Isoperimetric Hardy Type and Poincaré Inequalities on Metric Spaces

Transcription

1 Isoperimeric Hardy Type and Poincaré Inequaliies on Meric Spaces Joaquim Marín and Mario Milman Absrac We give a general consrucion of manifolds for which Hardy ype operaors characerize Poincaré inequaliies. We also show a class of spaces where his propery fails. As an applicaion, we exend recen resuls of E. Milman o our seing. 1 Inroducion While working on sharp Sobolev Poincaré inequaliies in he classical Euclidean seing (cf. [17]) as well as he Gaussian seing (cf. [14]), we observed ha he symmerizaion mehods we were developing could be readily exended o he more general seing of meric spaces (cf. [6, 14, 15, 16]). However, in he meric seing we found ha we could no always decide if he resuls we had obained were sharp or bes possible. Indeed, generally speaking, he mehods ha we use o show sharpness require he consrucion of special rearrangemens and hus our spaces need o exhibi sufficien symmeries. In fac, in all he examples where we know how o prove sharpness, he winning rearrangemens are hose ha are somehow conneced wih he soluion of he underlying isoperimeric problems (for example, he symmeric decreasing rearrangemens in he Euclidean case, which are associaed wih balls (cf. [17]), while in he Gaussian case one uses special rearrangemens associaed wih half spaces (cf. [8, 14]) and, likewise, Joaquim Marín Deparmen of Mahemaics, Universia Auònoma de Barcelona, Bellaerra, Barcelona, Spain jmarin@ma.uab.ca Mario Milman Florida Alanic Universiy, Boca Raon, Fl USA exrapol@bellsouh.ne 285

2 286 J. Marín and M. Milman in he more general model cases of log concave measures (cf. [5, 7] and he more recen [1, 15, 16]). In paricular, hese special rearrangemens allow us o show ha here exis special symmerizaions ha do no increase he norm of he gradien, i.e., ha a suiable version of he Pólya Szegö principle holds. In preparaion for a sysemaic sudy we observed ha, in all he model cases we could rea, a key role was played by he boundedness of cerain Hardy operaors, which we ermed isoperimeric Hardy operaors. This led us o isolae he concep of isoperimeric Hardy ype spaces. This propery can be formulaed in very general meric spaces and can be applied if we have esimaes on he isoperimeric profiles. By formulaing he problem in his fashion, while we may lose informaion abou bes consans, we gain he possibiliy of obaining posiive resuls ha would be hard o obain by oher mehods. In his noe, we coninue his program and we address he quesion: which meric spaces are of Hardy isoperimeric ype? On he posiive side we show, using ideas of Ros [26], how o consruc a class of meric spaces of Hardy isoperimeric ype ha conains all he model cases menioned above. Therefore his consrucion provides us wih a large class of spaces where our inequaliies are sharp. As anoher applicaion we coninue he discussion of he connecion beween our resuls and he recen work of E. Milman [23, 22, 24]), who has shown he equivalence, under convexiy assumpions, of cerain esimaes for isoperimeric profiles. In [16], we exended and simplified E. Milman s resuls o he seing of meric spaces of isoperimeric Hardy ype. The consrucion presened in his paper hus gives a general concree class of model spaces where Milman s equivalences hold. Finally on he negaive side we also consruc spaces ha do no saisfy he isoperimeric Hardy ype condiion. In he spiri of his book, we now commen briefly on he influence of Maz ya s work in our developmen. Underlying he equivalences of Theorem 2.1 below are wo deep insighs due o Maz ya: Maz ya s fundamenal resul showing he equivalence beween he Gagliardo Nirenberg inequaliy and he isoperimeric inequaliy (cf. [18] and also [9]), and Maz ya s echnique of showing self improvemen of Sobolev s inequaliy via smooh cu-offs (cf. [20]). Indeed, one of he hemes of Theorem 2.1 is o develop he explici connecion of hese wo ideas using poinwise symmerizaion inequaliies ( symmerizaion by runcaion cf. [17, 14]). Anoher heme of our mehod is ha we formulae our inequaliies incorporaing direcly geomeric informaion, an idea ha one can also already find in Maz ya s fundamenal work characerizing Sobolev inequaliies in rough domains [20] as well as in Maz ya s mehod characerizing Sobolev Poincaré inequaliies via isocapaciory inequaliies 1 (cf. [21]). 1 A deailed discussion of he connecion beween Maz ya s isocapaciory inequaliies and symmerizaion inequaliies is given in [16].

