworthofmention: Theorem 6.3. Let be a comact Riemannian manifold with boundary of class C, let ϕ 1 > 0 be the first eigenfunction related to the first eigenvalue λ 1 of the -Lalacian, and 0 < s < 1. Then the following inequality holds λ 1 ( 1 s) ( 1) ϕ s 1 u dv g ϕ s 1 u dv g, u C 0 4 This means that (λ 1,ϕ 1 ) is a solution of the roblem { ϕ = λ ϕ 2 ϕ on, ϕ = 0 on, and λ 1 := min{λ : (λ,ϕ) solves (6.70) }. 25 (). (6.74) (6.70)

26 Proof. Set φ := ϕ s/( 1) 1. By comutation we have φ = φ φ ( ) 1 ( 1 s) s ( ) 1 s +λ 1 φ 1. 1 Choosing h := φ 2 φ and A h := φ, an alication of Lemma 2.10 yields ( ) 1 s λ 1 φ 1 u dv g ( φ) u dv g 1 ) 1 ( ( s 1 This last chain of inequalities concludes the roof. 1 1 s φ ( 1) ( φ) 1 u dv g ) 1 φ 1 u dv g. Proof of Theorem 6.2. For a fixed number γ > 0, we denote by γ and γ resectively the sets γ := {x : d(x) < γ} and γ := {x : d(x) γ}. Let ϕ 1 be such that ϕ 1 1. By continuity argument, we have that there exist ǫ,b > 0 such that ϕ 1 (x) b > 0, for x ǫ. Since ϕ 1 is a Lischitz continuous function, we obtain that there exist L > 0 such that We set ϕ 1 (x) Ld(x), x. l ǫ := min ǫ ϕ 1. From (6.72) and (6.74) we get resectively ( ) 1 ( ) u u 1 dv g ϕ 1 b u dv g L ǫ d dv g, ǫ ϕ 1 u ( 1 s) ( 1) ( 1) dv g λ 1 ϕ s ( 1 s) 1 u dv g λ 1 l s u ǫ ǫ ǫ ǫ d dv g. {( ) } 1 Choosing 2c := min b ( 1 s),λ ( 1) L 1 l s ǫǫ and summing u the above estimates we obtain the claim. 6.2 Hardy inequality for -hyerbolic manifold In this section we establish Hardy inequalities involving the Green function of the oerator. The case = 2 is already roved in [LW]. 26

27 Examles of -hyerbolic manifolds are the following. The Euclidean sace R N is -hyerbolic for N >. If is a Cartan-Hadamard manifold (see Section 6.3) whose sectional curvature K is uniformly negative, that is K a 2 < 0, then is - hyerbolic for any > 1 (see [Ho1] and [HP]). We have the following. Theorem 6.4. Let (,g) be a -hyerbolic manifold, let G x be the Green function for with ole at x, and α R. Then the following inequality holds ( ) 1 α G α G x x \{x} G u dv g G α x u dv g, u C 0 ( \{x}). x \{x} (6.75) In articular, we have ( ) 1 G x u dv g u dv g, u C0 ( \{x}). (6.76) \{x} G x \{x} and, if < N, ( ) 1 G x u dv g G x u dv g, u C 0 (). (6.77) Proof. We know that G x W 1, loc () is a nonnegative function on. oreover the hyotheses of Theorem 3.1 are fulfilled; in fact i) G x = 0 in \{x} ii) Gx,G α G α x L1 loc ( \{x}). x Then, by Theorem 3.1, inequality (6.75) holds. In articular, taking α = 0, we obtain the inequality (6.76). oreover, if < N, we are in the osition to aly Corollary 2.3, because {x} is a set of zero -caacity (see Theorem 2.27 in [HK]), and then we can use Proosition A-1; this roves that also (6.77) holds. 6.3 Hardy inequality on Cartan-Hadamard manifold In what follows (, g) will denote a Cartan-Hadamard manifold, that is, a connected, simly connected, comlete Riemannian manifold of dimension N 2, of nonositive sectional curvature (see [Ca2, Ho1, LW] for further details). Examles of Cartan-Hadamard manifolds are the Euclidean sace R N with the usual metric (which has constant sectional curvature equal to zero), and the standard N-dimensional hyerbolic sace H N (which has constant sectional curvature equal to 1). Let o be a fixed oint and denote by r the distance function from o. We have the following. 27

