Infinitesimally Lipschitz functions on metric spaces

INFINITESIMALL Y LIPSCHITZ FUNCTIONS ON M ETRIC SP A CES E. DURAND AND J.A. JARAMILLO Abstract. F or a metric space X , we study the space D ∞ ( X ) of b ounded functions on X whose inﬁnitesimal Lipschitz constan t is uniformly b ounded. D ∞ ( X ) is compared with the space LIP ∞ ( X ) of b ounded Lipsch itz functions on X , in terms of diﬀeren t prop erties regarding the geometry of X . W e also obtain a Ban ac h-Stone the orem in this con text. In the case of a metric measure space, w e also compare D ∞ ( X ) with the Newtonian-Sobolev space N 1 , ∞ ( X ). In particular, if X supp orts a doubling measure and satisﬁes a local P oincar ´ e inequalit y , we obt ain that D ∞ ( X ) = N 1 , ∞ ( X ). 1. In troduction Recent y ears hav e seen many adv ances in geometry and analysis , w here ﬁrs t o rder diﬀerential calculus has b een extended to the setting of s paces with no a prio ri smo oth structure; see for instance [Am , He1, He2, S]. The notion of der iv ative measures the inﬁnitesima l oscillations of a function at a given p oint, a nd giv es information concerning for instance monotono city . In gener al metric spaces we do not have a deriv ative, even in the weak sense of Sob olev spaces. Nevertheless, if f is a r eal-v a lued function on a metric space ( X , d ) and x is a p oint in X , one can use similar mea surements of sizes of ﬁr st-order os cillations o f f a t small scales around x , such as D r f ( x ) = 1 r sup n | f ( y ) − f ( x ) | : y ∈ X, d ( x, y ) ≤ r o . On one hand, this q uantit y do es not co nt ain as m uch infor mation as standar d deriv atives on Euclidea n spa ces does (since we omit the signs) but, on the o ther hand, it makes sense in more g eneral settings since we do not need any sp ecial behavior of t he underlying space to deﬁne it. In fact, if we lo o k at the sup erior limit of the a bove expression as r tends to 0 we a lmost r ecov er in many ca ses, as in the Euclidean or Riemannian setting, the standar d notion of deriv ative. More precisely , given a contin uous function f : X → R , the inﬁnitesimal Lipschitz c onstant at a po int x ∈ X is deﬁned a s follows: Lip f ( x ) = lim sup r → 0 D r f ( x ) = lim sup y → x y 6 = x | f ( x ) − f ( y ) | d ( x, y ) . Recently , this functional ha s play ed a n imp ortant r ole in several con texts. W e just men tion here the construction of diﬀerentiable structur es in the setting of metric measure spaces [Ch, K] , the theory of upper gra dient s [HK , Sh2], or the Stepanov’s diﬀerentiabilit y theorem [BRZ]. This concept gives rise to a class of function spaces , inﬁnitesimal ly Lipschitz function s p ac es , whic h contains in some sense inﬁnitesimal informatio n ab out the 1991 Mathematics Subje ct Classiﬁc ation. 46E15, 46E35. Researc h partially suppor ted by DGES (Spa in) M TM2006-03531. 1 2 E. DURAND AND J.A. JARAMILLO functions, D ( X ) = { f : X − → R : k Lip f k ∞ < + ∞} . This space D ( X ) clearly co nt ains the space LIP( X ) of Lipschitz function and a ﬁrst approach should b e co mparing such spaces . In Corolla ry 2 .6 we give suﬃcient conditions on the metr ic spac e X to guara ntee the equality b etw een D ( X ) a nd LIP( X ). A p ow erful to ol which tr ansforms bounds on inﬁnitesimal oscillation to bo unds on maximal os cillation is a kind of mea n v alue theor em (see Lemma 2 . 5 in [S]). In fact, the la rgest class of spa ces for which we obtain a p ositive answer is the class of q uasi-length spaces, which has a characterization in terms of such mean type v alue theorem. In particular , this class includes qua si-conv ex spaces. In addition, we pre sent some examples for which LIP ( X ) 6 = D ( X ) (see Examples 2.7 and 2.8). A t this p oint, it seems natural to approa ch the problem o f determining which kind of spa ces can b e classiﬁed by their inﬁnitesimal Lipsc hitz structur e. Our strategy will be to follow the pro o f in [GJ2] where the authors ﬁnd a large class of metric space s for which the alg ebra of b ounded Lipschitz functions determines the Lipsc hitz structure for X . A crucial point in the pro of is the use of the Ba- nach space structure o f LIP( X ). Th us, we endow D ( X ) with a norm which aris es naturally from the deﬁnition o f the op era tor Lip. This norm is not co mplete in the gener al case, as it ca n b e se en in Example 3.3 . Ho wev er , ther e is a wide class of spa ces, the lo cally r adially q uasiconv ex metric spaces (see Deﬁnition 3.1 ), for which D ∞ ( X ) (b ounded inﬁnitesimally Lipschitz functions) admits the desir ed Ba- nach space s tructure. Mor eov er, for such spaces, we o btain a kind o f B anach-Stone theorem in this framework (se e Theo rem 4.7). If w e hav e a measur e on the metric space, w e can dea l wit h man y more pro blems. In this line, there ar e for example genera lizations of classica l Sob olev spa ces to the setting of arbitrar y metric mea sure spac es. It seems that Ha js laz w as the ﬁrst who in tro duced So bo lev type spaces in this context [Ha2]. He deﬁned the spaces M 1 ,p ( X ) for 1 ≤ p ≤ ∞ in connection with max imal op era tors. It is well known that M 1 , ∞ ( X ) is in fact the space of b ounded Lipschitz functions on X . Shanmugalingam in [Sh2] intro duced, using the notion of upp er gra dient (and more generally w eak upper gradients) the Newtonia n spaces N 1 ,p ( X ) for 1 ≤ p < ∞ . The generaliza tion to the case p = ∞ is straightforward and we will co mpare the function spaces D ∞ ( X ) and LIP ∞ ( X ) with such Sob olev space, N 1 , ∞ . F ro m Cheeg er’s work [Ch], metric spaces with a doubling measur e and a Poincar´ e inequality admit a diﬀerentiable structure with which Lipschitz functions ca n b e diﬀerentiated a lmost everywhere. Under the same hypotheses we prov e in C orollar y 5.1 6 the equality of all the mentioned spaces . F urther more, if we just require a lo cal Poincar ´ e inequality we obtain M 1 , ∞ ( X ) ⊆ D ∞ ( X ) = N 1 , ∞ ( X ). F or further information ab o ut more generaliza tions of Sob olev spaces on metric mea sure spac es see [Ha 1]. W e organize d the work a s follows. In Section 2 we will introduce inﬁn itesimal ly Lipschitz function sp ac es D ( X ) and we lo ok for conditions re garding the geometry of the metric spaces we are working with in order to under stand in which ca ses the inﬁnitesimal Lipschitz information yields the glo bal Lipschitz b ehavior of a function. Moreov er, we show the e xistence of metric spaces for which LIP( X ) ( D ( X ). In Section 3 w e in tr o duce the class of lo c al ly r adially qu asic onvex metric sp ac es a nd w e prov e tha t the spa ce of b ounded inﬁnitesimally Lipschitz function can b e endow e d with a natural Banach space structure. The purp ose of Section 4 is to state a kind of Banach-Stone theo rem in this context while the aim of Sec tion 5 is to compar e INFINITESIMALL Y LIPSCHITZ FUNCTIONS ON METRIC SP ACES 3 the function spac es D ∞ ( X ) a nd LIP ∞ ( X ) w ith Sob olev space s in metr ic measure spaces. 