3 Hardy ype and Poincaré Inequaliies 287 As his ouline shows, and we hope he res of he paper proves, our mehods owe a grea deal o he pioneering work of Professor Maz ya and we are graeful and honored by he opporuniy o conribue o his book. 2 Background Le (Ω, d, µ) be a meric probabiliy space equipped wih a separable Borel probabiliy measure µ. Le A Ω be a Borel se. Then he boundary measure or Minkowski conen of A is by definiion µ + (A) = lim inf h 0 µ (A h ) µ (A), h where A h = {x Ω : d(x, y) < h} denoes he h-neighborhood of A. The isoperimeric profile I (Ω,d,µ) is defined as he poinwise maximal funcion I (Ω,d,µ) : [0, 1] [0, ) such ha holds for all Borel ses A. µ + (A) I (Ω,d,µ) (µ(a)) Condiion. We assume hroughou ha our meric spaces have isoperimeric profile funcions I (Ω,d,µ) which are: coninuous, concave, increasing on (0, 1/2), symmeric abou he poin 1/2, and vanish a zero 2. A coninuous funcion I : [0, 1] [0, ), wih I(0) = 0, concave, increasing on (0, 1/2) and symmeric abou he poin 1/2, and such ha I I (Ω,d,µ), is called an isoperimeric esimaor on (Ω, d, µ). For measurable funcions u : Ω R he disribuion funcion of u is given by λ u () = µ{x Ω : u(x) > } ( > 0). The decreasing rearrangemen u of u is defined, as usual, by u µ(s) = inf{ 0 : λ u () s} ( (0, µ(ω)]), and we se u µ () = 1 0 u µ(s)ds. Given a locally Lipschiz real funcion, f defined on (Ω, d) (we wrie in wha follows f Lip(Ω)), he modulus of he gradien of f is defined by 2 For a large class of examples, where hese assumpions are saisfied we refer o [6, 23] and he references herein.

4 288 J. Marín and M. Milman f(x) f(y) f(x) = lim sup d(x,y) 0 d(x, y) and zero a isolaed poins 3. A Banach funcion space X = X(Ω) on (Ω, d, µ) is called a rearrangemeninvarian (r.i.) space if g X implies ha all µ-measurable funcions f wih he same rearrangemen funcion wih respec o he measure µ, i.e., such ha f µ = g µ, also belong o X; moreover, f X = g X. An r.i. space X(Ω) can be represened by an r.i. space X = X(0, 1) on he inerval (0, 1), wih Lebesgue measure, such ha f X = f µ X, for every f X. Typical examples of r.i. spaces are he L p -spaces, Lorenz spaces, and Orlicz spaces. For more informaion we refer o [4]. In our recen work on symmerizaion of Sobolev inequaliies, we showed he following general heorem (cf. [15, 16] and he references herein) Theorem 2.1. Le I : [0, 1] [0, ) be an isoperimeric esimaor on (Ω, d, µ). The following saemens hold and are in fac equivalen. 1. Isoperimeric inequaliy A Ω, Borel se, µ + (A) I(µ(A)). 2. Ledoux s inequaliy f Lip(Ω), I(λ f (s))ds 0 Ω f(x) dµ(x). 3. Maz ya s inequaliy 4 f Lip(Ω), ( fµ) (s)i(s) d f(x) dµ(x). ds { f >fµ (s)} 4. Pólya Szegö s inequaliy f Lip(Ω), (( f µ) (.)I(.)) (s)ds 0 0 f µ (s)ds. (The second rearrangemen on he lef hand side is wih respec o he Lebesgue measure). 3 In fac, i is enough in order o define f ha f will be Lipschiz on every ball in (Ω, d) cf. [6, pp. 184,189] for more deails. 4 See [19]; one can also find his inequaliy in [27, 28].