28 Theorem 6.5. Let (,g) be a Cartan-Hadamard manifold and α R. If (N )( 1 α) > 0, we have ( ) (N )( 1 α) r α N u 1 ( 1) \{o} r dv g r α N 1 u dv g, \{o} u C0 ( \{o}). (6.78) If 1 < < N, we have ( ) N u r dv g u dv g, u C0 (). (6.79) If 1 < N, setting := r 1 ([0,1[), we have ( ) 1 u rlnr dv g u dv g, u C0 (). (6.80) Let 1 < N. If 1 > α we set := r 1 (]0,1[), else if 1 < α we set := r 1 (]1,+ [). We have ( ) 1 α lnr α u rlnr dv g lnr α u dv g, u C 0 (). (6.81) Proof. In \{o} we define ρ = r β with β R that will be chosen later. The function ρ W 1, loc ( \{o}) is nonnegative on \{o}. The function r satisfies the relations (see [Ca2]). By comutation we obtain and hence r = 1, ρ = div ( ρ 2 ρ ) r N 1 r = div ( β 2 r (β 1)( 2) βr β 1 r ) = β 2 βdiv ( r (β 1)( 1) r ) = β 2 β [ (β 1)( 1)r (β 1)( 1) 1 +r (β 1)( 1) r ] = β 2 βr (β 1)( 1) 1 [(β 1)( 1)+r r], (6.82) ( 1 α) ρ = ( 1 α) β 2 βr (β 1)( 1) 1 [(β 1)( 1)+r r]. (6.83) 28

29 Next choosing β = N, using (6.82), we have that (β 1)( 1) + r r 0, and 1 then ( 1 α) ρ 0. That is the hyothesis i) of Theorem 3.1 is fulfilled. Since ρ = β 1 L 1 ρ α r α loc ( \{o}) and ρα = r βα L 1 loc ( \{o}), we are in a osition to aly Theorem 3.1, obtaining the inequality (6.78). In articular, taking α = 0 and using Corollary 2.3 we get (6.79). Indeed, since N, {o} is a set of zero -caacity (see Theorem 2.27 in [HK]), and then we can use Proosition A-1. Next we rove (6.81). To this end, by choosing ρ := (α + 1)lnr, we have that ρ > 0 in (according to the different cases α > (<) 1). By comutation we have ( 1 α) ρ = (α +1) div ( ρ 2 ρ ) = (α +1) div(r 1 r) ( 1 = (α +1) + r r ) r r ( ) N (α +1) 0. r The claim follows alying Theorem 3.1. We conclude the roof by roving (6.80). Choosing α = 0 in (6.81), we have that inequality (6.80) holds for every C0 (\{o}). However in this case {o} is a set of zero -caacity and, alying Corollary 2.3, we comlete the roof. Inequality (6.79) is resent in [Ca2] for = 2. In [KO] the authors rove (6.78) for = 2 and for a secial case of manifold, namely, when is the unit ball modeling the standard hyerbolic sace H N. For this case the authors rove that the constant in (6.78) is shar and they show that a remainder term can be added. 6.4 Hardy inequalities involving the distance from the soul of a manifold Let (,g) be a comlete non comact Riemannian manifold, of dimension N 2, with nonnegative sectional curvatures. A result due to Cheeger and Gromoll asserts that there exists a comact embedded totally convex submanifold S with emty boundary, whose normal bundle is diffeomorhic to (see [CG]). The submanifold S, called soul of, is not necessarily unique but every two souls of are isometric. Totally convex means that any geodesic arc in connecting two oints in S (which may coincide) lies entirely in S. In articular, S is connected, totally geodesic in, and has nonnegative sectional curvature. oreover 0 dims < dim. Denote by r : \ S R the distance function to S. We have that r is smooth on \S and r = 1 on \S. Now we suose that radial sectional curvature K r, that 29

