GRADIENT ESTIMATES, POINCARÉ INEQUALITIES, DE GIORGI PROPERTY AND THEIR CONSEQUENCES

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1 GRADIENT ESTIMATES, POINCARÉ INEQUALITIES, DE GIORGI PROPERTY AND THEIR CONSEQUENCES FRÉDÉRIC ERNICOT, THIERRY COULHON, DOROTHEE FREY Absac. On a doubling meic measue space endowed wih a caé du champ, we conside L p esimaes (G p ) of he gadien of he hea kenel and scale-invaian L p Poincaé inequaliies (P p ) fo p 2. We show ha he combinaion of (G p ) and (P p ) always implies he Gaussian hea kenel uppe bound and he L p boundedness of he Riesz ansfom (R p ). Moeove, his combinaion is shown o also yield he maching hea kenel lowe bound if p > ν, whee ν is he doubling exponen. If p ν, he same implicaion holds unde he addiional assumpion of a L p De Giogi ype popey. As a by-poduc, we give a shoe poof of he well-known fac ha he L 2 Poincaé inequaliy implies Gaussian uppe and lowe bounds of he hea kenel as well as of he main esul in [47]. Insumenal in ou appoach is a new noion of L p Hölde egulaiy fo a semigoup. Finally we impove known esuls on he L p boundedness of he Riesz ansfom fo p > 2. Conens 1. Inoducion 2 2. Fom Poincaé and gadien esimaes o hea kenel uppe bounds 9 3. L p Hölde egulaiy of he hea semigoup and hea kenel lowe bounds The case ν < p < + : fom Poincaé and gadien esimaes o hea kenel lowe bounds Poincaé inequaliies and hea kenel bounds: he L 2 heoy De Giogi popey and hea kenel bounds: he case 2 < p ν Gadien esimaes, Poincaé inequaliy and Riesz ansfom 37 Appendix A. Abou he p-independence of (Hp,p) η 45 Appendix. Fom Poincaé o De Giogi 48 Appendix C. A self-impoving popey fo evese Hölde inequaliies 55 Refeences 56 Dae: July 15, Mahemaics Subjec Classificaion. 58J35, Key wods and phases. Hea kenel lowe bounds, Hölde egulaiy of he hea semigoup, gadien esimaes, Poincaé inequaliies, De Giogi popey, Riesz ansfom. F. enico s eseach is suppoed by ANR pojecs AFoMEN no JS and HA no. ANR-12-S T. Coulhon s eseach is suppoed by he Ausalian Reseach Council (ARC) gan DP D. Fey s eseach is suppoed by he Ausalian Reseach Council (ARC) gans DP and DP

2 2 FRÉDÉRIC ERNICOT, THIERRY COULHON, DOROTHEE FREY 1. Inoducion Le M be a locally compac sepaable meisable space equipped wih a oel measue µ, finie on compac ses and sicly posiive on any non-empy open se. Fo Ω a measuable subse of M, we shall ofen denoe µ (Ω) by Ω. Le L be a non-negaive self-adjoin opeao on L 2 (M, µ) wih dense domain D L 2 (M, µ). Denoe by E he associaed quadaic fom E(f, g) = flg dµ, fo f, g D, and by F is domain, which conains D. Assume ha E is a songly local and egula Diichle fom (see [37,45] fo pecise definiions). As a consequence, hee exiss an enegy measue dγ, ha is a signed measue depending in a bilinea way on f, g F such ha E(f, g) = dγ(f, g) fo all f, g F. A possible definiion of dγ is hough he fomula (1.1) ϕ dγ(f, f) = E(ϕf, f) 1 2 E(ϕ, f 2 ), valid fo f F L (M, µ) and ϕ F C 0 (M). Hee C 0 (M) denoes he space of coninuous funcions on M ha vanish a infiniy. Accoding o he euling- Deny-Le Jan fomula, he enegy measue saisfies a Leibniz ule, namely (1.2) dγ(fg, h) = fdγ(g, h) + gdγ(f, h), fo all f, g, h F, see [37, Secion 3.2]. One can define a pseudo-disance d associaed wih E by (1.3) d(x, y) := sup{f(x) f(y); f F C 0 (M) s.. dγ(f, f) dµ}. Thoughou he whole pape, we assume ha he pseudo-disance d sepaaes poins, is finie eveywhee, coninuous and defines he iniial opology of M (see [68] and [45, Subsecion 2.2.3] fo deails). When we ae in he above siuaion, we shall say ha (M, d, µ, E) is a meic measue (songly local and egula) Diichle space. Noe ha his eminology is slighly abusive, in he sense ha in he above pesenaion d follows fom E. Fo all x M and all > 0, denoe by (x, ) he open ball fo he meic d wih cene x and adius, and by V (x, ) is measue (x, ). Fo a ball of adius and λ > 0, denoe by λ he ball concenic wih and wih adius λ. Finally, we will use u v o say ha hee exiss a consan C (independen of he impoan paamees) such ha u Cv and u v o say ha u v and v u. We shall assume ha (M, d, µ) saisfies he volume doubling popey, ha is (VD) V (x, 2) V (x, ), x M, > 0. I follows ha hee exiss ν > 0 such ha ( ) ν (VD ν ) V (x, ) V (x, s), x M, s > 0, s M M

3 DE GIORGI PROPERTY AND HEAT KERNEL OUNDS 3 which implies ( ) ν d(x, y) + V (x, ) V (y, s), x, y M, s > 0. s An easy consequence of (VD) is ha balls wih a non-empy inesecion and compaable adii have compaable measues. We shall say ha (M, d, µ, E) is a doubling meic measue Diichle space if i is a meic measue space endowed wih a songly local and egula Diichle fom and saisfying (VD). The Diichle fom E gives ise o a songly coninuous semigoup (e L ) >0 of self-adjoin conacions on L 2 (M, µ). In addiion (e L ) >0 is submakovian, ha is 0 e L f 1 if 0 f 1. I follows ha he semigoup (e L ) >0 is unifomly bounded on L p (M, µ) fo p [1, + ]. Also, (e L ) >0 is bounded analyic on L p (M, µ) fo 1 < p < + (see [67]), which means ha (Le L ) >0 is bounded on L p (M, µ) unifomly in > 0. Moeove, due o he doubling popey (VD), he semigoup has he consevaion popey (see [41,68]), ha is e L 1 = 1, > 0. Such a semigoup may o may no have a kenel, ha is fo all > 0 a measuable funcion p : M M R + such ha e L f(x) = p (x, y)f(y) dµ(y), a.e. x M. M If i does, p is called he hea kenel associaed wih L (in fac wih (M, d, µ, E)). Then p (x, y) is nonnegaive and symmeic in x, y since e L is posiiviy peseving and self-adjoin fo all > 0. One may naually ask fo uppe and lowe esimaes of p (fo uppe esimaes, see fo insance he ecen aicle [13] and he many elevan efeences heein; fo lowe esimaes, we will give moe efeences below). A ypical uppe esimae is (DUE) p (x, y) 1 V (x, )V (y,, > 0, a.e. x, y M. ) This esimae is called on-diagonal because if p happens o be coninuous hen (DUE) can be ewien as (1.4) p (x, x) 1 V (x,, > 0, x M. ) Unde (VD), (DUE) self-impoves ino a Gaussian uppe esimae (see [43, Theoem 1.1] fo he Riemannian case, [24, Secion 4.2] fo a meic measue space seing): ( ) 1 (UE) p (x, y) V (x, ) exp d2 (x, y), > 0, a.e. x, y M. C The poof of his fac in [24, Secion 4.2] elies on he following Davies-Gaffney esimae which was poved in ou seing in [69]: fo all open subses E, F M,