5 Hardy ype and Poincaré Inequaliies Oscillaion inequaliy f Lip(Ω), (fµ () fµ()) I() f µ (). (2.1) Given any rearrangemen invarian space X(Ω), i follows readily from (2.1) ha for all f Lip(Ω) f LS(X) := ( f µ () fµ() ) I() f X. X One salien characerisic of hese spaces is ha hey explicily incorporae in heir definiion he isoperimeric profiles associaed wih he geomery in quesion and hus hey can auomaically selec he correc opimal spaces for differen geomeries (cf. [14, 15, 16] for more on his). While he LS(X) spaces are no necessarily normed, ofen hey are equivalen o normed spaces (cf. [25]), and, in he classical cases, lead o opimal Sobolev Poincaré inequaliies and embeddings (cf. [17, 13, 14] as well as [3, 2, 29] and he references herein). 3 Hardy Isoperimeric Type Le Q I be he operaor defined on measurable funcions on (0, 1) by Q I f() = f(s) ds I(s), where I is an isoperimeric esimaor. We consider he possibiliy of compleely characerizing Poincaré inequaliies in erms of he boundedness of Q I as an operaor from X o Y. In order o moivae wha follows, we recall he following resul obained in [15, 16] for classical seings (cf. [14]). Theorem 3.1. Le X, Y be wo r.i. spaces on Ω. Suppose ha here exiss a consan c = c(x, Y ) such ha for every posiive funcion f X wih supp f (0, 1/2) Q I f() Y c f µ X. Then for all g Lip(Ω) 5 g Ω gdµ g X. (3.1) Y 5 We noe for fuure use ha Poincaré inequaliies can be equivalenly formulaed replacing Ω gdµ by a median value m of g, i.e., µ (g m) 1/2 and µ (g m) 1/2.

6 290 J. Marín and M. Milman Furhermore, if he space X is such ha f µ X f µ X, hen f Y f LS(X) + f L 1. In fac, LS(X) is an opimal space in he sense ha if (3.1) holds, hen for all g Lip(Ω) g gdµ g gdµ g X. Y LS(X) Ω We give a simple, bu nonrivial example ha illusraes how he preceding developmens allow us o ransplan Sobolev Poincaré inequaliies o he meric seing. Example. Suppose ha (Ω, µ) has an isoperimeric esimaor I(s) s 1 1/n (0 < s < 1/2). I follows ha, on funcions suppored on (0, 1/2), Q I f() Ω /2 s 1/n f() ds s. Since he condiions for he boundedness Q I on r.i. spaces are well undersood, we can ransplan he classical Sobolev inequaliies o (Ω, µ). Furhermore, we noe ha, in he borderline case q = n, he corresponding resul using he opimal LS(L n ) spaces is sharper han he classical Sobolev heorems (cf. [2]). As was menioned, i is known ha he converse o Theorem 3.1 is rue in a number of imporan classical cases. In oher words, he operaor Q I in hose cases gives a complee characerizaion of he Poincaré inequaliies (for he mos recen resuls cf. [15, 16]). This led us o inroduce he following definiion. Definiion. We say ha a probabiliy meric space (Ω, d, µ) is of isoperimeric Hardy ype if for any given isoperimeric esimaor I, he following asserions are equivalen for all r.i. spaces X = X(Ω), Y = Y (Ω). 1. There exiss c = c(x, Y ) such ha f Lip(Ω), f fdµ c f X. (3.2) Y 2. There exiss c = c(x, Y ) such ha Q I f Y f X, f X wih supp (f) (0, 1/2). Ω