30 is sectional curvature of two-lanes containing the direction r, satisfies where c N = N 2 ; then we have N where s = dims (see [EF]). We have the following: 0 K r c N (1 c N ) r 2, (6.84) r c N (N s 1), (6.85) r Theorem 6.6. Let (, g) be a Riemannian manifold with nonnegative curvature. Suose that (6.84) is fulfilled. Let G := c N (N s 1) +1. If G ( 1 α) > 0, we have ( ) G ( 1 α) ( 1) \S r α G 1 u r dv g \S r α G 1 u dv g, u C 0 ( \S). (6.86) oreover, if G > 0, we have ( ) G u r dv g u dv g, u C0 (). (6.87) Proof. Let ρ = r β in \ S with β = G. Arguing as in the roof of Theorem 1 6.5, using (6.83) and (6.85), we obtain ( 1 α) ρ 0. Since ρ W 1, loc ( \ S), ρ = β 1 L 1 ρ α r α loc ( \S) and ρα = r βα L 1 loc ( \S), the hyotheses of Theorem 3.1 are fulfilled, and (6.86) follows. In articular, if α = 0, (6.86) becomes ( G ) \S u r dv g \S u dv g, u C 0 ( \S). Now, the hyothesis G > 0 imlies that N > s. In fact, by the fact that G = c N (N s 1) +1 = N 2 (N s 1) +1 > 0, by simle comutations we get N (N 2)(N s 1) N (+1) = (N 2)(N s) N = N (N s) 2(N s) N = N (N s ) 2(N s) > 0, which imlies N s > 2 N s > 0. Then S is a set of zero -caacity (see Theorem N 2.27 in [HK]), and we can use Proosition A-1 and Corollary 2.3 to obtain inequality (6.87). 30

31 6.5 Hardy-Poincaré inequality for the hyerbolic lane Let C + = {z = x+iy : Imz = y > 0} betheuer half-laneequied withthe Poincaré metric ds 2 = dx2 +dy 2. This sace is a Riemannian manifold modeling the two dimensional y 2 hyerbolic sace. In this case, the gradient H, the divergence div H, the Lalacian H and the volume dv g related to the metric are resectively the following H u = y E u, div H = y 2 div E, H u = y 2 E u, dv g = dxdy y 2, (6.88) where we have denoted with E, div E, E the related oerator in the Euclidean setting, and dxdy is the Lebesgue measure in R 2. By using Theorem 3.1 with = 2, we deduce a Hardy inequality on the uer halflane. Theorem 6.7. Let α R. For every u C0 (C + ) we have (1 α) 2 y α u 2dxdy y α 4 C + y 2 H u 2dxdy, u C C + y 2 0 (C +). Proof. We consider the function ρ(z) = y, where z = x + iy. Clearly, ρ belongs to W 1,2 loc (C +), and ρ α = y α belongs to L 1 loc (C +). oreover, from (6.88), we have that and H ρ 2 ρ 2 α = y2 E ρ 2 y 2 α = y α L 1 loc(c + ), H ρ = y 2 E ρ = 0. Therefore, the hyotheses of Theorem 3.1 are satisfied and this concludes the roof. 6.6 The Euclidean case In this last section we show that our main results, Theorems 2.1 and 3.1, yield some well known shar Hardy inequalities in the Euclidean sace. Since R N is -hyerbolic for N > and it is a Cartan-Hadamard manifold, Theorems 6.4 and 6.5 hold also on R N with G x = x N 1 and r = x resectively. However, the function x N 1 is -harmonic in R N \{0} also for > N. Therefore we have the following: 31