4 4 FRÉDÉRIC ERNICOT, THIERRY COULHON, DOROTHEE FREY f L 2 (M, µ) suppoed in E, and > 0, ( ) 1/2 ( (1.5) e L f 2 dµ e d2 (E,F ) 4 F E f 2 dµ) 1/2, whee d(e, F ) denoes he disance beween E and F. Fo moe on he Davies- Gaffney esimae, see fo insance [24, Secion 3]. I is well-known on he conay ha he maching Gaussian lowe bound (LE) p (x, y) 1 V (x, ) exp ( d2 (x, y) c ), > 0, a.e. x, y M does no always follow fom (DUE) (see [9]). Convesely, unde (VD), (LE) implies (UE) (see [11] and [26]). I is no oo difficul o pove in ou siuaion ha he conjuncion of he uppe and lowe bounds (UE) and (LE) (ha is, (VD) and (LE)) is equivalen o a unifom paabolic Hanack inequaliy, see [35] as well as [8, Secion 1]. One also knows ([45, Thm 2.32]) ha his Hanack inequaliy self-impoves ino a Hölde egulaiy esimae fo he hea kenel: hee exiss η (0, 1] ( ) η d(x, y) (H η ) p (x, z) p (y, z) p (x, z), fo all > 0 and a.e. x, y, z M such ha d(x, y). Noe ha, if (H η ) holds, p admis in paicula a coninuous vesion. Wha is also ue, bu much moe difficul o pove (see [42], [62], [63], [64], [65], [70], as well as [45, Theoem 2.31]) is ha (UE) + (LE) is also equivalen o (VD) ogehe wih he following scale-invaian Poincaé inequaliy (P 2 ): (P 2 ) f fdµ 2 1/2 dµ) dγ(f, f), fo evey f D and evey ball M wih adius. Hee fdµ = 1 fdµ denoes he aveage of f on. A somewha simplified poof of he main implicaion, namely he one fom (VD) + (P 2 ) o (UE) + (LE), has been given in [47]. One of he oucomes of he pesen aicle will be o povide a fuhe simplificaion (see he poof of Theoem 5.3 below) as well as a sho poof of he main esul in [47] (see Theoem 5.4 below). In he pesen wok, we shall sudy in paicula he ansiion fom (UE) o (LE) in he spii of [20] and [12]. The main novely hee will be a noion of L p η-hölde egulaiy of he hea kenel: fo p [1, + ] and η (0, 1], we shall say ha popey (Hp,p) η holds if fo evey 0 <, evey pai of concenic balls, wih especive adii and, and evey funcion f L p (M, µ), (H η p,p) el f e L fdµ p dµ) 1/p ( ) η 1/p f p, wih he obvious modificaion fo p = +. Cucial o ou appoach is Theoem 3.5 below whee we pove he equivalence, unde (VD) and (UE), beween he lowe Gaussian bound (LE) and he exisence

5 DE GIORGI PROPERTY AND HEAT KERNEL OUNDS 5 of some p [1, + ) and η > 0 such ha (H η p,p) holds, a popey which uns ou o be independen of p [1, + ). The above scale-invaian Poincaé inequaliy (P 2 ) quanifies he conol of he oscillaion of funcions by he Diichle fom. As we have jus seen fo he Hölde egulaiy of he hea semigoup, i is impoan o have a hand a full scale of condiions fo p [1, + ], no jus p = 2. This equies, beyond he noion of L 2 nom of he gadien povided by he Diichle fom, o have a noion of L p nom of he gadien, hence a poinwise noion of lengh of he gadien. The elevan noion in ou geneal seing is he one of caé du champ (see fo insance [45] and he efeences heein). The Diichle fom (o is enegy measue) admis a caé du champ if fo all f, g F he enegy measue dγ(f, g) is absoluely coninuous wih espec o µ. Then he densiy Υ(f, g) L 1 (M, µ) of dγ(f, g) is called he caé du champ and saisfies he following inequaliy (1.6) Υ(f, g) 2 Υ(f, f)υ(g, g). In he sequel, when we assume ha (M, d, µ, E) admis a caé du champ, we shall abusively denoe [Υ(f, f)] 1/2 by f. This has he advanage o sick o he moe inuiive and classical Riemannian noaion, bu one should no foge ha one woks in a much moe geneal seing (see fo insance [45] fo examples), and ha one neve uses diffeenial calculus in he classical sense. We shall summaise his siuaion by saying ha (M, d, µ, E) is a (doubling) meic measue Diichle space wih a caé du champ. We can now fomulae he L p vesions of he scale-invaian Poincaé inequaliies, which may o may no be ue and, conay o he Hölde egulaiy condiions fo he hea semigoup, do depend on p [1, + ). Moe pecisely, fo p [1, + ), one says ha (P p ) holds if p (P p ) f 1/p 1/p fdµ dµ) f dµ) p, f F, whee anges ove balls in M of adius. Recall ha (P p ) is weake and weake as p inceases, ha is (P p ) implies (P q ) fo q > p, see fo insance [46], and he p = vesion is ivial in he Riemannian seing (see howeve ineesing developmens fo moe geneal meic measue spaces in [33]). On he Euclidean space, (P p ) holds fo all p [1, + ]. On he conneced sum of wo copies of R n, (P p ) is valid if and only if p > n, as one can see by adaping he poof of [46, Example 4.2]. Moe ineesing examples follow fom [48, Theoem 6.15], see also [34, Secion 5]. On conical manifolds wih compac basis, (P p ) holds a leas fo p 2 (see [22]). A deep esul fom [53] saes if ha (P p ) holds fo some p (1, + ), hen (P pε ) holds fo some ε > 0. Finally he se {p [1, + ]; (P p ) holds on M} may be eihe {+ }, o [1, + ], o of he fom (p M, + ] fo some p M > 1. We will also use esimaes on he gadien (o caé du champ ) of he semigoup, which wee inoduced in [4]: fo p [1, + ], conside (G p ) sup e L p p < +, >0

6 6 FRÉDÉRIC ERNICOT, THIERRY COULHON, DOROTHEE FREY which is equivalen o he inepolaion inequaliy (1.7) f 2 p Lf p f p, f D (see [25, Poposiion 3.6]). Up o an abiaily small loss in p, one can efomulae (G p ) in ems of inegal esimaes of he gadien of he hea kenel. Moe pecisely, in pesence of (VD) and (DUE), fo 2 < p 0 +, (G p ) fo all p (2, p 0 ) is equivalen o (1.8) x p (., y) p C p [ ] 1 1, a.e. y M, > 0, V (y, ) p fo all p (2, p 0 ), see [4, Poposiion 1.10]. Also, unde (DUE), (G ) is equivalen o he songe esimae ( ) 1 ( G ) x p (x, y) exp d2 (x, y), a.e. x, y M, > 0 V (x, ) C (see [25, Secion 4.4], [32, Theoem 1.1]). As fa as examples ae concened, ( G ) holds on manifolds wih non-negaive Ricci cuvaue ([54]), Lie goups wih polynomial volume gowh ([61]), and co-compac coveing manifolds wih polynomial gowh deck ansfomaion goup ([31], [32]). On he ohe hand, conical manifolds wih a compac basis povide a family of doubling spaces (M, d, µ, E) wih a caé du champ saisfying (UE) and (LE) such ha fo evey p 0 > 2 hee exis examples in his family whee (G p ) holds fo 1 < p < p 0 and no fo p p 0, see [55],[56],[22]. Popey (G p ) is closely elaed o he L p boundedness of he Riesz ansfom R = L 1/2. One says ha (R p ) holds if he Riesz ansfom is bounded on L p (M, µ), which means ha (R p ) f p Lf p, f D. Since by definiion f 2 2 = E(f, f) = Lf 2 2, f F, (R 2 ) ivially holds and (G 2 ) follows fom he analyiciy of (e L ) >0 on L 2 (M, µ). Moeove fo p (1, 2), (G p ) also holds always, see [16, Poposiion 2.7], and (R p ) holds due o (DUE), see [21]. Now, by inepolaion, (G p ) as well as (R p ) ae songe and songe as p inceases above 2. I is easy o see ha (R p ) implies (G p ) (due o he fac ha he semigoup is bounded analyic on L p ) o equivalenly (1.7). Convesely, [4, Theoem 1.3] (which is saed fo Riemannian manifolds, bu does exend o ou cuen seing, as hined on p.122) says ha unde (VD) and Poincaé inequaliy (P 2 ), fo any p 0 (2, + ) one has (G p ) fo all 2 < p < p 0 (R p ) fo all 2 < p < p 0. Anohe oucome of he pesen aicle will be o pove he same equivalence unde (P p0 ) insead of (P 2 ) (see Theoem 7.1 below). In he pesen pape, we ae going o look a he combinaion (G p ) + (P p ) fo 1 p +, and especially fo 2 p < +. Fo 1 p 2, (G p ) + (P p ) is nohing bu (P p ) and heefoe is weake and weake as p goes fom 1 o 2. On he