7 Hardy ype and Poincaré Inequaliies Model Riemannian Manifolds In his secion, we consruc a class of spaces of isoperimeric Hardy ype spaces ha includes he n-sphere S n (R n, γ n ) (R n wih Gaussian measure) and symmeric log-concave probabiliy measures on R. We follow he consrucion of Ros (cf. [26]). Le M 0 be a complee smooh oriened n 0 -dimensional Riemannian manifold wih disance d. An absoluely coninuous probabiliy measure µ 0 w.r. o dv in M 0 is called a model measure if here exiss a coninuous family (in he sense of he Hausdorff disance on compac subses) D = {D : 0 1} of closed subses of M 0 saisfying he following condiions. 1. µ 0 (D ) = and D s D, for 0 s < D is a smooh isoperimeric domain of µ 0 and I µ0 () = µ + 0 (D ) is posiive and smooh for 0 < < 1, where I µ0 denoes he isoperimeric profile of M The r-enlargemen of D, defined by (D ) r = {x M 0 : d(x, D ) r} verifies (D ) r = D s for some s = s(, r), D 1 = M 0 and D 0 is eiher a poin or he empy se. Theorem 4.1. Le (M 0, d) be an n 0 -dimensional Riemannian manifold endowed wih a model measure µ 0. Then (M 0, d) is of isoperimeric Hardy ype. Proof. Consider he funcion defined by p : M 0 [0, 1], x D. Le x, y M 0 be such ha 0 < p(y) < p(x). Le D D be such ha y / D. Consider he funcion h(r) = µ 0 (D r ), which is coninuous and smooh for 0 < h(r) < 1 and, in his range (cf. [26]), h (r) = I µ0 (h(r)). (4.1) From he definiion of p i follows ha p(x) = h(d(x, D)) and p(y) = h(d(y, D)). Since d(x, D) d(y, D) d(x, y), we see ha p(x) p(y) d(x, y) h(d(x, D)) h(d(y, D)) d(x, D) d(y, D) sup h (s), s i.e., p Lip(M 0 ) and p(x) = lim sup p(x) p(y) d(x, y) y x

8 292 J. Marín and M. Milman is finie, i follows ha p(x) exiss a.e. w.r. o dv (cf. [6, p. 2]) and hence a.e. w.r. µ 0. Le us now compue p. Given x M 0 such ha p(x) = < 1, le D D so ha x / D and, as above, consider he funcion h(r) = µ 0 (D r ). Le z(x) M 0 be such ha d(x, D) = d(x, z(x)). Selec y n on he geodesic joining z(x) and x such ha y n x. Then lim p(x) p(y n ) n d(x, y n ) = lim h(d(x, D)) h(d(y n, D)) n d(x, D) d(y n, D) = h (d(x, D)) = I µ0 (h(d(x, D))) [by (4.1)] = I µ0 (p(x)). (4.2) Le f X be a posiive funcion wih suppf (0, 1/2). Define F (x) = p(x) ds f(s) I µ0 (s). I is obvious ha F Lip(M 0 ) and, by (4.2), 1 F (x) = f(p(x)) p(x) = f(p(x)) a.e. I µ0 (p(x)) We claim ha he map p : (M 0, µ 0 ) ([0, 1], ds) is a measure-preserving ransformaion. To prove his claim, we need o see ha for any measurable subse R [0, 1] ( µ 0 p 1 (R) ) = ds. (4.3) I is enough o see (4.3) for a closed inerval. Le [a, b] [0, 1] (0 a < b 1). Then µ 0 ( p 1 ([a, b]) ) = µ 0 (D b ) µ 0 (D a ) = b a. Using his claim (cf. [4, Proposiion 7.2, p. 80]) hen a.e. we have R F µ 0 (s) = ds f(s) I µ0 (s), F µ 0 (s) = f (s). I is obvious ha he condiion (3.2) is equivalen o u m Y u X, where m is a median 6 of f, now since µ 0 (F = 0) 1/2, 0 is a median of F, and from 6 i.e., µ 0 (f m) 1/2 and µ 0 (f m) 1/2.