32 ( Theorem 6.8. Let N. For any u C0 R N \{0} ) we have ( ) N 1 α x α n u 1 ( 1) x dx x α n 1 u dx. In the half sace R N + R N \{0} there holds the following: R N \{0} Theorem 6.9. Let α R, let N 2, let R N + = { (x 1,...,x N ) R N : x 1 > 0 }, and let ρ(x) := d ( x, R+) N be the distance from the boundary of R N +. Then we have ( ) 1 α ρ α u R N ρ dx ρ α u ( ) dx, u C 0 R N +. (6.89) + R N + ( ) Proof. The distance ρ(x) = x 1 W 1, loc R N + is nonnegative on R N + and it is easy to verify that the hyotheses of Theorem 3.1 are satisfied. Therefore the thesis follows. From Theorem 6.9, we can deduce, as a articular case, a well-known Hardy inequality for the uer half-lane C + = {z = x+iy : Imz = y > 0} (see for instance [He] and references therein). Corollary Let α R. Then, for every u C 0 (C +), we have the following: ( ) 1 α u da(z) ( α 2/2 (Imz) α u(z) 2 + u(z) 2) /2 da(z), C + (Imz) C + (6.90) where da(z) := dxdy, and, are the Wirtinger oerators, that is π := 1 ( 2 x i ), := 1 ( y 2 x +i ). (6.91) y Actually a more general theorem for convex domains holds. Theorem Let D be a roer convex oen subset of R N, let d := dist(, D) be the distance from D and let α < 1. Then we have ( ) 1 α d α u d dx d α u dx, u C 0 (D), and, in articular, ( 1 ) D D D u d dx u dx, u C 0 (D). D Proof. The thesis will follow by alying Theorems 2.1 and 3.1. To this end it suffices to rove that the function d(x) := dist(x, D) is -suerharmonic. Indeed, D = Π, and d(x) = inf Π dist(x, Π) where the intersection and the infimum are taken over all the half-saces Π containing D. Since dist(x, Π) is continuous and -harmonic, we have that d is -suerharmonic (see [HK]). This concludes the roof. 32

33 A Aendix Let us recall that -caacity of a comact set K is defined as { ca (K,) = inf u dv g : u C0 (), 0 u 1, u = 1 in a neighborhood of K }. (A-1) Proosition A-1. Let be a -hyerbolic manifold of dimension N. Let K be a comact set of zero -caacity. Then D 1, () D 1, ( \K), that is every function u D 1, () can be aroximated by function C 0 norm D 1,. ( \K) in the Proof. Let ϕ C 0 (). In order to rove the claim it is sufficient to rove that ϕ D 1, ( \K). Since ca (K,) = 0, there exists a sequence (u j ) j 1 such that, for any j 1, u j C0 (), 0 u j 1, u j = 1 in a neighborhood of K and u j 0 in D 1, (). For every j 1 the function ϕ j := (1 u j )ϕ belongs to C 0 ( \K) D1, ( \K). We shall rove that ϕ j ϕ in D 1, ( \K), that is ϕ j ϕ dv g 0, (as j + ). (A-2) \K In fact, we have ( ) 1/ ϕ j ϕ dv g = \K ( \K ( \K ϕ(1 u j ) ϕ u j ϕ dv g ) 1/ ) 1/ ( ϕ u j dv g + \K ϕ u j dv g ) 1/. (A-3) The second term in (A-3) converges to 0 for j +. Indeed, since u j 0 in D 1, (), we obtain ϕ u j dv g ϕ u j dv g \K ϕ u j dv g 0, (as j + ). It remains to rove that the first term in (A-3) converges to 0 as well. Let D be the suort of ϕ; then we get ϕ u j dv g ϕ u j dv g = ϕ u j dv g \K D ϕ u j dv g ϕ C u j dv g 0, (as j + ), D 33