7 DE GIORGI PROPERTY AND HEAT KERNEL OUNDS 7 conay, fo 2 p +, since (G p ) is songe as p inceases, wheeas (P p ) is weake, (G p ) + (P p ) does no exhibi a pioi any monooniciy. A one end of he ange, (G ) + (P ), a leas in he Riemannian seing, is nohing bu (G ), which does no seem o have consequences in iself. Howeve, i has been shown in [25, Coollay 2.2] ha, in pesence of (VD), he songe vesion ( G ) implies (UE) and (LE), heefoe, by [21] and [4], (R p ) fo all 1 < p < +, and finally (E p ) f p Lf p, f D. A he ohe end of he ange, fo p = 2, we aleady ecalled he fundamenal fac ha (G 2 ) + (P 2 ) = (P 2 ) implies (UE) + (LE). We ae going o complee he picue fo 2 < p < + and by he same oken simplify he poof of he case p = 2. Fis, assuming (VD ν ), we will pove ha fo ν < p < +, (G p ) + (P p ) also has song consequences. We obain in Poposiion 2.1 ha unde he doubling popey (VD), fo any p [2, + ) he combinaion (G p ) + (P p ) implies he uppe esimaes (DUE) and heefoe (UE). Using his sep, we fuhe show in Theoem 4.1, in he spii of [20] and using Theoem 3.5, ha unde (VD ν ) and fo ν < p < +, (1.9) (G p ) + (P p ) = (LE). Puing ogehe hese wo esuls, (G p ) + (P p ) = (UE) + (LE) fo ν < p < + (i is known anyway ha (LE) = (UE), see [11, Theoem 1.3] and [26]). As a by-poduc, we shall see in Coollay 4.5 ha finally (G p ) + (P p ) fo some p > ν implies (G q ) + (P q ) fo all q [2, p). If ν < 2, he elevan ange p [2, + ) is coveed by he above. One expecs hings o be easie in his case (see fo insance [13, Coollay 2.3.6, Poposiion 4.1.8]. This is no only a folkloe case (see [23] fo a discee example), bu ceainly a maginal case, and we ceainly have o conside he moe common siuaion whee ν 2. The case p [2, ν] is moe complicaed. This is no a pioi obvious, and i means ha in he couple (G p ) + (P p ) i is moe efficien o have a songe (G p ) a he expense of a weake (P p ), han he opposie. In his ange, we will have o inoduce an exa assumpion in ode o ensue he validiy of he implicaion (1.9), namely a non-local L p -vesion of De Giogi popey (DG p ) (see Secion 6 fo deails and definiions) and we shall pove in Theoem 6.4 ha, again unde (VD ν ), fo p [2, ν], (G p ) + (P p ) + (DG p ) = (LE). I is easy o see ha (DG p ) always holds fo p > ν, so his esul is an exension of (1.9). This can be undesood in he following way: fo p > ν, he coesponding Sobolev inequaliy, ha is he so-called Moey inequaliy (4.1), suffices o deduce fom L p gadien bounds L Hölde esimaes on he hea kenel (see he sho poof afe Remak 4.3 below); fo p (2, ν], howeve, one has o use an ellipic ieaive agumen o ge fom L p gadien bounds up o L Hölde esimaes, and he popey (DG p ) pecisely incopoaes such an ieaion. We do no know whehe his new assumpion (DG p ) is eally necessay in his ange; i may well

8 8 FRÉDÉRIC ERNICOT, THIERRY COULHON, DOROTHEE FREY follow fom (G p )+(P p ) as in he case p = 2 (see Appendix A), bu we have no been able o pove i. Noe ha along he way we inoduce and pove an L p Caccioppoli inequaliy (6.1) which may be of independen inees. In any case, (DG p ) is no fa fom being opimal since unde (G p ) fo some p > 2, (P 2 ) (hence (LE) as we aleady explained) implies (DG q ) fo evey q (2, p] (see Poposiion 6.7 below). Again, ogehe wih Poposiion 2.1 (o by [11, Theoem 1.3] adaped o ou seing), we can conclude, fo p [2, ν], (G p ) + (P p ) + (DG p ) = (UE) + (LE). Fo p = 2 he well-known implicaion (P 2 ) = (UE) + (LE) hen follows fom he fac ha (G 2 ) is always ue and ha (P 2 ) = (DG 2 ). The lae can be seen by an ellipic Mose ieaion, much easie han is paabolic counepa. Since (UE) + (LE) implies (P 2 ) (see [62 65]), one can a poseioi summaise he above by saying ha unde (VD ν ) and (G p ), (P p ) self-impoves o (P 2 ), wihou any fuhe condiion if p > ν, and ogehe wih (DG p ) if p ν. In paicula, unde (P p ) if p > ν and (P p ) + (DG p ) if p ν, (R p ) can only hold if (P 2 ) holds. Noe ha in he ange p > ν, (P p ) is paiculaly simple: fo insance, if V (x, ) ν, i is equivalen o he Moey inequaliy f(x) f(y) [d(x, y)] 1 ν p f p, f D, see [19, Théoème 7.3]. In paicula, i is sable unde he opeaion of glueing, say, wo manifolds wih his volume gowh along a compac. Now, if we glue wo such manifolds saisfying (R p ) and (P p ) fo p > ν > 2, he implicaion (R p ) + (P p ) = (G p ) + (P p ) = (P 2 ), shows ha (R p ) canno hold on he new manifold, since i is easy o see ha (P 2 ) is false on a manifold wih a leas wo ends having polynomial gowh of exponen ν > 2. This emak is nohing bu a sysemaisaion of he coune-example in [21, Secion 5]. This also explains why glueing is no allowed in he second asseion of [29, Theoem 1.1]. Fo a diffeen saemen in his diecion, see [14, Coollay 7.5]. Finally, we will see in Theoem 7.7 ha fo any p 0 2, ha is even when one canno use Theoem 4.1 o ely on (LE), o (P 2 ), he combinaion (G p0 ) + (P p0 ) implies (R p ) fo p (p 0, p 0 + ε). This impoves he main esul of [4], which gives, up o an abiay small loss in p, he equivalence beween (G p ) and (R p ) fo p > 2, unde a L 2 -Poincaé inequaliy (P 2 ), as well as he main esul of [3] which eas he case p 0 = 2. I follows ha, unde he assumpion (G p0 )+(P p0 ) fo some p 0 2, he ange of exponens p (1, ) fo which (G p ) holds coincides wih he one fo which (R p ) holds, and his ange is an open ineval of he fom (1, p 1 ), fo some exponen p 1 (p 0, + ] (see Theoem 7.8). We can summaise mos of ou esuls in he following way: Theoem. Le (M, d, µ, E) be a meic measue Diichle space wih a caé du champ saisfying (VD ν ). Assume (G p0 ) and (P p0 ) fo some p 0 2. Then (DUE)

9 DE GIORGI PROPERTY AND HEAT KERNEL OUNDS 9 holds. Moeove hee exiss p 1 (p 0, + ] such ha {p (1, ), (R p ) holds} = {p (1, ), (G p ) holds} = (1, p 1 ). If eihe p 0 > ν o else 2 < p 0 ν and (DG p0 ) holds, hen (LE), and heefoe (P 2 ), holds. The plan of he pape is as follows. Secion 2 is devoed o he poof of uppe esimaes (UE) unde (G p ) + (P p ) fo any p [2, + ). In Secion 3, we obain he cucial self-impovemen popey of L p Hölde egulaiy esimaes (H η p,p) fo he semigoup (Poposiion 3.1), and hei equivalence wih (LE). As an applicaion, in Secion 4, (LE) is shown o be implied by (G p ) and (P p ) fo p > ν (Theoem 4.1). The counepa p ν (Theoem 6.4) is invesigaed in Secion 6, hough he sudy of a suiable De Giogi popey called (DG p ). In Secion 5, we give a simple poof of he implicaion fom (VD) + (P 2 ) o (UE) + (LE); he only emaining non ivial pa is he implicaion fom (VD) + (P 2 ) o he mos classical De Giogi popey (DG 2 ), which is ecalled in Appendix. Wih simila agumens, we also obain a new poof of he esul fom [47] ha he ellipic egulaiy ogehe wih a scale-invaian local Sobolev inequaliy imply he paabolic Hanack inequaliy (Theoem 5.4). Finally, in Secion 7, we impove he main esuls of [3] and [4] by poving he equivalence beween he gadien esimae (G p ) and he boundedness of he Riesz ansfom (R p ), unde he Poincaé inequaliy (P p ). In Appendix A, we sudy moe closely he p-independence of popey (H η p,p). Appendix C spells ou a self-impoving popey of evese Hölde esimaes which is used in he poof of Theoem 7.1. Moeove, we efe he eade o a fohcoming wok of he auhos [10], whee hese new noions (L p Hölde egulaiy popeies fo he hea semigoup and L p De Giogi ype esimaes) will be used o esablish he fac ha, unde ceain assumpions on he hea kenel, he spaces {f L (M, µ), L α/2 f L p (M, µ)} ae algebas fo he poinwise poduc fo α (0, 1) and p (1, + ). Since ou esuls avoid paabolic Mose ieaion, which is vey had o un diecly in a discee ime seing (see [28]), hey ae well suied o an exension o andom walks on discee gaphs. As a mae of fac, ou Appendix is inspied by [3], bu on he ohe hand ou appoach below gives a simple poof of he main esul in [3] by avoiding he ieaion sep in [3, Poposiion 4.5]. Fo he discee vesion of ou esuls on Riesz ansfom one can ely on [7]. We leave his fo fuue wok. 2. Fom Poincaé and gadien esimaes o hea kenel uppe bounds In his secion we shall need a vesion of he Davies-Gaffney esimae (1.5) which also includes he gadien, namely ( 1/2 (2.1) e L f dµ) 2 + ( ) 1/2 ( 1/2 e L f 2 d(e,f )2 c dµ e f dµ) 2, F F E fo some c > 0, all open subses E, F M, f L 2 (M, µ) suppoed in E, and > 0, d(e, F ) being he disance beween E and F. The poof of his fac in