2. infinitesimall y Lipschitz functions Let ( X, d ) be a metric spac e. Giv en a function f : X → R , the inﬁnitesimal Lipschitz c onstant of f a t a non iso lated p oint x ∈ X is deﬁned a s follows: Lip f ( x ) = lim sup y → x y 6 = x | f ( x ) − f ( y ) | d ( x, y ) . If x is an isolated p oint we deﬁne Lip f ( x ) = 0 . This v alue is a lso known a s u pp er sc ale d oscil lation (se e [BRZ]) or as p ointwise inﬁnitesimal Lipschitz nu mb er (see [He2]). Examples 2.1. (1) If f ∈ C 1 (Ω) where Ω is a n o p en subs et o f E uclidean space, o r of a Riemannian manifo ld, then Lip f = |∇ f | . (2) Let H b e the ﬁrs t Heisenber g group, a nd consider an op en subset Ω ⊂ H . If f ∈ C 1 H (Ω), that is, f is H − contin uo usly diﬀerentiable in Ω, then Lip f = |∇ H f | where ∇ H f deno tes the hor izontal g radient o f f . F or further details see [Ma]. (3) If ( X, d, µ ) is a metric measure spa ce which admits a measura ble diﬀerentiable structure { ( X α , x α ) } α and f ∈ LIP( X ), then Lip f ( x ) = | d α f ( x ) | µ − a.e. , where d α f denotes the Cheeger’s diﬀerential. F o r further information ab o ut measur able diﬀerentiable structures see [Ch, K]. Lo osely sp eak ing, the op er ator Lip f estimates s ome kind of inﬁnitesimal lips- chit zian pr op erty around each point. Our ﬁr st aim is to see under which conditions a function f : X → R is Lipschitz if a nd only if Lip f is a b ounded functional. It is clear that if f is a L − Lipschitz function, then Lip f ( x ) ≤ L for every x ∈ X . Mor e precisely , we consider the following s paces of functions: ⋄ LIP( X ) = { f : X − → R : f is Lips chit z } ⋄ D ( X ) = { f : X − → R : sup x ∈ X Lip f ( x ) = k Lip f k ∞ < + ∞} . W e denote by L IP ∞ ( X ) (res pe ctively D ∞ ( X )) the s pace o f bo unded Lipschitz functions (resp ectively , bo unded functions which are in D ( X )) and C ( X ) will denote the s pace of co nt inuous functions on X . It is not diﬃcult to see that for f ∈ D ( X ), Lip f is a Borel function on X and that k Lip( · ) k ∞ yields a seminorm in D ( X ). In what follows, k · k ∞ will denote the s upremum norm whereas k · k L ∞ will denote the essential supr emum nor m, provided we have a measure o n X . In a ddition, LIP( · ) will denote the Lipschitz co nstant. Since functions with unifor mly b ounded inﬁnitesima l Lipschitz constant hav e a ﬂav o ur of diﬀere n tiability it seems reasonable to determine if the inﬁnitesima lly Lipschitz functions ar e in fact contin uous. Namely , Lemma 2.2. L et ( X , d ) b e a metric sp ac e. Then D ( X ) ⊂ C ( X ) . Pr o of. Let x 0 ∈ X b e a non isolated po int and f ∈ D ( X ). W e are going to see that f is contin uous at x 0 . Since f ∈ D ( X ) we have that k Lip f k ∞ = M < ∞ , in particular, Lip f ( x 0 ) ≤ M . By deﬁnition we hav e that Lip f ( x 0 ) = inf r > 0 sup d ( x 0 ,y ) ≤ r y 6 = x 0 | f ( x 0 ) − f ( y ) | d ( x 0 , y ) . 4 E. DURAND AND J.A. JARAMILLO Fix ε > 0. Then, ther e ex ists r > 0 such that | f ( x 0 ) − f ( z ) | d ( x 0 , z ) ≤ sup d ( x 0 ,y ) ≤ r y 6 = x 0 | f ( x 0 ) − f ( y ) | d ( x 0 , y ) ≤ M + ε ∀ z ∈ B ( x 0 , r ) , and so | f ( x 0 ) − f ( z ) | ≤ ( M + ε ) d ( x 0 , z ) ∀ z ∈ B ( x 0 , r ) . Thu s, if d ( x 0 , z ) → 0 then | f ( x 0 ) − f ( z ) | → 0, and so f is contin uo us at x 0 .  Now we lo ok for conditions regarding the geometry of the metric space X under which L IP( X ) = D ( X ) (resp ectively L IP ∞ ( X ) = D ∞ ( X )). As it can b e exp ected, we need some kind o f c onne cte dness . In fact, we are going to obtain a p o sitive answer in the class of length sp ac es or , more generally , of quasi-c onvex sp ac es . Recall that the length o f a contin uo us curve γ : [ a, b ] → X in a metric s pace ( X , d ) is deﬁned as ℓ ( γ ) = sup n n − 1 X i =0 d ( γ ( t i ) , γ ( t i +1 )) o where the supremum is taken over all partitions a = t 0 < t 1 < · · · < t n = b of the interv al [ a, b ]. W e will say that a cur ve γ is es r e ctiﬁable if ℓ ( γ ) < ∞ . No w, ( X, d ) is s aid to b e a length sp ac e if for each pair of points x, y ∈ X the distance d ( x, y ) coincides with the inﬁmum of all le ngths of curves in X connecting x with y . Another in teresting class o f metric spa ces, whic h cont ains leng th spaces, are the so c alled quasi-c onvex spaces. Reca ll that a metric spa ce ( X , d ) is quasi-c onvex if there exists a constant C > 0 such that for eac h pair of points x, y ∈ X , there exists a curve γ connecting x and y with ℓ ( γ ) ≤ C d ( x, y ) . As one can exp ect, a metric space is quasi-c onvex if, and only if, it is bi-Lipschitz homeomorphic to s ome length space. W e b egin o ur analy sis with a technical result. Lemma 2.3. L et ( X , d ) b e a metric s p ac e and let f ∈ D ( X ) . L et x, y ∈ X and supp ose that t her e exists a r e ctiﬁable curve γ : [ a, b ] → X c onne cting x and y , t hat is, γ ( a ) = x and γ ( b ) = y . Then, | f ( x ) − f ( y ) | ≤ k Lip f k ∞ ℓ ( γ ) . Pr o of. Since f ∈ D ( X ), we have that M = k Lip f k ∞ < + ∞ . Fix ε > 0 . F or each t ∈ [ a, b ] ther e exists ρ t > 0 such that if z ∈ B ( γ ( t ) , ρ t ) \ { γ ( t ) } then | f ( γ ( t )) − f ( z ) | ≤ ( M + ε ) d ( γ ( t ) , z ) . Since γ is contin uo us, there ex ists δ t > 0 such that I t = ( t − δ t , t + δ t ) ⊂ γ − 1 ( B ( γ ( t ) , ρ t )) . The family o f interv als { I t } t ∈ [ a,b ] is an op en cov ering of [ a, b ] and by compactness it a dmits a ﬁnite s ubcovering which will b e denote by { I t i } n +1 i =0 . W e may as sume, reﬁning the s ubcovering if nec essary , tha t an interv al I t i is not contained in I t j for i 6 = j . If we relab el the indices o f the po ints t i in no n-decreasing o rder, we can now choose a po int p i,i +1 ∈ I t i ∩ I t i +1 ∩ ( t i , t i +1 ) for each 1 ≤ i ≤ n − 1. Using the auxiliary p oints that we have just chosen, w e deduce that: d ( x, γ ( t 1 )) + n − 1 X i =1 h d ( γ ( t i ) , γ ( p i,i +1 )) + d ( γ ( p i,i +1 ) , γ ( t i +1 )) i + d ( γ ( t n ) , y ) ≤ ℓ ( γ ) , and so | f ( x ) − f ( y ) | ≤ ( M + ε ) ℓ ( γ ). Finally , since this is true for each ε > 0 , we conclude that | f ( x ) − f ( y ) | ≤ k Lip f k ∞ ℓ ( γ ), a s wan ted.  