9 Hardy ype and Poincaré Inequaliies 293 we obain as we wished o show. F 0 Y F X ds f(s) I µ0 (s) f X Y 5 E. Milman s Equivalence Theorems In he nex resul, we have a lis of progressively weaker saemens ha neverheless have been shown by E. Milman (cf. [23, 22, 24]) o be equivalen under cerain convexiy assumpions. Likewise, E. Milman also has formulaed similar resuls in he conex of Orlicz spaces. In [16], we have simplified and exended Milman s resuls o he conex of meric spaces wih Hardy isoperimeric ype, as well as considering general r.i. spaces. Theorem 5.1. Le (Ω, d, µ) be a space of Hardy isoperimeric ype. Then he following saemens are equivalen: (E1) Cheeger s inequaliy (E2) Poincaré inequaliy C > 0 s.. I (Ω,d,µ) C, (0, 1/2]. P > 0 s.. f m L 2 (Ω) P f L 2 (Ω). (E3) Exponenial concenraion: for all f Lip(Ω) wih f Lip(Ω) 1 c 1, c 2 > 0 s.. µ{ f m > } c 1 e c2, (0, 1). (E4) Firs momen inequaliy: for all f Lip(Ω) wih f Lip(Ω) 1 F > 0 s.. f m L 1 (Ω) F. Theorem 5.2. Le (Ω, d, µ) be a space of isoperimeric Hardy ype. Le 1 q,, and le N be a Young funcion such ha N()1/q is nondecreasing and here exiss α > max{ 1 q 1 2, 0} such ha N(α ) is nonincreasing. Then he following saemens are equivalen. (E5) (L N, L q ) Poincaré inequaliy holds P > 0 s.. f m LN (Ω) P f L q (Ω).

10 294 J. Marín and M. Milman (E6) Any isoperimeric profile esimaor I saisfies: here exiss a consan c > 0 such ha 1 1/q I() c N 1, (0, 1/2]. (1/) The consrucion of he previous secion hus provides a class of spaces were he previous heorems apply. 6 Some Spaces Tha Are no of Isoperimeric Hardy Type In his secion, we show ha, unforunaely, no all meric spaces are of isoperimeric Hardy ype. Le I : [0, 1] [0, ) be concave, coninuous, increasing on (0, 1/2), symmeric abou he poin 1/2, and such ha I(0) = 0. Le 0 β 1. We say ha I has β-asympoic behavior if lim s 0 + (0, ). I(s) s 1 β exiss and lies on Theorem 6.1. Suppose ha I is of β-asympoic behavior. Then he following asserions hold. (i) Given 0 < β < 1/2, here is a meric space (Ω 0, d, µ) wih I(s) I (Ω0,d 0,µ 0)(s), a pair of r.i. spaces X, Y on Ω 0, and a consan c = c(x, Y ) such ha g gdµ 0 c g X, g Lip(Ω 0 ), Ω 0 Y bu Q I : X Y is no bounded. (ii) Given 0 < β < 1, here is a meric space (Ω 1, d 1, µ 1 ) such ha I(s) I (Ω1,d 1,µ 1 )(s) and (Ω 1, d, µ) is of isoperimeric Hardy ype. Proof. (i) (cf. [13] for a more general resul). Le 1 < α < 2,, and le Ω be an α-john domain on R 2 ( Ω = 1). Then (cf. [11]) I Ω (s) s α/2 = s 1 (1 α/2), 0 s 1/2. 2 Le > 1 be such ha α > 1, and le r = α+(1 ). Noe ha 1 < < r. Then (cf. [12]) g g g L. Ω L r In his case, he operaor Q IΩ is given by

11 Hardy ype and Poincaré Inequaliies 295 Q IΩ f() = u α/2 f(u)du. Noe ha Q IΩ is no bounded from L o L r. Indeed, he boundedness of Q IΩ can be reformulaed as a weighed norm inequaliy for he operaor g g(u)du; namely, x g(u)du c g(x)x α/2 L. (6.1) x L r I is well known ha (6.1) holds if and only if (cf. [20]) ( a sup 1 a>0 0 ) 1/r ( Now, since α < 2, i follows ha ( a ) 1/r ( 1 0 a a (u α/2) 1 ) 1 1 du <. (6.2) α 2( 1) + 1 < 0, and for a near zero we have (u α/2) 1 ) 1 ( ) 1 du a 1/r a α+2( 1) 1 2( 1) 1 a (1 )(α 1) 2. Consequenly, since (1 )(α 1) 2 < 0, (6.2) canno hold. (ii) We follow Gallo s mehod (cf. [10]) in order o consruc (Ω 1, d 1, µ 1 ). Le B(r) = r ds, 0 r 1. I(s) Since I is of β-asympoic behavior, we see ha L = B(0) <. Since B is decreasing, i has an inverse which we denoe by A. Consider he revoluion surface M = (0, L) S 1 (compacified by adjoining he poins {0} S 1 and {L} S 1 )) provided wih he Riemannian meric g = dr 2 + I(A(r)) 2 dθ 2, where θ S 1 and dθ 2 is he canonical Riemannian meric on ( S 1, can ). Noice ha I(A(0)) = I(A(L)) = 0. We denoe by Vol M he volume of (M, g). Muliplying he meric g by a consan, we can and will assume wihou loss ha Vol M (M) = 1. Denoe by I M he isoperimeric profile of (M, g, Vol M ). Then (cf. [10, Appendix A.1.]) we can find a consan c, depending only on I, such ha ci(s) I M (s) I(s). Le X, Y be wo r.i. spaces on M such ha