10 10 FRÉDÉRIC ERNICOT, THIERRY COULHON, DOROTHEE FREY [4, Secion 3.1] woks in ou seing of a Diichle space wih a caé du champ. Indeed, he poof elies on he following inequaliy: fo ϕ a non-negaive cu-off funcion wih suppo S, ( ) 1/2 ( ) 1/2 (2.2) ϕ e L f 2 dµ e L f 2 ϕ 2 dµ e L f 2 dµ S + ϕ e L f Le L f dµ, which follows fom (1.1), (1.2), and (1.6). Poposiion 2.1. Le (M, d, µ, E) be a meic measue Diichle space wih a caé du champ saisfying (VD). Then he combinaion of (G p ) wih (P p ) fo some p [2, + ) implies (UE). Poof. Assume fis 2 < p < +. Fom he self-impoving popey of (P p ) (see [53]), hee exiss p (2, p) such ha (P p ) holds. Then, by inepolaing beween he L 2 Davies-Gaffney esimae fo e L conained in (2.1) and (G p ), one obains ha fo > 0 he opeao e L saisfies L p -L p off-diagonal esimaes a he scale. Similaly, by inepolaing he unifom L boundedness wih (2.1), one sees ha he semigoup e L also saisfies such esimaes. Namely, fo some c > 0, (2.3) e L L p () L p ( ) + el L p () L p ( ) exp ( c d2 (, ) fo evey > 0 and all balls, of adius. On he ohe hand, he (P p ) Poincaé inequaliy self-impoves ino a (P p,q ) inequaliy fo some q > p (given by q 1 = p 1 ν 1 if p < ν, q = + if p < ν and any q > ν if p = ν, see [36]). Tha is, fo evey ball of adius, one has q f 1/q fdµ dµ) 1/ p dµ) f p. Hence el f e L fdµ q 1/q dµ) e L f dµ) p 1/ p, fo all > 0 and f L p (M, µ). I follows by Jensen s inequaliy ha e L f 1/q dµ) q e L f 1/ p dµ) p + e L f dµ) p 1/ p. Then fom (2.3), we deduce ha fo evey pai of balls, of adius one has ( (2.4) e L L p () L q ( ) exp c d2 (, ) ) 1 q 1 p. We now use [13], and efe o i fo moe deails. Se V (x) := V (x, ), and denoe abusively by w he opeao of muliplicaion by a funcion w. Using doubling, ),

11 DE GIORGI PROPERTY AND HEAT KERNEL OUNDS 11 (2.4) may be wien as ( V 1 p 1 q e L L p () L q ( ) exp c d2 (, ) ). y he doubling popey, we may sum his inequaliy ove a coveing of he whole space a he scale and deduce (V E p,q ), which is sup >0 y dualiy, one obains (EV q, p ), ha is sup >0 V 1 p 1 q e L p q < +. e L V 1 p 1 q q p < +. Then by inepolaion [13, Poposiion 2.1.5] beween (V E p,q ) and (EV q, p ), one obains (V EV,, 1 1 ), ha is 2 sup V e L V < +, >0 whee 1 < 2 is given by 1 = 1( 1 p + 1 ) = 1 + ( 1 p 1). Then (EV 2 q 2 q,2) holds by [13, Remak 2.1.3]. Thanks o he L 1 -unifom boundedness of he semigoup, he exapolaion [13, Poposiion 4.1.9] yields (EV 1,2 ), hence (DUE) by [13, Poposiion 2.1.2] and (UE) by [24, Secion 4.2]. Finally, if p = 2, one can un he above poof by seing diecly p = 2. Alenaively, one can see by [26, Secion 5] ha (P 2 ) and (VD) imply he so-called Nash inequaliy (N) and apply [13, Theoem 1.2.1]. Remak 2.2. The case 1 p < 2 of Poposiion 2.1 follows ivially fom he case p = 2. Remak 2.3. One may avoid he use of he highly non-ivial esul fom [53] by assuming diecly (G p ) and (P q ) fo some q (2, p). Noe ha his vesion does wok fo p = L p Hölde egulaiy of he hea semigoup and hea kenel lowe bounds The following saemen is valid in a moe geneal seing han he one pesened in Secion 1 and used in Secion 2: i is enough o conside a meic measue space (M, d, µ) saisfying (V D), endowed wih a semigoup (e L ) >0 acing on L p (M, µ), 1 p +. Fo 1 p + le us wie he L p -oscillaion fo u L p loc (M, µ) and a ball: if p < + and p- Osc (f) := - Osc (f) := ess sup 1/p f f dµ dµ) p, f f dµ. Poposiion 3.1. Le (M, d, µ, L) as above. Le p [1, + ] and η (0, 1]. Then he following wo condiions ae equivalen:

12 12 FRÉDÉRIC ERNICOT, THIERRY COULHON, DOROTHEE FREY (a) fo all 0 <, evey pai of concenic balls, adii and, and evey funcion f L p (M, µ), ( ) η (Hp,p) η p- Osc (e L f) 1/p f p. wih especive (b) fo all 0 <, evey pai of concenic balls, wih especive adii and, and evey funcion f L p (M, µ), (Hp, ) η ess sup e L f(x) e L f(y) ( ) η 1/p f p. x,y Remak 3.2. I is easy o see ha (Hp, ) η is equivalen o he following condiion, which jusifies is name: fo all 0 <, evey pai of concenic balls, wih especive adii and, and evey funcion f L p (M, µ), ( ) η (3.1) - Osc (e L f) 1/p f p. Poposiion 3.1 is an easy consequence of a well-known chaaceisaion of Hölde coninuous funcions in ems of he gowh of hei L p oscillaions on balls. This esul is due o Meyes [58] in he Euclidean space, and is poof was lae simplified, see e.g. [39, III.1]. I can be fomulaed in ems of embeddings of Moey- Campanao spaces ino Hölde spaces. The poof goes hough in a doubling meic measue space seing (see [2, Poposiion 2.6] fo an L 2 vesion). A paicula case of he following lemma will be used in he poof of Poposiion 6.7 below. We give a poof fo he sake of compleeness. Lemma 3.3. Le (M, d, µ) be a meic measue space saisfying (VD). Le 1 p < + and η > 0. Then fo evey funcion f L p loc (M, µ) and evey ball in (M, d, µ), f C η () := ess sup x,y x y f(x) f(y) d η (x, y) f C η,p () := sup 6 p- Osc (f) η ( ). Poof. Le x, y be Lebesgue poins fo f. Le i (x) = (x, 2 i d(x, y)), fo i N. Noe ha fo all i N, i (x) 0 (x) 3. Wie f(x) fdµ fdµ fdµ 0 (x) i 0 i (x) i+1 (x) f fdµ dµ i 0 i 0 i 0 i+1 (x) i+1 (x) i (x) f p- Osc i (x)(f), i (x) fdµ p ) 1/p dµ

13 DE GIORGI PROPERTY AND HEAT KERNEL OUNDS 13 whee he las inequaliy uses doubling. I follows ha ( ) f(x) fdµ [ ( i (x))] η f C η,p () 0 (x) i 0 ( ) = 2 ηi d η (x, y) i 0 d η (x, y) f C η,p ( ), f C η,p () as well as he simila esimae wih he oles of x, y exchanged. Finally, since 0 (y), 0 (x) 2 0 (x) wih compaable measues by doubling, fdµ fdµ fdµ fdµ + fdµ fdµ 0 (x) 0 (y) 0 (x) 2 0 (x) 0 (y) 2 0 (x) f fdµ dµ 2 0 (x) 2 0 (x) f = p- Osc 20 (x)(f), hence fdµ fdµ 0 (x) 0 (y) The claim follows by wiing f(x) f(y) f(x) fdµ + 0 (x) 0 (x) 2 0 (x) 2 0 (x) fdµ p ) 1/p dµ dη (x, y) f C η,p ( ). fdµ 0 (y) fdµ + f(y) 0 (y) fdµ. Poof of Poposiion 3.1. The implicaion fom (H η p, ) o (H η p,p) is obvious by inegaion. The case p = + of he convese is simila o Remak 3.2. Assume (H η p,p) fo 1 p < +. Le > 0, and a ball of adius 0 <. Fom Lemma 3.3, we deduce ha fo f L p (M, µ), a.e. x, y, (3.2) e L f(x) e L f(y) η p- Osc (e L f) sup 6 η ( ). Now (H η p,p) yields p- Osc (e L f) η ( ) η/2 1/p f p, whee is he ball concenic o wih adius. The balls and have he same adius and, if 6, i follows by doubling ha and ae compaable, hence (3.3) sup 6 p- Osc (e L f) η ( ) and (3.2) ogehe wih (3.3) yield (H η p, ). η/2 1/p f p,

14 14 FRÉDÉRIC ERNICOT, THIERRY COULHON, DOROTHEE FREY The pevious poof elies on he fac ha a poinwise Hölde egulaiy follows fom Hölde esimaes in ems of oscillaion. This key obsevaion equies a sufficienly apid decay a 0 of he Hölde modulus of coninuiy unde consideaion. Unde he addiional assumpion of Gaussian uppe esimaes fo he hea kenel, we shall now give anohe poof ha (Hp,p) η implies (Hp, ) η ha does no ake ino accoun he decay of he modulus of coninuiy. In ohe wods, he second poof explains how he gain of inegabiliy due o he Gaussian esimaes of he hea kenel allows o pass fom a egulaiy in ems of L p -oscillaion o a poinwise egulaiy. ( ) The second poof holds fo any doubling modulus of egulaiy insead η, of fo insance a logaihmic modulus of coninuiy (1log( )) α wih any α > 0, wheeas he above agumen fails if α (0, 1). Poposiion 3.4. Le (M, d, µ) as above and assume ha (e L ) >0 is a nonnegaive self-adjoin semigoup on L 2 (M, µ) wih a measuable kenel p saisfying ( ) 1 (3.4) p (x, y) V (x, ) exp d2 (x, y), > 0, a.e. x, y M, C which we will abusively sill call (UE). Assume also consevaiveness: e L 1 = 1. Le p [1, + ] and ω be a doubling modulus of coninuiy, which is a nondeceasing funcion ω : [0, ) [0, ) wih lim ω(x) = 0 x 0 saisfying fo some D > 0: fo evey x 0 and 1 (3.5) ω(x) D ω(x). Then he following wo condiions ae equivalen: (a) fo all 0 <, evey pai of concenic balls, wih especive adii and, and evey funcion f L p (M, µ), ( ) (3.6) p- Osc (e L f) ω 1/p f p. (b) fo all 0 <, evey pai of concenic balls, wih especive adii and, and evey funcion f L p (M, µ), (3.7) ess sup e L f(x) e L f(y) ( ) ω 1/p f p. x,y Noe ha as a consequence of (3.4), (e L ) >0 acs and is unifomly bounded on all L p (M, µ), 1 p +. Recall ha (DUE) implies (UE) also in his seing as soon as (e L ) >0 saisfies a Davies-Gaffney esimae (see [24, Secion 4.2]). Poof. Again, he implicaion fom (3.7) o (3.6) is obvious by inegaion. In ode o pove (3.7), fix a ball of adius (again, he emaining case follows 4 obviously by doubling); hen i is sufficien o show ) ( ) (3.8) ess sup el f(x) e (2 )L fdµ ω 1/p f p, x which we ae now going o pove.

15 DE GIORGI PROPERTY AND HEAT KERNEL OUNDS 15 Sep 1: We claim ha fo evey ball of adius, we have ) (3.9) e L fdµ e fdµ) L ( 1 + d(, ) D+ ν p ( ) ) ω 1/p f p. Indeed, conside a ball ρ conaining boh and wih adius ρ +d(, ). We have ) e L fdµ e fdµ) L ) ) ( ) ) e L fdµ e L fdµ ρ + e L fdµ e L fdµ ρ. These wo ems can be eaed similaly, so le us focus on he fis one. ) ) ( ) e L fdµ e L fdµ ρ el f e L fdµ ρ dµ ( ) p el f e L fdµ dµ ρ ( 1 ) ρ el f e L fdµ ρ ( ) 1/p ρ = p- Osc ρ(e L f). y (VD ν ) i follows ha ) ) ( ρ ) ν (3.10) e L fdµ e L fdµ ρ p p- Osc ρ (e L f). Then if ρ, applying (H η p,p) o ρ and ρ ρ gives ( ) ρ p- Osc ρ (e L f) ω ρ ρ 1/p f p. ) 1/p p dµ ) 1/p Now ρ ρ and have he same adius and a non-empy inesecion since hey boh conain, hence by doubling hey have compaable measues. Hence ( ) ρ p- Osc ρ (e L f) ω 1/p f p.

16 16 FRÉDÉRIC ERNICOT, THIERRY COULHON, DOROTHEE FREY If ρ, hen, using he L p -boundedness of he semigoup and he fac ha ρ, we can wie p- Osc ρ (e L f) e L f p dµ ρ ) 1/p 1/p f p ( ) ρ ω 1/p f p, whee he implici consan depends on ω hough ω(1) (since ω is nondeceasing). In all cases, one has ( ) ρ p- Osc ρ (e L f) ω 1/p f p, which wih (3.10) and (3.5) yields ) ) ( ρ ) ( ) D+ ν e L fdµ e L fdµ ρ p ω 1/p f p. The claim follows. Sep 2: Conclusion of he poof of (3.8). Fo x and f L p (M, µ), one can wie, hanks o he consevaion popey, ) ( )) e L f(x) e (2 )L fdµ = e 2 L e (2 )L f e (2 )L f dµ (x). Now (UE) yields, fo g L p loc (M, µ) and x, e 2L g(x) e c d 2 (, i ) i I ( 2 i g p dµ) 1/p, whee ( ) i i I is a boundedly ovelapping ( coveing of he whole space by balls of ) adius. Taking g = e (2 )L f e (2 )L f dµ gives ) (3.11) el f(x) e (2 )L f dµ ) e c d 2 (, i ) p 1/p ( 2 )L e(2 f e (2 )L f dµ dµ). i I i We hen decompose ) p ) 1/p )L f e e(2 (2 )L fdµ dµ i ) ) p- Osc i (e (2 )L f) + e (2 )L fdµ e (2 )L fdµ. i