As a str aightforw ard consequence of the prev ious res ult, we de duce INFINITESIMALL Y LIPSCHITZ FUNCTIONS ON METRIC SP ACES 5 Corollary 2.4. If ( X , d ) is a quasi-c onvex sp ac e t hen LIP( X ) = D ( X ) . The pro of of the previous result is bas ed on the existence of curves connecting each pair of p oints in X and who se length can be estimated in ter ms of the distance betw een the po int s. A rea sonable kind of spaces in which we can appro ach the problem of determining if LIP( X ) and D ( X ) coincide, are the so called chainable sp ac es . It is an interesting class of metric spaces containing length spaces and quasi-convex spaces. Recall that a metric space ( X , d ) is said to b e wel l-chaine d o r chainable if for ev ery pair of p oints x, y ∈ X and for every ε > 0 there exists a n ε − chain joining x and y , that is, a ﬁnite se quence of po in ts z 1 = x, z 2 , . . . , z ℓ = y such that d ( z i , z i +1 ) < ε , for i = 1 , 2 , . . . , ℓ − 1 . In such spa ces there exis t “chains” of p oints which co nnect tw o g iven p oints, and for which the distance betw een the no des , which ar e the p oints z 1 , z 2 , . . . , z ℓ , is a rbitrary sma ll. Howev er, throughout some examples we will se e that there ex ists chainable spaces for which the spaces of functions LIP( X ) and D ( X ) do not coincide (see Example 2.7). Nev ertheless, if we w ork with a metr ic space X in which we can control the num ber of no des in the c hain betw een tw o p oints in terms of the distance b etw een that p oints, then we will o btain a p ositive answer to our problem. A chainable space fo r which ther e exists a co nstant K (whic h only depends on X ) such that for every ε > 0 and for every x, y ∈ X there ex ists an ε − chain z 1 = x, z 2 , . . . , z ℓ = y such that ( ℓ − 1) ε ≤ K ( d ( x, y ) + ε ) is called a quasi-length sp ac e . In Lemma 2.5. [S], Semmes gave a characterization of qua si-length s paces in terms of a condition which reminds a k ind o f “ mean v alue theorem”. Lemma 2.5. A metric sp ac e ( X , d ) is a quasi-length sp ac e if and only if ther e exists a c onstant K such that for e ach ε > 0 and e ach function f : X − → R we have that | f ( x ) − f ( y ) | ≤ K ( d ( x, y ) + ε ) sup z ∈ X D ε f ( z ) for e ach x, y ∈ X , wher e D ε f ( z ) = 1 ε sup n | f ( y ) − f ( z ) | : y ∈ X , d ( z , y ) ≤ ε o . The previo us characterization allows us to give a p os itive answ er to o ur problem for quas i-length space s. More precisely , we ha ve the following: Corollary 2.6. L et ( X, d ) b e a quasi-length sp ac e. Then, LIP( X ) = D ( X ) . Pr o of. W e hav e to check that D ( X ) ⊂ LIP( X ). Let f ∈ D ( X ) and deno te by M = k Lip f k ∞ < + ∞ . Since X is a quasi-length space we obtain, aplying Lemma 2.5, that there ex ists a c onstant K ≥ 1 such that | f ( x ) − f ( y ) | ≤ K ( d ( x, y ) + ε ) sup z ∈ X D ε f ( z ) for each x, y ∈ X and each ε > 0. Thus, if we ta ke the sup erior limit when ε tends to zero we deduce: | f ( x ) − f ( y ) | ≤ K d ( x, y ) sup z ∈ X Lip f ( z ) = K M d ( x, y ) for each x, y ∈ X . Thus, f is a K M − Lipschitz function and we are done.  W e will see in 3.5 that the co n verse of Corollar y 2.6 is true under more restrictive hypothesis. 6 E. DURAND AND J.A. JARAMILLO Next, let us see that there exist metric space s for which LIP( X ) ( D ( X ). W e will appro ach this b y constructing tw o metric spa ces for which LIP ∞ ( X ) 6 = D ∞ ( X ). In the ﬁrst example w e see that the equality fails “fo r la rge dis tances” while in the second one it fails “for inﬁnitesimal dista nces”. Example 2.7. Deﬁne X = [0 , ∞ ) = S n ≥ 1 [ n − 1 , n ] , a nd write I n = [ n − 1 , n ] for each n ≥ 1. Consider the sequence of functions f n : [0 , 1] → R given by f n ( x ) =    x if x ∈  0 , 1 n  nx + n − 1 n 2 if x ∈  1 n , 1  . F o r each pa ir of po int s x, y ∈ I n , w e write d n ( x, y ) = f n ( | x − y | ) , and w e deﬁne a metric on X as follows. Given a pa ir o f points x, y ∈ X with x < y , x ∈ I n , y ∈ I m we deﬁne d ( x, y ) =    d n ( x, y ) if n = m d n ( x, n ) + P m − 1 i = n +1 d i ( i − 1 , i ) + d m ( m − 1 , y ) if n < m A stra ightforw ard co mputation shows that d is in fact a metric and it coincides lo cally with the Euclidean metric d e . More pr ecisely , if x ∈ I n , on J x =  x − 1 n + 1 , x + 1 n + 1  we have that d | J x = d e | J x . Next, cons ider the b ounded function g : X → R g iven by g ( x ) = ( 2 k − x if x ∈ I 2 k , x − 2 k if x ∈ I 2 k +1 . Let us chec k that g ∈ D ∞ ( X ) \ LIP ∞ ( X ). Indeed, le t x ∈ X a nd assume that there exists n ≥ 1 such that x ∈ I n . Then, we hav e that if y ∈ J x , Lip f ( x ) = lim sup y → x y 6 = x | g ( x ) − g ( y ) | d ( x, y ) = lim sup y → x y 6 = x | x − y | | x − y | = 1 . Therefore, g ∈ D ∞ ( X ). On the other hand, for ea ch positive integer n we hav e | g ( n − 1 ) − g ( n ) | = 1 and d ( n − 1 , n ) = f n (1) = 2 n − 1 n 2 . Thus, we obtain that lim n →∞ | g ( n − 1) − g ( n ) | d ( n − 1 , n ) = lim n →∞ 1 2 n − 1 n 2 = ∞ and so g is not a Lipschitz function. In particular, since LIP( X ) 6 = D ( X ), we deduce b y Coro llary 2 .6 that X is not a quasi- conv ex s pace. How ever, it ca n b e check ed that X is a chainable space.  Example 2.8. Consider the set X = { ( x, y ) ∈ R 2 : y 3 = x 2 , − 1 ≤ x ≤ 1 } = { ( t 3 , t 2 ) , − 1 ≤ t ≤ 1 } , and let d be the restrictio n to X of the Euclidean metric o f R 2 . W e deﬁne the bo unded function g : X → R , ( x, y ) 7→ g ( x, y ) = ( y if x ≥ 0, − y if x ≤ 0. Let us see that g ∈ D ∞ ( X ) \ LIP ∞ ( X ). INFINITESIMALL Y LIPSCHITZ FUNCTIONS ON METRIC SP ACES 7 Indeed, if t 6 = 0, it ca n b e check ed that Lip g ( t 3 , t 2 ) ≤ 1. On the other hand, at the or igin we hav e Lip g (0 , 0) = lim sup ( x,y ) → (0 , 0) | g ( x, y ) − g (0 , 0) | d (( x, y ) , (0 , 0)) = lim sup t → 0 t 2 p ( t 3 ) 2 + ( t 2 ) 2 = 1 . Thu s, we o btain that k Lip f k ∞ = 1 and so g ∈ D ∞ ( X ). T a ke now tw o symmetric po int s from the cusp with resp ect to the y − a xis, that is, A t = ( t 3 , t 2 ) and B t = ( − t 3 , t 2 ) for 0 < t < 1. In this c ase, we get d ( A t , B t ) = 2 t 3 and | f ( A t ) − f ( B t ) | = t 2 − ( − t 2 ) = 2 t 2 . If t tends to 0, we hav e lim t → 0 + | f ( A t ) − f ( B t ) | d ( A t , B t ) = lim t → 0 + 2 t 2 2 t 3 = lim t → 0 + 1 t = + ∞ . Thu s, g is not a Lipschitz function.  In ge neral, if X is non compact space we ha ve that LIP lo c ( X ) ( ( LIP( X ) ( LIP lo c ( X ) ∩ D ( X ) C ( X ) ( ( D ( X ) where LIP lo c ( X ) denotes the s pace of lo cally Lipschitz functions . Recall that in 2.7 we have cons tructed a function f ∈ LIP lo c ( X ) ∩ D ( X ) \ LIP ( X ). In a ddition, there is no inclusio n relation betw een LIP lo c ( X ) and D ( X ). Indeed, consider fo r instance the metric spa ce X = S ∞ i =1 B i ⊂ R with the Euclidean distance where B i = B ( i, 1 / 3) denotes the open ball centered at ( i, 0) and radius 1 / 3. One can chec k that the function f ( x ) = ix if x ∈ B i is lo cally Lipschitz wherea s f / ∈ D ( X ) bec ause k Lip f k ∞ = ∞ . On the other ha nd, the function g in E xample 2.8 b elo ngs to D ( X ) \ LIP lo c ( X ). 3. A Ba n ach sp a ce structure f or infinitesimall y Lipschitz functions In this section we searc h for suﬃcien t conditions to hav e a con v erse for Corollary 2.6. W e b eg in in tro ducing a kind of metric space s which will play a central r ole throughout this section. In addition, for such spaces, w e will endow the spa ce of functions D ∞ ( X ) and D ( X ) with a Banach s tructure. Deﬁnition 3. 1. Let ( X , d ) be a metric space. W e say that X is lo c al ly r adial ly quasi-c onvex if for ea ch x ∈ X , ther e e xists a neig hborho o d U x and a constant K x > 0 suc h that for each y ∈ U x there exists a rectiﬁable curve α in U x connecting x and y suc h that ℓ ( γ ) ≤ K x d ( x, y ). Note that the spaces in tro duced in the Ex amples 2.7 a nd 2 .8 are lo ca lly r adially quasi-convex. Observe that there exis t lo cally ra dially quasi-conv ex space s which are not lo cally quasi-convex. Indeed, let X = S ∞ n =1  ( x, x n ) : x ∈ R  and d b e the restriction to X of the Euclidean metric of R 2 . It can b e chec k ed that ( X, d ) is lo cally r adially quasi-c onv e x but it is not lo cally qua si-conv ex. Next, we endow the space D ∞ ( X ) with the following norm: k f k D ∞ = max {k f k ∞ , k Lip f k ∞ } for each f ∈ D ∞ ( X ). Theorem 3. 