12 296 J. Marín and M. Milman Y g gdvol M g X, g Lip(M). M Le f be a posiive Lebesgue measurable funcion on (0, 1) wih suppf (0, 1/2). Define u(r, θ 1, θ 2 ) = A(r) f(s) ds I(s), (r, θ 1, θ 2 ) M. I is plain ha u is a Lipschiz funcion on M such ha Vol M {u = 0} 1/2. Hence 0 is a median of u. On he oher hand, recall ha (cf. [10, Page 57]) for any domain of revoluion Ω(λ) = (0, λ) S 1 M we have ha In oher words, Vol + M ( Ω(λ)) = I (Vol M (Ω(λ))). A (r) = I(A(r)). Therefore, u(r, θ 1, θ 2 ) = r u(r) = f(a(r)) A (r) I(A(r)) = f(a(r)). Now, since from we deduce ha u Vol M () = as we wished o show. f(s) ds I(s), u 0 Y u X f(s) ds I(s) f X, Y u Vol M () = f (), By Theorem 6.1 he verificaion of a Sobolev Poincaré inequaliy canno be reduced, in general, o esablish he boundedness of he associaed isoperimeric Hardy operaor. However, if he profiles are of β-asympoic behavior, we have he following weaker posiive resul. Theorem 6.2. Le I be of β-asympoic behavior (0 < β < 1). Le M I be he se of meric probabiliy spaces (Ω, d, µ) such ha I (Ω,d,µ) I.

13 Hardy ype and Poincaré Inequaliies 297 Le X, Y be wo r.i. spaces on [0, 1]. Then he following saemens are equivalen inf inf (Ω,d,µ) M h g Lip(Ω) g µ X ( g Ω gdµ) = c > 0. Y µ Q I : X Y is bounded. (6.3) Proof Given I of β-asympoic behavior, consider he revoluion surface M consruced in par (ii) of he previous heorem. Since M M I, by hypohesis: ( ) g gdµ c ( g) µ, g Lip(M), X Y Ω µ which is equivalen o (6.3) since M is of Hardy isoperimeric ype The implicaion is a direc consequence of Theorem 3.1. Acknowledgemen. The firs auhor was suppored in par by Grans MTM , MTM C02-02 and by 2005SGR References 1. Barhe, F., Caiaux, P., Robero, C.: Inerpolaed inequaliies beween exponenial and Gaussian, Orlicz hyperconraciviy and isoperimery. Rev. Ma. Iberoam. 22, (2006) 2. Basero. J., Milman, M., Ruiz, F.: A noe on L(, q) spaces and Sobolev embeddings. Indiana Univ. Mah. J. 52, (2003) 3. Benne, C., DeVore, R., Sharpley, R.: Weak L and BMO. Ann. Mah. 113, (1981) 4. Benne, C., Sharpley, R.: Inerpolaion of Operaors. Academic Press, Boson (1988) 5. Bobkov, S.G.: Exremal properies of half-spaces for log-concave disribuions. Ann. Probab. 24, (1996) 6. Bobkov, S.G., Houdré, C.: Some connecions beween isoperimeric and Sobolev ype inequaliies. Mem. Am. Mah. Soc. 129, no. 616 (1997) 7. Borell, C.: Inrinsic bounds on some real-valued saionary random funcions. In: Lec. Noes Mah. 1153, pp Springer (1985) 8. Ehrhard, A.: Symérisaion dans le space de Gauss. Mah. Scand. 53, (1983) 9. Federer, H.: Geomeric Measure Theory. Springer, Berlin (1969)

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