17 DE GIORGI PROPERTY AND HEAT KERNEL OUNDS 17 The fis em is esimaed by (Hp,p): η ( ) p- Osc i (e (2 )L f) ω i 1/p f p, whee i is dilaed fom i a scale, and one uses he fac ha 2, since we have chosen ( ). Now by doubling i 1/p 4 1/p 1 + d(, ν/p i ) and since his is esimaed by ( 1/p 1 + d(, i ) The second em is esimaed by (3.9): ) e (2 )L fdµ ( i 1 + d(, i ) ) D+ ν p ω ( ) ν/p. ) e (2 )L fdµ ) 1/p f p (again one uses he fac ha 2 ). Coming back o (3.11), we obain ) el f(x) e (2 )L fdµ ( e c d 2 (, i ) d(, ) D+ ν ) i p ( ) ω 1/p f p. i I Since by doubling d 2 (, i ) i I ec yields (3.8). 2 (1 + d(, i ) ) D+ ν p is unifomly bounded, his Following some ideas in [20, Theoem 3.1], we can now idenify (H η p,p) as he popey needed o pass fom (UE) o (LE). Theoem 3.5. Le (M, d, µ, E) be a meic measue Diichle space saisfying (VD) and he uppe Gaussian esimae (UE). If hee exis p [1, + ] and η (0, 1] such ha (H η p,p) is saisfied, hen he lowe Gaussian bound (LE) holds. Convesely (LE) implies (H η p,p) fo all p [1, + ) and some η (0, 1]. Remak 3.6. Le us emphasise wo by-poducs of Theoem 3.5: (LE) is equivalen o he exisence of some p [1, + ) and some η (0, 1] such ha (Hp,p) η holds; The popey hee exiss η > 0 such ha (Hp,p) η holds is independen of p [1, + ). We efe he eade o Appendix A, whee one poves he independence of his popey on p [1, + ] (including he infinie exponen) by a diec agumen. In fac, we shall pove ha he popey (Hp,p) η iself is p-independen of p [1, + ], up o an abiaily small loss on η. Poof of Theoem 3.5. Fis assume (H η p,p) fo some p [1, + ] and some η > 0. y Poposiion 3.1, we know ha his esimae self-impoves ino (H η p, ). Fix a

18 18 FRÉDÉRIC ERNICOT, THIERRY COULHON, DOROTHEE FREY poin z M and conside he funcion f = p (, z). Then (Hp, ) η yields ( ) η d(x, y) p 2 (x, z) p 2 (y, z) 1/p p (, z) p, unifomly fo a.e. x, y wih d(x, y) and whee is any ball of adius conaining x, y (in paicula, p is coninuous and p (x, x) has a meaning). I follows fom (VD) and (UE) ha (3.12) p (, z) p and ha [ V (z, ] 1 p ) 1 V 1 (z, ) p 2 (z, z). Fo hese wo classical facs, see fo insance [20, Theoem 3.1]. Hence ( ) ( η d(x, y) V (z, ) 1/p ) p 2 (x, z) p 2 (y, z) V (z, ) 1 ( ) ( η d(x, y) V (z, ) 1/p ) p 2(z, z). Noe ha his esimae is nohing bu a slighly weake fom of he classical Hölde esimae (H η ) fom he inoducion. In paicula, fo x = z and evey y (x, ) we deduce ha (3.13) p 2 (x, x) p 2 (y, x) ( d(x, y) ) η p 2 (x, x). I is well-known ha (LE) follows (see fo insance [20, Theoem 3.1]). Assume now (LE). Since we have assumed (UE), i follows, hough he equivalence of (UE) + (LE) wih he paabolic Hanack inequaliy (see [64, Poposiion 3.2] o [45, Theoems ]), ha hee exis θ (0, 1) such ha, fo a.e. x, y, 0 <, and a.e. z M (3.14) p (x, z) p (y, z) ( 1 V (z, )V (x, ) ) θ (his is ye anohe vesion of (H η )). On he ohe hand, (UE) and doubling imply ha ( ) 1 (3.15) p (x, z) p (y, z) p (x, z) + p (y, z) V (x, ) exp c d2 (x, z). If d(x, z), V (z, ) V (x, ) and (3.14) yields p (x, z) p (y, z) 1 ( ) θ.

19 DE GIORGI PROPERTY AND HEAT KERNEL OUNDS 19 If d(x, z), one muliplies he squae oos of (3.14) and (3.15) o obain ( ) θ/2 ( ) 1 p (x, z) p (y, z) V 1/4 (z, )V 3/4 (x, exp c d2 (x, z) ) ( ) θ/2 ( ) 1/4 ( ) 1 V (x, ) = V (x, ) V (z, exp c d2 (x, z) ) ( ) θ/2 1, whee he las inequaliy uses again doubling. Now we poceed as in [20, Theoem 3.1]. We have jus shown ha p (x,.) p (y,.) 1 ( ) θ/2. The hea semigoup being submakovian, p (x,.) p (y,.) 1 2. I follows by Hölde inequaliy ha fo 1 p < + (3.16) p (x,.) p (y,.) p 1/p ( fo a.e. x, y, 0 <. Now e L f(x) e L f(y) M ) θ/2p, p (x, z) p (y, z) f(z) dµ(z) p (x,.) p (y,.) p f p, which ogehe wih (3.16) yields (H η p, ) wih η = θ/2p, hence (H η p,p) by Poposiion 3.1. Remak 3.7. In he case of a bounded space, ha is diam(m) < (which unde (VD) is equivalen o a finie measue M < ), o ge (LE) i is sufficien o have (Hp,p) η fo some p [1, + ] and η (0, 1] whee we conside only scales wih δ diam(m), fo some δ (0, 1). Indeed, le us assume his esiced (Hp,p) η popey. Following he above poof, we deduce he Hölde egulaiy (3.13) hence (LE) fo δ diam(m). Then define M = δ 2 diam 2 (M)/2 and conside 2 M. Fis, he self-impovemen given by Poposiion 3.1 sill holds in he same ange whee (Hp,p) η is assumed, so in paicula a he scale δ diam(m). This yields 2 (Hη p, ), ha is, fo evey funcion f L p (M, µ) and 0 < < δ diam(m), 2 ess sup e M L f(x) e M L f(y) ( ) η M 1/p f p, x,y M whee we have used he fac ha by doubling (x, M ) M fo evey x M. Since 2 M, we may apply his inequaliy o e ( M )L f insead of f. y he

20 20 FRÉDÉRIC ERNICOT, THIERRY COULHON, DOROTHEE FREY L p -boundedness of he semigoup wih M (x,, we ge ess sup e L f(x) e L f(y) ( ) η 1/p f p. x,y M Following he above poof of (3.13), one deduces ha fo evey x, y M wih d(x, y) δ diam(m) fo some δ < δ: ( ) δ η p (x, x) p (y, x) p (x, x) 1 δ 2 p (x, x), whee we have chosen δ small enough such ha he las inequaliy comes wih an exac consan smalle han 1/2. This gives fo d(x, y) δ diam(m) p (y, x) (x, ) 1 M 1. Then he sandad ieaion and he doubling popey allow us o exend his inequaliy fo evey x, y M, which gives (LE): p (y, x) (x, ) 1 d(x,y)2 c e. Remak 3.8. Poposiion 3.1 and Theoem 3.5 sill hold in he conex of sub- Gaussian esimaes. Insead of (UE), le us assume ha he hea kenel saisfies fo some m > 2 (UE m ) ( ( ) ) 1 d(x, y) m 1/(m1) p (x, y) V (x, 1/m ) exp, > 0, a.e. x, y M. C Then one can easily check ha he above emains ue by eplacing eveywhee he scaling faco by 1/m. One could also conside he moe geneal hea kenel esimaes fom [47, Secion 5], whee he equivalence wih maching Hanack inequaliies is poved, see also [8]. An alenaive poof of he second saemen in Theoem 3.5 can be given using [12, Theoem 6] insead of Poposiion 3.1. We leave he deails o he eade. Convesely, a naual follow-up of he end of he poof of Theoem 3.5 is o ge he esuls fom [12], ha is he exension of [20, Theoem 4.1] o he doubling seing. Thee is nohing essenially new hee, bu we shall give a poof of [12, Poposiion 10] fo he sake of compleeness. We fis need o inoduce he noion of evese doubling. I is known (see [44, Poposiion 5.2]), ha, if M is unbounded, conneced, and saisfies (VD ν ), one has a so-called evese doubling volume popey, namely hee exis 0 < ν ν and c > 0 such ha, fo all s > 0 and x M ( ν c s) V (x, ) V (x, s). Le us say ha (M, d, µ) saisfies (VD ν,ν ) if, fo all s > 0 and x M, ( ν V (x, ) ( ) ν c s) V (x, s) C. s