2. L et ( X , d ) b e a lo c al ly r adial ly quasi-c onvex metric sp ac e. Then, ( D ∞ ( X ) , k · k D ∞ ) is a Banach sp ac e. 8 E. DURAND AND J.A. JARAMILLO Pr o of. Let { f n } n be a Ca uch y sequence in ( D ∞ ( X ) , k · k D ∞ ). Since { f n } n is uni- formly Cauch y , there exists f ∈ C ( X ) such that f n → f with the no rm k · k ∞ . Let us see that f ∈ D ( X ) and that { f n } n conv erges to f with resp ect to the seminorm k Lip( · ) k ∞ . Indeed, let x ∈ X . Since ( X , d ) is lo cally radially quasi-co n vex, there exist a neighborho o d U x and a constant K x > 0 s uch that for ea ch y ∈ U x there e xists a rectiﬁable curve γ which c onnects x and y such that ℓ ( γ ) ≤ K x d ( x, y ). B y Lemma 2.3, we ﬁnd that for each y ∈ U x and for each n, m ≥ 1 | f n ( x ) − f m ( x ) − ( f n ( y ) − f m ( y )) | ≤ k Lip( f n − f m ) k ∞ K x d ( x, y ) . Let r > 0 b e such that B ( x, r ) ⊂ U x and let y ∈ B ( x, r ). W e have that    f n ( x ) − f m ( x ) r − f n ( y ) − f m ( y ) r    ≤ k Lip( f n − f m ) k ∞ K x d ( x, y ) r ≤ k Lip( f n − f m ) k ∞ K x . Let ε > 0. Since { f n } n is a Cauch y sequence with r esp ect to the seminor m k Lip( · ) k ∞ , there exists n 1 ≥ 1 such that if n, m ≥ n 1 , then k Lip( f n − f m ) k ∞ < ε 4 K x . Thu s, for e ach r > 0 such that B ( x, r ) ⊂ U x and for ea ch n, m ≥ n 1 , we hav e the following chain of inequalities    | f n ( x ) − f n ( y ) | r − | f m ( x ) − f m ( y ) | r    ≤    f n ( x ) − f m ( x ) r − f n ( y ) − f m ( y ) r    ≤ k Lip( f n − f m ) k ∞ K x < ε 4 for each y ∈ B ( x, r ). In particular , for ea ch n ≥ n 1 , we obtain that | f n ( x ) − f n ( y ) | r ≤    | f n ( x ) − f n ( y ) | r − | f n 1 ( x ) − f n 1 ( y ) | r    + | f n 1 ( x ) − f n 1 ( y ) | r < | f n 1 ( x ) − f n 1 ( y ) | r + ε 4 . Thu s, the previous inequalit y implies, upon ta king the supre m um o ver B ( x, r ), that sup y ∈ B ( x,r ) n | f n ( x ) − f n ( y ) | r o ≤ sup y ∈ B ( x,r ) n | f n 1 ( x ) − f n 1 ( y ) | r o + ε 4 for each r > 0 such that B ( x, r ) ⊂ U x . On the other hand, for Lip( f n 1 )( x ), there exists r 0 > 0, s uch that if 0 < r < r 0 , then B ( x, r ) ⊂ U x and sup y ∈ B ( x,r ) n | f n 1 ( x ) − f n 1 ( y ) | r o ≤ Lip( f n 1 )( x ) + ε 4 . Hence, for each n ≥ n 1 and each 0 < r < r 0 , we obtain that sup y ∈ B ( x,r ) n | f n ( x ) − f n ( y ) | r o ≤ Lip( f n 1 )( x ) + 2 ε 4 . Since f n is a Cauch y sequence with resp ect to the seminorm k Lip( · ) k ∞ , then the sequence of real num b er s k Lip( f n ) k ∞ is a Cauch y sequence to o and so there INFINITESIMALL Y LIPSCHITZ FUNCTIONS ON METRIC SP ACES 9 exists M > 0 suc h that k Lip( f n ) k ∞ < M for each n ≥ 1 . In particular, for ea ch n ≥ n 1 and 0 < r < r 0 , we obtain the following: sup x ∈ B ( x, r ) n | f n ( x ) − f n ( y ) | r o < Lip( f n 1 )( x ) + 2 ε 4 ≤ k Lip( f n 1 ) k ∞ + ε 2 ≤ M + ε 2 . Now, let us see what happ ens with f . If n ≥ n 1 , 0 < r < r 0 and y ∈ B ( x, r ), we hav e that | f ( x ) − f ( y ) | r ≤ | f ( x ) − f n ( x ) | r + | f n ( x ) − f n ( y ) | r + | f n ( y ) − f ( y ) | r ≤ | f ( x ) − f n ( x ) | r + | f n ( y ) − f ( y ) | r + M + ε 2 . Since { f n } n conv erges unifor mly to f , it co nv er ges p oint wise to f a nd so there exists n ≥ n 1 such that | f ( x ) − f n ( x ) | + | f n ( y ) − f ( y ) | < εr 2 . Putting all ab ov e together we deduce that | f ( x ) − f ( y ) | r ≤ M + ε. Thu s, tha t inequality implies, upo n taking the inﬁmum ov er B ( x, r ) and letting r tending to 0 that Lip( f )( x ) ≤ M + ε for each x ∈ X . No w, if ε → 0, we have tha t Lip( f )( x ) ≤ M for ea ch x ∈ X . And so k Lip f k ∞ ≤ M < + ∞ which implies f ∈ D ( X ). T o ﬁnish the pr o of, let us see that k Lip( f n − f ) k ∞ − → 0. Using the a b ov e notation we hav e that if n, m ≥ n 1 and 0 < r < r 0 | f n ( x ) − f ( x ) − ( f n ( y ) − f ( y )) | r ≤ | f n ( x ) − f m ( x ) − ( f n ( y ) − f m ( y )) | r + | f m ( x ) − f ( x ) | r + | f ( y ) − f m ( y ) | r ≤ | f m ( x ) − f ( x ) | r + | f ( y ) − f m ( y ) | r + ε 4 . The sequence { f n } n conv erges u niformly to f a nd, in particular, it converges p oint- wise to f . Thus, there e xists n ≥ n 1 such that | f ( x ) − f n ( x ) | + | f n ( y ) − f ( y ) | < εr 2 . Hence, we hav e | f n ( x ) − f ( x ) − ( f n ( y ) − f ( y )) | r < ε. Thu s, we deduce that if n ≥ n 1 , then Lip( f n − f )( x ) ≤ ε. This is true for ea ch x ∈ X , and so we o btain that k Lip( f n − f ) k ∞ ≤ ε if n ≥ n 1 . Therefore, we have that k Lip( f n − f ) k ∞ − → 0. Th us, we conclude that ( D ∞ ( X ) , k · k D ∞ ) is a Banach space as wan ted.  Let us see how ever that in g eneral ( D ∞ ( X ) , k · k D ∞ ) is not a Banach space. Example 3.3 . Co nsider the co nnected metric spa ce X = X 0 ∪ S ∞ n =1 X n ∪ G ⊂ R 2 with the metric induced by the Euclidea n one, where X 0 = { 0 } × [0 , + ∞ ), X n = { 1 n } × [0 , n ], n ∈ N and G = { ( x, 1 x ) : 0 < x ≤ 1 } . F or ea ch n ∈ N consider the sequence of functions f n : X → [0 , 1] g iven by f n  1 k , y  = ( k − y k √ k if 1 ≤ k ≤ n 0 if k > n, 10 E. DURAND AND J.A. JARAMILLO and f n ( x, y ) = 0 if x 6 = 1 k ∀ k ∈ N . O bserve that f n ( 1 k , 0) = 1 √ k and f n ( 1 k , k ) = 0 if 1 ≤ k ≤ n . Since Lip f n ( 1 k , y ) = 1 k √ k and Lip f n ( x, y ) = 0 if x 6 = 1 k ∀ k ∈ N , we have that f n ∈ D ∞ ( X ) for each n ≥ 1. In addition, if 1 < n < m , k f n − f m k ∞ = 1 √ n + 1 and k Lip( f n − f m ) k ∞ = 1 ( n + 1) √ n + 1 . Thu s, we deduce that { f n } n is a Ca uch y sequence in ( D ∞ ( X ) , k · k D ∞ ). Howev e r, if f n → f in D ∞ then f n → f p oint wis e. Then f m ( 1 n , 0) = 1 √ n for each m ≥ n and so f ( 1 n , 0) = 1 √ n and f (0 , 0) = 0. Thus, we obtain that Lip( f )(0 , 0) ≥ lim n →∞ | f (( 1 n ) , 0) − f (0 , 0) | d ( 1 n , 0) = lim n →∞ 1 √ n 1 n = + ∞ , and so f / ∈ D ∞ ( X ). This means that ( D ∞ ( X ) , k · k D ∞ ) is not a Ba nach spa ce. Theorem 3 .4. L et ( X , d ) b e a c onne cte d lo c al ly ra dial ly quasi-c onvex metric sp ac e and l et x 0 ∈ X . If we c onsider on D ( X ) the norm k · k D = max {| f ( x 0 ) | , k Lip( · ) k ∞ } , then ( D ( X ) , k · k D ) is a Banach sp ac e. Pr o of. By hypo thesis, for each y ∈ X , there ex ists a neighbor ho o d U y such that for each z ∈ U y , there exists a r ectiﬁable curve in U y connecting z and y . Since X is connected, there exists a ﬁnite seque nce o f p oints y 1 , . . . , y m such that U y k ∩ U y k +1 6 = ∅ for k = 1 , . . . , m − 1 , x ∈ U y 1 and x 0 ∈ U y m . Now, fo r each k = 1 . . . m , choose a p oint z k ∈ U y k ∩ U y k +1 . T o simplify notation w e write z 0 = x 0 and z n +1 = x . F o r each k = 1 . . . m , we choo se a curve γ k which connects z k with z k +1 . T aking γ = γ 0 ∪ . . . γ m we o btain a rectiﬁable curve γ which connects x 0 and x . Let us see now that ( D ( X ) , k · k D ) is a Ba nach s pace. Indeed, let { f n } n be a Cauch y seq uence. W e consider the case on which f n ( x 0 ) = 0 for each n ≥ 1. The general case can b e do ne in a similar w ay . By combining the previous argument with Lemma 2.3, we obtain that for n, m ≥ 1 and for each x ∈ X , we hav e that | f n ( x ) − f m ( x ) | ≤ k Lip( f n − f m ) k ∞ ℓ ( γ ) where γ is a rectiﬁable curve connecting x and x 0 . Since { f n } n is a Cauch y se- quence with resp ect to the se minorm k Lip( · ) k ∞ , the sequence { f n ( x ) } n is a Ca uch y sequence for ea ch x ∈ X , and therefor e, it conv erges to a po int y = f ( x ). The n, in particular, { f n } n conv erges point wise to a function f : X → R . Next, one ﬁnds using the sa me str ategy as in Theorem 3.2 (where we have just used the p oint wise co n vergence) tha t a Cauch y sequence { f n } n ⊂ D ( X ) such that f n ( x 0 ) = 0 for each n ≥ 1, conv er ges in ( D ( X ) , k · k D ) to a function f ∈ D ( X ).  W e are now prepar ed to state the co nv er se of Coro llary 2.6. Corollary 3 .5. L et ( X, d ) b e a c onne cte d lo c al ly r adial ly qu asi-c onvex metric s p ac e such that LIP ( X ) = D ( X ) . Then X is a qu asi-length sp ac e. Pr o of. In view of Lemma 2 .5 we have to prove that there exists K > 0 such that for each ε > 0 and each function f : X − → R we hav e that: | f ( x ) − f ( y ) | ≤ K ( d ( x, y ) + ε ) sup z ∈ X D ε f ( z ) ∀ x, y ∈ X ( ∗ ) . Indeed, let ε > 0. If sup z ∈ X D ε f ( z ) = ∞ , then ( ∗ ) is trivia lly tr ue. Thus, we may assume that sup z ∈ X D ε f ( z ) < ∞ . Since k Lip f k ∞ ≤ sup z ∈ X D ε f ( z ) then f ∈ D ( X ) and we distinguish tw o cases: INFINITESIMALL Y LIPSCHITZ FUNCTIONS ON METRIC SP ACES 11 (1) If k Lip f k ∞ = 0 then f is loca lly consta nt and so constant b eca use X is connected. Therefore, the inequality trivially holds. (2) If k Lip f k ∞ 6 = 0 , using that f ∈ D ( X ) = LIP( X ), we hav e the following inequality | f ( x ) − f ( y ) | ≤ LIP ( f ) d ( x, y ) ∀ x, y ∈ X . Now, ﬁx a p o int x 0 ∈ X . Since LIP( X ) = D ( X ) is a Banach spac e with bo th nor ms k f k LIP = max { LIP( f ) , | f ( x 0 ) |} a nd k f k D = max {k Lip f k ∞ , | f ( x 0 ) |} , (see Theorem 3.4 a nd e.g. [W]) and k · k D ≤ k · k LIP , then there exists a constant K > 0 s uch that k · k LIP ≤ K k · k D . Thus, if we co nsider the function g = f − f ( x 0 ) we hav e tha t LIP( f ) = LIP( g ) = k g k LIP ≤ K k g k D = K k Lip g k ∞ = K k Lip f k ∞ ( ♥ ) . Thu s, we obtain that | f ( x ) − f ( y ) | ≤ LIP( f ) d ( x, y ) ≤ LIP( f )( d ( x, y ) + ε ) ( ♥ ) ≤ K k Lip f k ∞ ( d ( x, y ) + ε ) ≤ K sup z ∈ X D ε f ( z )( d ( x, y ) + ε ) ∀ x, y ∈ X , as wan ted.  4. A Banach-Stone Theorem for infinitesimall y Lipschitz functions There exist many results in the litera ture rela ting the top olo gical structure of a top ological space X with the alg ebraic or topolo gical-a lgebraic struc tures of certain function spa ces deﬁned on it. The classica l Banach-Stone theorem asserts that for a c ompact space X , the linear metric str ucture of C ( X ) endow ed with the sup- norm determines the top ology o f X . Results a long this line for space s of Lipschitz functions hav e b een r ecently obtained in [GJ2, GJ3]. In this section we prov e tw o versions of the Banach-Stone theore m for the function spaces D ∞ ( X ) and D ( X ) resp ectively , where X is a lo cally radially q uasi-conv ex space. Since in general D ( X ) has not an alg ebra structure we will consider on it its natural unita l vector lattice structure. On the other ha nd, on D ∞ ( X ) we will consider b oth, its alge bra and its unital vector lattice structures. The concept of r eal-v a lued inﬁnitesimally Lipschitz function can b e g eneralized in a natural way when the target s pace is a metric space. Deﬁnition 4.1. Let ( X, d X ) and ( Y , d Y ) b e metric spaces. Given a function f : X → Y we deﬁne Lip f ( x ) = lim sup y → x y 6 = x d Y ( f ( x ) , f ( y )) d X ( x, y ) for e ach non-iso lated x ∈ X . If x is an iso lated p oint we deﬁne Lip f ( x ) = 0. W e consider the following space of functions D ( X , Y ) = { f : X − → Y : k Lip f k ∞ < + ∞} . As w e hav e seen in Lemma 2.2 we may obs erve that if f ∈ D ( X , Y ) then f is con- tin uous. It can b e easily chec k ed that w e hav e also a Leibniz’s r ule in this context, that is, if f , g ∈ D ∞ ( X ), then k Lip( f · g ) k ∞ ≤ k Lip f k ∞ k g k ∞ + k Lip g k ∞ k f k ∞ . In this wa y , we ca n alwa ys endow the space D ∞ ( X ) w ith a natural alg ebra str uc- ture. Note tha t D ∞ ( X ) is uniformly sep ar ating in the sense that for every pair of 12 E. DURAND AND J.A. JARAMILLO subsets A a nd B of X with d ( A, B ) > 0, ther e exists some f ∈ D ∞ ( X ) such that f ( A ) ∩ f ( B ) = ∅ . In o ur c ase, if A and B ar e s ubsets o f X with d ( A, B ) = α > 0, then the function f = inf { d ( · , A ) , α } ∈ LIP ∞ ( X ) ⊂ D ∞ ( X ) satisﬁes that f = 0 on A a nd f = α on B . In a ddition, we can endow either D ∞ ( X ) o r D ( X ) with a natural unital vector lattice s tructure. W e deno te by H ( D ∞ ( X )) the s et of a ll nonzero algebra homomor phisms ϕ : D ∞ ( X ) → R , that is, the set of all nonzer o multiplicativ e linea r functionals on D ∞ ( X ). No te that in pa rticular e very algebra homo morphism ϕ ∈ H ( D ∞ ( X )) is po sitive, that is, ϕ ( f ) ≥ 0 when f ≥ 0. Indeed, if f a nd 1 /f are in D ∞ ( X ), then ϕ ( f · (1 / f )) = 1 implies that ϕ ( f ) 6 = 0 a nd ϕ (1 /f ) = 1 /ϕ ( f ). Th us, if we assume that ϕ is not pos itive, then there exists f ≥ 0 with ϕ ( f ) < 0. The function g = f − ϕ ( f ) ≥ − ϕ ( f ) > 0, satisﬁes g ∈ D ∞ ( X ), 1 /g ∈ D ∞ ( X ) and ϕ ( g ) = 0 which is a contradiction. Now, we endow H ( D ∞ ( X )) with the top ology of p oint wise conv ergence (that is, considered as a to po logical subspace of R D ∞ ( X ) with the pro duct topo logy). This construc tion is sta ndard (see for instance [I]), but w e give some details fo r completeness. It is ea sy to chec k that H ( D ∞ ( X )) is closed in R D ∞ ( X ) and therefore is a compact space. In addition, since D ∞ ( X ) separ ates p oints and closed sets, X can b e e m b edded as a top olog ical subspace of H ( D ∞ ( X )) identif ying each x ∈ X with the po int e v aluation ho momorphism δ x given by δ x ( f ) = f ( x ), for every f ∈ D ∞ ( X ). W e are going to see that X is dense in H ( D ∞ ( X )). Indeed, given ϕ ∈ H ( D ∞ ( X )), f 1 , . . . , f n ∈ D ∞ ( X ), and ε > 0, there exists s ome x ∈ X such that | δ x ( f i ) − ϕ ( f i ) | < ε , for i = 1 , . . . , n . Otherwise, the function g = P n i =1 | f i − ϕ ( f i ) | ∈ D ∞ ( X ) w ould satisfy g ≥ ε and ϕ ( g ) = 0, a nd this is imp os sible since ϕ is p ositive. It follows that H ( D ∞ ( X )) is a compactiﬁcation of X . Mor eov er, ev ery f ∈ D ∞ ( X ) admits a contin uous extension to H ( D ∞ ( X )), namely by deﬁning b f ( ϕ ) = ϕ ( f ) for all ϕ ∈ H ( D ∞ ( X )). Lemma 4.2. L et ( X , d ) b e a met ric sp ac e and ϕ ∈ H ( D ∞ ( X )) . Then, ϕ : D ∞ ( X ) → R is a c ontinuous map. Pr o of. Let f ∈ D ∞ ( X ). W e know that it admits a contin uous extensio n b f : H ( D ∞ ( X )) → R so that b f ( ϕ ) = ϕ ( f ). Thus, s ince X is dense in H ( D ∞ ( X )), | ϕ ( f ) | = | b f ( ϕ ) | ≤ sup η ∈ H ( D ∞ ( X )) | b f ( η ) | = sup x ∈ X | f ( x ) | ≤ k f k D ∞ and we are done.  