21 DE GIORGI PROPERTY AND HEAT KERNEL OUNDS 21 Fo he sake of simpliciy we shall se ouselves in he Gaussian case (m = 2 in he noaion of Remak 3.8), bu he geneal case is simila. Theoem 3.9. Le (M, d, µ, E) be a meic measue Diichle space saisfying (VD ν,ν ) and he uppe Gaussian esimae (UE). Then (LE) holds if and only if, fo some (all) p (1, + ), some α > ν p and α > ν f(x) f(y) f D, x, y M. 1 V 1/p (x, d(x, y)) p, (d α (x, y) L α/2 f p + d α (x, y) L α /2 f p ), Poof. Assume (LE). Le 1 < p < +, α, α > 0 o be chosen lae, and k N such ha k > max ( α, ) α 2 2. Le f D. Thanks o (UE) and o he fac ha by evese doubling V (x, ) + as +, e L f 0 in L 2 (M, µ), as + (see [17, Secion 3.1.2] fo deails). Since e L f is bounded in L 1 (M, µ), e L f 0 in L p (M, µ) by dualiy and inepolaion. Thus one can wie hence f(x) f(y) c(k) = c(k) f = c(k) + 0 k1 L k e L f d, k1 L k e L f(x) L k e L f(y) d k1 e (/2)L L k e (/2)L f(x) e (/2)L L k e (/2)L f(y) d. Now fo = d(x, y) and 0 <, we ge fom (3.12) and (VD ν ) e (/2)L L k e (/2)L f(x) e (/2)L L k e (/2)L f(y) p (x,.) p (y,.) p L k e (/2)L f p ( p (x,.) p + p (y,.) p ) L k e (/2)L f p V (x, ) 1/p (k α 2 ) L α/2 f p V (x, ) 1/p ( ) ν/p (k α 2 ) L α/2 f p, whee he las inequaliy uses he analyiciy of (e L ) >0 on L p (M, µ). Fo 0 <, we can wie as in he end of he poof of Theoem 3.5, e (/2)L L k e (/2)L f(x) e (/2)L L k e (/2)L f(y) p (x,.) p (y,.) p L k e (/2)L f p 1/p ( Now evese doubling yields e (/2)L L k e (/2)L f(x) e (/2)L L k e (/2)L f(y) V (x, ) 1/p ( ) θ 2p + ν p (k α 2 ) L α /2 f p. ) θ/2p α (k 2 ) L α /2 f p.

22 22 FRÉDÉRIC ERNICOT, THIERRY COULHON, DOROTHEE FREY Finally 2 f(x) f(y) V (x, ) 1/p L α/2 f p ν/p k1 ν 2p k+ α 2 d + V (x, ) 1/p L α /2 f p θ 2p + ν p The above inegals convege if α > ν p and α < θ ν 2p p k1 θ 4p ν α k+ 2p 2 d., in which case one obains f(x) f(y) V (x, ) 1/p ( α L α/2 f p + α L α /2 f p ). One can choose any α > ν and some p α > ν. The convese is easy, see [12, Theoem p 6]. Remak One can ake α = α if ν = ν, ecoveing in paicula he polynomial volume gowh case fom [20, Theoem 4.1]. 4. The case ν < p < + : fom Poincaé and gadien esimaes o hea kenel lowe bounds In [20, Thm. 5.2], i is poved ha if he volume gowh is polynomial of exponen ν 2, hen (R p ) and (P p ) fo ν < p < + imply (LE). Using he equivalence beween (G p ) and (1.7), i is easy o see ha he same poof woks wih (G p ) insead of (R p ). Ou nex heoem exends his esul o he doubling case, and in addiion is poof is moe diec. Theoem 4.1. Le (M, d, µ, E) be a meic measue Diichle space wih a caé du champ saisfying (VD ν ). Assume (G p ) and (P p ) fo some p (ν, + ). Then (LE) holds. Poof. Replacing f wih e L f in (P p ), we have, fo evey > 0 and evey ball of adius > 0, ) 1/p p- Osc (e L f) e L f p dµ. If is concenic wih and, ) 1/p ( ) e L f p 1/p dµ e L f p dµ) 1/p, hence by (VD ν ) ) 1/p ( ) ν e L f p p dµ 1/p e L f p ( ) ν p 1/p f p, whee he las inequaliy follows fom (G p ). Gaheing he wo above esimaes yields ( ) 1 ν p- Osc (e L p f) 1/p f p,

23 DE GIORGI PROPERTY AND HEAT KERNEL OUNDS 23 ha is (Hp,p) η wih η = 1 ν (0, 1) since p > ν. y Poposiion 2.1, (UE) also p holds. We conclude by applying Theoem 3.5. Remak 4.2. In he above saemen, (P p ) is necessay, since (LE) implies (P 2 ), bu (G p ) is no, as he example of conical manifolds shows (see [22]). Remak 4.3. Fo p = +, he above poof wih he obvious modificaions shows ha (G ) ogehe wih (UE) implies (LE). This also follows fom [25, Coollay 2.2] and he fac ha (G ) and (UE) imply ( G ). Le us give an alenaive poof of Theoem 4.1, which is moe diec, bu does no shed he same ligh on he ange 2 p ν (see Secion 5 below) as he above one. We shall sa wih a lemma which is close o [46, Theoem 5.1] and o seveal saemens in [19] (fo he polynomial volume gowh case), bu we find i useful o fomulae and pove i in he following simple and naual way, which is in fac inspied by [46, Theoem 3.2]. Lemma 4.4. Le (M, d, µ, E) be a meic measue Diichle space wih a caé du champ saisfying (VD ν ). Then (P p ) fo some p > ν implies he following Moey inequaliy: fo evey funcion f D and almos evey x, y M, d(x, y) (4.1) f(x) f(y) V 1/p (x, d(x, y)) f p. Poof. Le x, y be Lebesgue poins fo f. Le i (x) = (x, 2 i d(x, y)), fo i N 0. As in Lemma 3.3, one has f(x) fdµ p- Osc i (x)(f), 0 (x) i 0 Then using (P p ) and (VD ν ) which yields 0 (x) 2 iν i (x), we can wie f(x) fdµ ) 1/p 2 i d(x, y) f p dµ 0 (x) i 0 i (x) ( ) 2 2 i iν 1/p ( d(x, y) f p dµ i 0 0 (x) i (x) ( ) 2 i(1 ν p ) d(x, y) 0 (x) 1/p f p i 0 d(x, y) 0 (x) 1/p f p, whee we used p > ν. Similaly we have f(y) fdµ d(x, y) 0 (y) 1/p f p 0 (y) d(x, y) 0 (x) 1/p f p, whee 0 (x) 0 (y) follows fom doubling. Finally, as in Lemma 3.3, p fdµ fdµ ( f 1/p fdµ dµ), 0 (x) 0 (y) 2 0 (x) 2 0 (x) ) 1/p

24 24 FRÉDÉRIC ERNICOT, THIERRY COULHON, DOROTHEE FREY and by (P p ) The claim follows. 0 (x) fdµ 0 (y) fdµ d(x, y) 0(x) 1/p f p. We can now deive Theoem 4.1 easily. Replacing f wih e L f in he conclusion of Lemma 4.4 and applying (G p ) yields e L f(x)e L f(y) hence by (VD ν ) (4.2) e L f(x) e L f(y) d(x, y) V 1/p (x, d(x, y)) el f p ( d(x, y) d(x, y) [V (x, d(x, y))] 1/p f p, ) 1 ν p 1/p f p, fo evey f D and evey ball wih adius d(x, y) and conaining x. Le now be concenic o wih adius such ha 0 <. Since p > ν, i follows fom (4.2) ha ess sup e L f(x) e L f(y) x,y ( ) 1 ν p 1/p f p. This is nohing bu (Hp, ) η wih η = 1 ν (0, 1), and we conclude by using (UE) p as in he beginning of he poof of Theoem 3.5. An obvious by-poduc of Theoem 4.1 is he following monooniciy popey fo (G p ) + (P p ). Coollay 4.5. Le (M, d, µ, E) be a meic measue Diichle space wih a caé du champ saisfying (VD ν ). Then (G p ) + (P p ) fo some p > ν implies (P 2 ), hence (G q ) + (P q ) fo all q [2, p). Moeove, if (G p ) holds fo some p (ν, + ), hen fo evey q [2, p]. (P q ) (P 2 ) Poof. We have jus poven ha (G p ) + (P p ) fo p > ν implies (UE) and (LE). y [62], [70], (P 2 ) follows, hence (P q ) fo all q > 2. On he ohe hand, (G p ) implies (G q ) fo all q [2, p) by inepolaion. The las saemen follows in he same way. 5. Poincaé inequaliies and hea kenel bounds: he L 2 heoy The so-called De Giogi popey o Diichle popey on he gowh of he Diichle inegal fo hamonic funcions was inoduced by De Giogi in [27], fo L a second ode divegence fom diffeenial opeao wih eal coefficiens on R n : hee exiss ε (0, 1) such ha fo all R, evey pai of concenic balls, R wih adii, R and all funcions u W 1,2 (R n ) hamonic in 2 R, i.e. Lu = 0 in 2 R, one has ) 1/2 ( ) ε ) 1/2 R (5.1) u 2 dµ u 2 dµ. R