Recall that we hav e s hown in Theorem 3.2 that if X is a lo ca lly radially quasi- conv ex space then ( D ∞ ( X ) , k · k D ∞ ) is a Banach space. Using this in a c rucial wa y , w e next give some r esults which will give ris e to a Bana ch-Stone theorem for D ∞ ( X ). Lemma 4.3 . L et ( X , d X ) and ( Y , d Y ) b e lo c al ly ra dial ly quasi-c onvex met r ic sp ac es. Then, every unital algebr a homomorphi sm T : D ∞ ( X ) → D ∞ ( Y ) is c ontinuous for the r esp e ct ive D ∞ -norms. Pr o of. In order to prov e the contin uit y of the linea r map T , we a pply the closed graph theo rem. It is enough to chec k that given a sequence { f n } n ⊂ D ∞ ( X ) with k f n − f k D ∞ conv ergent to z ero a nd g ∈ D ∞ ( X ) such that k T ( f n ) − g k D ∞ also conv ergent to zero, then T ( f ) = g . Indeed, let y ∈ Y , and let δ y ∈ H ( D ∞ ( Y )) be the homomorphis m g iven by the ev aluatio n at y , that is, δ y ( h ) = h ( y ). By Lemma INFINITESIMALL Y LIPSCHITZ FUNCTIONS ON METRIC SP ACES 13 4.2, we hav e that δ y ◦ T ∈ H ( D ∞ ( X )) is co nt inuous a nd so T ( f n )( y ) = ( δ y ◦ T )( f n ) → ( δ y ◦ T )( f ) = T ( f )( y ) when n → ∞ . On the other hand, since co nv er gence in D ∞ − norm implies p oint w ise conv er- gence, then T ( f n )( y ) conv erges to g ( y ). That is, T ( f )( y ) = g ( y ), for e ach y ∈ Y . Hence, T ( f ) = g a s wan ted.  As a cons equence, we obtain the following result conce rning the co mpo sition o f inﬁnitesimally Lipschitz functions. Prop ositi on 4.4. L et ( X , d X ) and ( Y , d Y ) b e lo c al ly r adial ly quasi-c onvex metric sp ac es and let h : X → Y . Supp ose t hat f ◦ h ∈ D ∞ ( X ) for e ach f ∈ D ∞ ( Y ) . Then h ∈ D ( X , Y ) . Pr o of. W e b egin by chec king that h is a co ntin uous function, that is, h − 1 ( C ) is closed in X for each closed s ubset C in Y . Let C b e a closed subset o f Y a nd x 0 ∈ Y \ C . T a ke f = inf { d ( · , C ) , d ( x 0 , C ) } ∈ D ∞ ( Y ) which satisﬁes that f ( x 0 ) = 1 and f ( y ) = 0 for each y ∈ C . Let us obs erve that f ( x ) = 0 if and o nly if x ∈ C and so f − 1 ( f ( C )) = C . Th us, since f ◦ h is contin uous and f ( C ) = 0 is closed in R , h − 1 ( C ) = h − 1 ( f − 1 ( f ( C ))) = ( f ◦ h ) − 1 ( f ( C )) is closed in Y . Now, let x 0 ∈ X such that h ( x 0 ) is not an isola ted p oint. Note that if a ll po int s belong ing to h ( X ) are iso lated we hav e that Lip h ( x ) = 0 for each x ∈ X and so k Lip h k ∞ = 0, which implies that h ∈ D ( X , Y ). Thus, we may as- sume that there exists x 0 ∈ X s uch that h ( x 0 ) is not an isola ted p oint. Let f x 0 = min { d Y ( · , h ( x 0 )) , 1 } ∈ D ∞ ( Y ), since it is a Lipschit z function. W e have that Lip( f x 0 ◦ h )( x 0 ) = lim sup y → x 0 y 6 = x 0 | f x 0 ◦ h ( y ) − f x 0 ◦ h ( x 0 ) | d X ( x 0 , y ) = lim sup y → x 0 y 6 = x 0 | f x 0 ◦ h ( y ) | d X ( x 0 , y ) = lim sup y → x 0 y 6 = x 0 | min { d Y ( h ( y ) , h ( x 0 )) , 1 }| d X ( x 0 , y ) ( ∗ ) = Lip h ( x 0 ) . The equality ( ∗ ) holds b ecause, a s we hav e chec ked ab ov e, the map h is c ontin uous. Thu s, we obtain that Lip h ( x 0 ) = Lip( f x 0 ◦ h )( x 0 ) ≤ k Lip( f x 0 ◦ h ) k ∞ ≤ k f x 0 ◦ h k D ∞ ( X ) ( ♥ ) ≤ K k f x 0 k D ∞ ( Y ) ( ∗ ) = K k Lip( f x 0 ) k ∞ ( † ) for a certain consta n t K > 0 dep ending only on g . F or ( ♥ ) we hav e used that, by Lemma 4.3 , the homomo rphism T : D ∞ ( Y ) → D ∞ ( X ), g → g ◦ h is contin uo us. The inequality ( ∗ ) ho lds true beca use k f x k D ∞ ( Y ) = max {k f x 0 k ∞ , k Lip( f x 0 ) k ∞ } , k f x 0 k ∞ ≤ 1 and k Lip( f x 0 ) k ∞ = 1. It remains to chec k that, k Lip( f x 0 ) k ∞ = 1. Indeed, Lip f x 0 ( z ) = lim sup z ′ → z z ′ 6 = z | f x 0 ( z ′ ) − f x 0 ( z ) | d Y ( z ′ , z ) . W e hav e to disting uish three diﬀerent cases: (i) If d Y ( z , h ( x 0 )) > 1 , there exists a neighborho o d V z where d Y ( z ′ , h ( x 0 )) > 1 for ea ch z ′ ∈ V z and f x 0 | V z = 1. Thus, Lip f x 0 ( z ) = 0. 14 E. DURAND AND J.A. JARAMILLO (ii) If d Y ( z , h ( x 0 )) < 1 , there exists a neighborho o d V z where d Y ( z ′ , h ( x 0 )) > 1 for ea ch z ′ ∈ V z and so Lip f x 0 ( z ) = lim sup z ′ → z z ′ 6 = z | d Y ( z ′ , h ( x 0 )) − d Y ( z , h ( x 0 )) | d Y ( z ′ , z ) ≤ lim sup z ′ → z z ′ 6 = z d Y ( z ′ , z ) d Y ( z ′ , z ) = 1 . (iii) If d Y ( z , h ( x 0 )) = 1, then Lip f x 0 ( z ) = lim sup z ′ → z z ′ 6 = z 1 − min { d Y ( z ′ , h ( x 0 )) , 1 } d Y ( z ′ , z ) . If d Y ( z ′ , h ( x 0 )) ≥ 1 , then 1 − min { d Y ( z ′ , h ( x 0 )) , 1 } = 0. On the other hand, if d Y ( z ′ , h ( x 0 )) < 1 , then 1 − min { d Y ( z ′ , h ( x 0 )) , 1 } = d Y ( z , h ( x 0 )) − d Y ( z ′ , h ( x 0 )) ≤ d Y ( z , z ′ ) . Hence, we deduce that Lip f x 0 ( z ) ≤ 1 . On the o ther hand, since h ( x 0 ) is not an isolated p o in t, Lip f x 0 ( z ) = 1 and so k Lip f k ∞ = 1 beca use we hav e seen that k Lip f k ∞ ≤ 1. Then, up on taking the supremum over X in b oth sides of the inequality ( † ) we conclude that k Lip h k ∞ ≤ K , a s wan ted.  Remark 4.5. If we lo o k at Theorem 3 . 12 in [GJ2], where a n ana logous result to Prop ositio n 4.4 for Lipschitz functions is obtained, we can see that the arg ument there is bas ed on the fac t that the distance can b e ex pressed in terms of Lipschitz functions. In our case , w e cannot use the same strateg y since we do not know how to compare the v alues of an inﬁnitesimally Lipschit z function at tw o arbitra ry po int s of the space. Finally , w e need the following useful Lemma, which shows that the p oints in X can be top o logically distinguished into H ( D ∞ ( X )). It is essentially known (see for instance [GJ1]) but we g ive a pro of for completeness . Lemma 4. 6. L et ( X , d ) b e a c omplete metric sp ac e and let ϕ ∈ H ( D ∞ ( Y )) . Then ϕ has a c ountable neighb orho o d b asis in H ( D ∞ ( X )) if, and only if, ϕ ∈ X . Pr o of. Supp ose ﬁrst that ϕ ∈ H ( D ∞ ( Y )) \ X has a countable neighbor ho o d basis. Since X is dens e in H ( D ∞ ( X )), there exists a sequence ( x n ) in X con verging to ϕ . The completeness o f X implies tha t ( x n ) has no d − Cauch y s equence, a nd therefore there exist ε > 0 a nd a subsequence ( x n k ) such that d ( x n k , x n j ) ≥ ε for k 6 = j . Now, the sets A = { x n k : k even } a nd B = { x n k : k o dd } satisfy d ( A, B ) ≥ ε , and since D ∞ ( X ) is unifor mly s eparating, there is a function f ∈ D ∞ ( X ) with f ( A ) ∩ f ( B ) = ∅ . But this is a co nt radiction since f ex tends con tinu ously to H ( D ∞ ( X )) and ϕ is in the closure o f b oth A and B . Conv ersely , if ϕ ∈ X , co nsider B n the op en ba ll in X with center ϕ and radius 1 /n . Then the family { B n } n of the clo sures of B n in H ( D ∞ ( X )) is easily seen to be a countable neighborho o d basis as re quired.  