25 DE GIORGI PROPERTY AND HEAT KERNEL OUNDS 25 The De Giogi popey was subsequenly used in many woks and in vaious siuaions o pove Hölde egulaiy fo soluions of inhomogeneous ellipic equaions and sysems (see fo insance [40]). The idea o look a he hea equaion as a Laplace equaion whee he RHS is a ime deivaive, and o deduce paabolic egulaiy esuls fom ellipic ones by using a non-homogeneous equivalen vesion of De Giogi popey was inoduced in [1] fo L a second ode opeao in divegence fom on R n. In [2], he same ideas ae applied in a discee geomeic seing, and he ole of Poincaé inequaliies clealy appeas o ensue he ellipic egulaiy and he equivalence beween he homogeneous and non-homogeneous vesions of De Giogi. This is he appoach we will follow hee, while aking full advanage of Theoem 3.5. We shall conside he following non-homogeneous vesion of De Giogi popey. Wih he help of Lemma 5.7 below, one shows ha his fomulaion is a pioi weake han he one in [2, Poposiion 4.4]. We shall see in he poof of Poposiion 5.2 in Appendix ha unde (UE) i is equivalen o (5.1). Definiion 5.1 (De Giogi popey). Le (M, d, µ, E) be a meic measue Diichle space wih a caé du champ and L he associaed opeao. We say ha (DG 2,ε ) holds if he following is saisfied: fo all R, evey pai of concenic balls, R wih especive adii and R, and fo evey funcion f D, one has (DG 2,ε ) ) 1/2 f 2 dµ ( ) [ ε ) 1/2 R f 2 dµ + R Lf L (R)]. R We someimes omi he paamee ε, and wie (DG 2 ) if (DG 2,ε ) is saisfied fo some ε (0, 1). Le us now sae he counepa of a esul of [2] in he discee seing. Fo he convenience of he eade, we give a poof in Appendix. Poposiion 5.2. Le (M, d, µ, E) be a doubling meic measue Diichle space wih a caé du champ. Then (P 2 ) implies (DG 2 ). We ae now in a posiion o give a simple poof of he main saemen of [42], [62], and [70]. Fo simpliciy le us denoe in wha follows, fo a ball and f L 2 loc (M, µ): Osc (f) := 2- Osc (f) = 1/2 f f dµ dµ) 2. Theoem 5.3. Le (M, d, µ, E) be a doubling meic measue Diichle space wih a caé du champ. Then (P 2 ) implies (LE). Poof. Applying (P 2 ) o e L f fo > 0 and f L 2 (M, µ) on a ball fo > 0 yields ) 1/2 (5.2) Osc (e L f) e L f 2 dµ.

26 26 FRÉDÉRIC ERNICOT, THIERRY COULHON, DOROTHEE FREY Accoding o Poposiion 5.2, (DG 2,ε ) holds fo some ε > 0, hence ) 1/2 (5.3) e L f 2 dµ ( ) ε ( 1/2 e L f dµ) 2 + ess sup Le L f(x) x ( ) ε ( 1/2 e L f 2 + ) ess sup Le L f(x) x fo some ε (0, 1), 0 < and wih adius concenic o. y (G 2 ), (5.4) e L f 2 f 2. Now ecall ha unde ou assumpions, (UE) holds hanks o Poposiion 2.1. y [43, Coollay 3.3] (one can also use he complex ime bounds of [15, Poposiion 4.1] and a Cauchy fomula) he kenel of he opeao Le L also saisfies poinwise Gaussian esimaes. I follow ha (5.5) ess sup Le L f(x) 1/2 f 2. x 2 Puing ogehe (5.2), (5.3), (5.4) and (5.5) yields ( ) 1ε Osc (e L f) 1/2 f 2, ha is, (H η 2,2) wih η = 1ε > 0. This implies (LE) accoding o Theoem 3.5. The oiginal poofs of Theoem 5.3 wen hough he paabolic Hanack inequaliy; some [62 65,70] used a paabolic Mose ieaion, anohe one [42] icky geomeic agumens. In [47, Secion 4.2], a shoe poof was given, which wen in hee seps (wih a fouh one, boowed fom [35], o deduce paabolic Hanack fom (LE)). The fis one is o deive an ellipic egulaiy esimae fom (VD) and (P 2 ). We do no change his sep, which elies on he ellipic Mose ieaion; we give a poof fo he sake of compleeness in Poposiion.4 below. The second sep is o obain (UE). Ou appoach in Poposiion 2.1 is paiculaly simple since p = 2. The hid sep is a lowe bound on he Diichle hea kenel inside a ball whose adius is he squae oo of he ime unde consideaion. This is no ivial (see [47, pp ]) and hee lies ou main simplificaion. We fis push sep one a lile fuhe by deducing (DG 2 ) fom he ellipic egulaiy. We could hen deduce he paabolic egulaiy as in [2, Secion 4]). Insead, we use he self-impovemen of Hölde egulaiy esimaes on he semigoup fom Poposiion 3.1 and Theoem 3.5. Inoduce he scale-invaian local Sobolev inequaliy (LS q ) f 2 q 1 V 1 2 q (x, ) ( f E(f) ),

27 DE GIORGI PROPERTY AND HEAT KERNEL OUNDS 27 fo evey ball = (x, ), evey f F suppoed in (x, ), and fo some q > 2. This inequaliy was inoduced in [62] and was shown, unde (VD), o be equivalen o (DUE) in he Riemannian seing. The equivalence was saed in ou moe geneal seing in [69]. See also [13] fo many efomulaions of (LS q ), an alenaive poof of he equivalence wih (DUE), and moe efeences. The main aim of [47] is o pove ha he ellipic Hanack inequaliy, o an equivalen ellipic egulaiy esimae, ogehe wih (LS q ), o equivalenly (UE), implies he paabolic Hanack inequaliy. I is enough in his espec o pove (LE), since as we aleady said he paabolic Hanack inequaliy follows fom (UE) + (LE). This phenomenon falls in he cicle of he ideas we ae developing in he pesen wok, and, using a ansiion ick fom esimaes fo hamonic funcions o esimaes fo all funcions ogehe wih Theoem 3.5, we will now offe a simple poof of [47, Theoem 3.1]. Le us say ha u F is hamonic on a ball if Lu = 0 in he weak sense on. Noe ha he following saemen involves diam(m) as we wan o ea by he same oken he cases M bounded and unbounded. In a fis eading one can ceainly assume diam(m) = +. Theoem 5.4. Le (M, d, µ, E) be a doubling meic measue Diichle space wih a caé du champ saisfying (LS q ) fo some q > 2. Assume ha he following ellipic egulaiy esimae holds: hee exiss α > 0 and δ (0, 1) such ha fo evey x 0 M, R > 0 wih R < δ diam(m), u F hamonic in (x 0, R) and x, y (x 0, R/2), one has ( ) α d(x, y) (ER) u(x) u(y) Osc (x0,r)(u). R Then (LE) follows. Remak 5.5. I is known ha (P 2 ) implies (LS q ) fo some q > 2, see fo insance [62, Theoem 2.1], [26, Secion 5]. We shall also see in Poposiion.4 below ha (P 2 ) implies (ER). Thus Theoem 5.4 gives back Theoem 5.3. efoe we sa he poof of Theoem 5.4, ecall ha (LS q ) fo some q > 2 implies he following elaive Fabe-Kahn inequaliy ( 1/2 ( ) β ( ) 1/2 Ω (F K) f dµ) 2 f 2 dµ Ω V (x, ) Ω fo some β > 0, all balls (x, ), x M, (0, δ diam(m)) wih some δ < 1, and all f F suppoed in Ω (x, ). See fo insance [47, Theoem 2.5], as well as [13, Secion 3.3]. In paicula, one has ( ( (5.6) 1/2 f dµ) 2 (x,) 1/2 f dµ) 2, (x,) fo all balls (x, ), x M, (0, δ diam(m)) wih some δ < 1, and all f F suppoed in (x, ). We will need he following esul inspied by [2, Lemma 4.2]. Noe ha he ole classically played by ellipiciy in such Lax-Milgam ype agumens is played hee by (5.6).