Now, we ar e in a positio n to show that the algebra structure of D ∞ ( X ) de- termines the inﬁnitesimal Lipschitz structure of a complete lo cally radially quasi- conv ex metric spac e. W e say that tw o metric spaces X and Y ar e inﬁn itesimal ly Lipschitz home omorphic if there exists a bijection h : X → Y such that h ∈ D ( X , Y ) and h − 1 ∈ D ( Y , X ). Theorem 4.7. (Banac h-Stone t yp e) L et ( X, d X ) and ( Y , d Y ) b e c omplete lo c al ly r adial ly quasi-c onvex metric sp ac es. The fol lowing ar e e quivalent: INFINITESIMALL Y LIPSCHITZ FUNCTIONS ON METRIC SP ACES 15 (a) X is inﬁ nitesimal ly Lipschitz home omorphi c t o Y . (b) D ∞ ( X ) is isomorphic to D ∞ ( Y ) as unital algebr as. (c) D ∞ ( X ) is isomorphic to D ∞ ( Y ) as unital ve ctor lattic es. Pr o of. ( a ) = ⇒ ( b ) If h : X → Y is an inﬁnitesimally Lipschitz homeo morphism, then it is easy to c heck the map T : D ∞ ( Y ) → D ∞ ( X ), f 7→ T ( f ) = f ◦ h , is an isomorphism of unital algebra s. ( b ) = ⇒ ( a ) Let T : D ∞ ( X ) → D ∞ ( Y ) b e an isomorphis m of unital a lgebras . W e deﬁne h : H ( D ∞ ( Y )) → H ( D ∞ ( X )), ϕ 7→ h ( ϕ ) = ϕ ◦ T . Let us see ﬁrst that h is an homeomor phism. T o reach that aim, it is enough to prov e that h is bijective, closed and contin uous. Since T is an isomor phism, h − 1 ( ψ ) = ψ ◦ T − 1 exists for every ψ ∈ H ( D ∞ ( X )), and so h is bijective. In addition, once we chec k that h is contin uous we will a lso have that h is closed b eca use H ( D ∞ ( Y )) is compact and H ( D ∞ ( X )) is a Hausdorﬀ s pace. Now co nsider the following diagram: Y T ( f )     / / H ( D ∞ ( Y )) h / / [ T ( f )   b f ◦ h ' ' P P P P P P P H ( D ∞ ( X )) b f   o o ? _ X f   R R R R Here, b f (re sp ectively [ T ( f )) deno tes the co nt inuous extension o f f (re sp ectively T ( f )) to H ( D ∞ ( X )). Thus, h is cont inuous if and o nly if b f ◦ h is contin uous for all f ∈ D ∞ ( X ). Hence, it is enough to prov e that b f ◦ h = [ T ( f ). Since X is dense in H ( D ∞ ( X )), it is suﬃces to chec k that [ T ( f )( δ x ) = b f ◦ h ( δ x ) , where δ x denotes the ev aluation homomor phism for each x ∈ X . It is clea r that, b f ◦ h ( δ x ) = ( h ◦ δ x )( f ) = ( δ x ◦ T )( f ) = δ T ( f )( x ) = δ x ( T f ) = [ T ( f )( δ x ) , and so h is contin uo us. By L emma 4 .6 we have that a p oint ϕ ∈ H ( D ∞ ( X )) ha s a co unt able neighbor- ho o d ba sis in H ( D ∞ ( X )) if and only if it co rresp onds to a p oint of X . Since the same holds for Y and H ( D ∞ ( Y )) we conclude that h ( Y ) = X and by Prop ositio n 4.4 we hav e that h | Y ∈ D ( Y , X ). Analogo usly , h − 1 | X ∈ D ( X , Y ) and so X a nd Y are inﬁnitesimally Lipschitz homeomor phic. T o prov e ( b ) ⇐ ⇒ ( c ) W e use that D ∞ ( X ) is close d under b ounde d inversion which means that if f ∈ D ∞ ( X ) and f ≥ 1, then 1 /f ∈ D ∞ ( X ). Indeed, if f ∈ D ∞ ( X ) and f ≥ 1, given ε > 0 there exists r > 0 such that | f ( x ) − f ( y ) | d ( x, y ) ≤ s up d ( x,y ) ≤ r y 6 = x | f ( x ) − f ( y ) | d ( x, y ) ≤ M + ε ∀ y ∈ B ( x, r ) ( ⋆ ) . Thu s, given x ∈ X ,    1 f ( y ) − 1 f ( x )    = | f ( x ) − f ( y ) | | f ( x ) f ( y ) | ( ∗ ) ≤ d ( x, y )( M + ε ) ∀ y ∈ B ( x, r ) , where inequality ( ∗ ) is obtained after applying ( ⋆ ) and the fa ct that | f ( x ) f ( y ) | ≥ 1. Thu s, the conclusion follows from Lemma 2 . 3 in [GJ2].  Corollary 4 .8. L et ( X, d X ) and ( Y , d Y ) b e c omplete lo c al ly r adial ly quasi-c onvex metric sp ac es. The fol lowing assertions ar e e quivalent: (a) X is inﬁ nitesimal ly Lipschitz home omorphi c t o Y . 16 E. DURAND AND J.A. JARAMILLO (b) D ( X ) is isomorph ic to D ( Y ) as u nital ve ctor lattic es. Pr o of. ( a ) = ⇒ ( b ) If h : X → Y is an inﬁnitesimally Lipschitz homeo morphism, then it is clear that the map T : D ( Y ) → D ( X ), f 7→ T ( f ) = f ◦ h , is an isomorphism of unital vector la ttices. ( b ) = ⇒ ( a ) It follows from Theorem 4.7, since each homomorphism of unital vector lattices T : D ( Y ) → D ( X ) takes bo unded functions to b ounded functions. Indeed, if | f | ≤ M then | T ( f ) | = T ( | f | ) ≤ T ( M ) = M .  Next we deal with what we call inﬁ nitesimal isometries b etw een metric spa ces, related to inﬁnitesimally Lipschitz functions. Deﬁnition 4.9. Let ( X , d X ) and ( Y , d Y ) be metric spaces . W e say that X and Y are inﬁn itesimal ly isometric if there exists a bijection h : X − → Y suc h that k Lip h k ∞ = k Lip h − 1 k ∞ = 1. Remark 4.10. W e deduce from the pro ofs o f Prop os ition 4.4 and Theorem 4.7 t hat t wo complete lo cally radia lly quas i-conv ex metric spaces X and Y ar e inﬁnitesimally isometric if, and only if, ther e exists an alg ebra isomorphism T : D ∞ ( Y ) → D ∞ ( X ) which is a n isometry for the k · k D ∞ -norms (that is, k T k = k T − 1 k = 1). It is c lear that if tw o metric spac es a re lo cally is ometric, then they a re inﬁnite- simally isometric. The conv erse is not true, as we can see througho ut the following example. Example 4 .11. Let ( X , d ) be the metric spac e intro duced in E xample 2.8 a nd let ( Y , d ′ ) b e the metric s pace deﬁned in the follo wing w ay . Consider the interv al Y = [ − 1 , 1] and let us deﬁne a metr ic o n it a s follows: d ′ ( t, s ) =          d (( t 3 , t 2 ) , ( s 3 , s 2 )) if t, s ∈ [ − 1 , 0] , d (( t 3 , t 2 ) , ( s 3 , s 2 )) if t, s ∈ [0 , 1] , d (( t 3 , t 2 ) , (0 , 0)) + d ((0 , 0) , ( s 3 , s 2 )) if t ∈ [ − 1 , 0] , s ∈ [0 , 1] . It is easy to see that d ′ deﬁnes a metric. W e deﬁne h : X → Y , ( t 3 , t 2 ) → t. Let us observ e that k Lip h k ∞ = k Lip h − 1 k ∞ = 1 a nd s o X and Y are inﬁnitesimally isometric. How ev er, at the origin (0 , 0), for e ach r > 0 we hav e that d ( z , y ) 6 = d ′ ( h ( z ) , h ( y )) ∀ z , y ∈ B ((0 , 0) , r ) . Thu s, h is an inﬁnitesimal isometry , but not a lo cal isometry . In fact, it ca n b e chec ked that there is no lo cal is ometry f : X − → Y . (4.12) Non compl ete case. If X is a metric s pace and e X denotes its completion, then b oth metric spa ces hav e the same uniformly contin uous functions. Therefor e, LIP( X ) = LIP( e X ), and completeness o f spaces ca nnot b e avoided in the Lips- chit zian cas e. W e are interested in ho w co mpleteness assumption works for the D -case. It w ould b e useful to analyze if there exists a Bana ch-Stone theorem for not co mplete metric spaces. Example 4.13. Let ( X , d ) b e the metric space given b y X = { ( x, y ) ∈ R 2 : y 3 = x 2 , − 1 ≤ x ≤ 1 } = { ( t 3 , t 2 ) , − 1 ≤ t ≤ 1 } , INFINITESIMALL Y LIPSCHITZ FUNCTIONS ON METRIC SP ACES 17 where d is the restrictio n to X of the Euc lidean metric of R 2 . Let ( Y , d ′ ) b e the metric spa ce given by Y = X \{ 0 } and d ′ = d | Y . Obser ve that ( X, d ) is the completion of ( Y , d ′ ). The function h : Y → R , ( x, y ) 7→  1 if x < 0 0 if x > 0 , belo ngs to D ( Y ) but h cannot b e ev en contin uously ex tended to X . Thu s, D ( Y ) 6 = D ( X ). In the follo wing example we construc t a metric spa ce X such that D ( X ) = D ( e X ), where e X deno tes the completion o f X , a nd so that X is not homeomo rphic to e X . This fact illustra tes that, a pr iori, o ne cannot exp ect a c onclusive result for the non complete case. Example 4.14. Let X b e a metric space deﬁned as fo llows: X = { ( t 3 , t 2 ) , − 1 ≤ t ≤ 1 } ∪ { ( x, 1) ∈ R 2 : 1 ≤ x < 2 } = A ∪ B . ✲ ✻ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ) Now, we consider the completion of X : e X = { ( t 3 , t 2 ) , − 1 ≤ t ≤ 1 } ∪ { ( x, 1) ∈ R 2 : 1 ≤ x ≤ 2 } = e A ∪ e B . Let f ∈ D ( X ). First of a ll, D ( B ) = LIP( B ), since B is a quasi-length spa ce, and so, b y McShane ’s theor em (see [He1]), there e xists F ∈ LIP( e B ) such that F | B = f . Thu s, G ( x, y ) = ( f ( x, y ) if ( x, y ) ∈ A = e A F ( x, y ) if ( x, y ) ∈ e B , is a D − extension o f f to the completion e X . And so D ( X ) = D ( e X ). How ever, X is not homeomor phic to e X since e X is co mpact but X is not. 5. So bolev sp aces on metric measure sp aces Along this section, w e alwa ys assume that ( X , d, µ ) is a metric mea sure space, where µ is a Bor el r e gular me asur e , tha t is, µ is an outer measure on a metric space ( X , d ) such that all Borel sets a re µ − measurable and for each set A ⊂ X there exists a Borel set B such that A ⊂ B a nd µ ( A ) = µ ( B ). Our a im in this sec tion is to compa re the function s paces D ∞ ( X ) and LIP ∞ ( X ) with c ertain Sobo lev spaces on metric-measure s paces. There are several p ossible extensions of the class ical theor y of Sob olev space s to the setting of metric spaces equipp ed with a B orel measure. F ollowing [Am] and [Ha1] w e record the de ﬁnition of M 1 ,p spaces: (5.1) Ha j lasz-Sob ol ev space. F or 0 < p ≤ ∞ the spa ce f M 1 ,p ( X, d, µ ) is deﬁned as the set of all functions f ∈ L p ( X ) for which there exis ts a function 0 ≤ g ∈ L p ( X ) such that | f ( x ) − f ( y ) | ≤ d ( x, y )( g ( x ) + g ( y )) µ − a.e. ( ∗ ) . 18 E. DURAND AND J.A. JARAMILLO As usua l, we get the spac e M 1 ,p ( X, d, µ ) after ide n tifying a ny tw o functions u, v ∈ f M 1 ,p ( X, d, µ ) such that u = v almost everywhere with r esp ect to µ . The space M 1 ,p ( X, d, µ ) is equipp ed with the norm k f k M 1 ,p = k f k L p + inf g k g k L p , where the inﬁmum is tak en ov er all functions 0 ≤ g ∈ L p ( X ) that satisfy the requirement ( ∗ ). In particula r, if p = ∞ it can be shown that M 1 , ∞ ( X, d, µ ) coincides with LIP ∞ ( X ) provided that µ ( B ) > 0 for every op en ball B ⊂ X (s ee [Am]) a nd that 1 / 2 k · k LIP ∞ ≤ k · k M 1 , ∞ ≤ k · k LIP ∞ . In this case we obtain that M 1 , ∞ ( X ) = LIP ∞ ( X ) ⊆ D ∞ ( X ). (5.2) Newtonian space. Another interesting generalizatio n o f Sob olev spaces to general metric spa ces a re the so -called Newtonian Spac es, intro duced by Shanm un- galingam [Sh1, Sh2]. Its deﬁnition is based on the notion of the upper g radient that we r ecall here for the sake of completeness . A non-neg ative Borel function g on X is said to b e an upp er gr adient for an extended r eal-v a lued function f on X, if | f ( γ ( a )) − f ( γ ( b )) | ≤ Z γ g ( ∗ ) for every rectiﬁable curve γ : [ a, b ] → X . W e see that the upp er gr adient plays the role of a deriv ative in the formu la ( ∗ ) which is s imilar to the one related to the fundamen tal theorem of ca lculus. The p oint is that using upp er gr adients w e may hav e many of the pro p erties of ordinary Sobolev spac es ev en though we do not hav e deriv atives of our functions. If g is an upper g radient of u and e g = g almost everywhere, then it may happ en that e g is no longer an upp er g radient for u . W e do not wan t our upp er gra dient s to be sensitive to c hanges o n small sets. T o avoid this unpleasant situation the no tion of we ak upp er gr adient is introduced as follows. First we need a wa y to measure how large a family o f cur ves is. The most imp or tant point is if a family of curves is small enoug h to be ignored. This kind o f pro blem was ﬁrst a pproached in [F u]. In what follows let Υ ≡ Υ( X ) denote the family o f all nonc onstant rectiﬁa ble curves in X . It may happ en Υ = ∅ , but we will b e mainly concer ned with metric spaces for which the space Υ is lar ge enough. Deﬁnition 5 .3. (Mo dulus of a family of curves) Let Γ ⊂ Υ . F o r 1 ≤ p < ∞ we deﬁne the p − mo dulus of Γ b y Mo d p (Γ) = inf ρ Z X ρ p dµ, where the inﬁm um is taken ov er all no n-negative B orel functions ρ : X → [0 , ∞ ] such that R γ ρ ≥ 1 for all γ ∈ Γ. If some pro pe rty ho lds for all cur ves γ ∈ Υ \ Γ, such that Mo d p Γ = 0, then we say tha t the prop erty holds for p − a.e. curve . Deﬁnition 5 .4. A non-neg ative Borel function g o n X is a p − we ak upp er gr adient of an extended real-v a lued function f on X, if | f ( γ ( a )) − f ( γ ( b )) | ≤ Z γ g for p − a .e. c urve γ ∈ Υ. INFINITESIMALL Y LIPSCHITZ FUNCTIONS ON METRIC SP ACES 19 Let e N 1 ,p ( X, d, µ ), wher e 1 ≤ p < ∞ , be the clas s o f all L p int egrable Bor el functions on X for which there exists a p − weak upp er gr adient in L p . F or f ∈ e N 1 ,p ( X, d, µ ) we de ﬁne k u k e N 1 ,p = k u k L p + inf g k g k L p , where the inﬁm um is taken ov er all p − weak up p er g radients g of u . Now, w e deﬁne in e N 1 ,p an equiv a lence relation by u ∼ v if and only if k u − v k e N 1 ,p = 0 . Then the space N 1 ,p ( X, d, µ ) is deﬁned as the quo tient e N 1 ,p ( X, d, µ ) / ∼ a nd it is e quipped with the norm k u k N 1 ,p = k u k e N 1 ,p . Next, we co nsider the case p = ∞ . W e will intro duce the corr esp onding deﬁnition of ∞− mo dulus of a family of rectiﬁable curves which will b e an imp orta nt ingr edient for the deﬁnition o f the So bo lev space N 1 , ∞ ( X ). Deﬁnition 5.5. F or Γ ⊂ Υ, let F (Γ) b e the f amily of all Bore l mea surable functions ρ : X → [0 , ∞ ] such that Z γ ρ ≥ 1 for all γ ∈ Γ . W e deﬁne the ∞− m o dulus of Γ by Mo d ∞ (Γ) = inf ρ ∈ F (Γ) {k ρ k L ∞ } ∈ [0 , ∞ ] . If some pr op erty holds for all cur ves γ ∈ Υ \ Γ, where Mod ∞ Γ = 0, then we say that the prop erty holds for ∞− a.e. cu rve . Remark 5. 6. It can b e eas ily chec ked tha t Mo d ∞ is an outer measure as it happ ens for 1 ≤ p < ∞ . See for example Theo rem 5 . 2 in [Ha1]. Next, we provide a characterization of path families whose ∞− mo dulus is zero. Lemma 5.7. L et Γ ⊂ Υ . The fol lowing c onditions ar e e quivalent: (a) Mod ∞ Γ = 0 . (b) Ther e exists a Bor el function 0 ≤ ρ ∈ L ∞ ( X ) such that R γ ρ = + ∞ , for e ach γ ∈ Γ . (c) Ther e exists a Bor el fun ction 0 ≤ ρ ∈ L ∞ ( X ) such that R γ ρ = + ∞ , for e ach γ ∈ Γ and k ρ k L ∞ = 0 . Pr o of. ( a ) ⇒ ( b ) If Mod ∞ Γ = 0, for each n ∈ N there exists ρ n ∈ F (Γ) such that k ρ n k L ∞ < 1 / 2 n . Let ρ = P n ≥ 1 ρ n . Then k ρ k L ∞ ≤ P ∞ n =1 1 / 2 n = 1 and R γ ρ = R γ P n ≥ 1 ρ n = ∞ . ( b ) ⇒ ( a ) On the other hand, let ρ n = ρ/n for all n ∈ N . By h yp othesis R γ ρ n = ∞ for all n ∈ N and γ ∈ Γ. Then ρ n ∈ F (Γ) and k ρ k L ∞ /n → 0 a s n → ∞ . Hence Mo d ∞ (Γ) = 0. ( b ) ⇒ ( c ) By hypo thesis there exists a Borel measura ble function 0 ≤ ρ ∈ L ∞ ( X ) such that, Z γ ρ = + ∞ for every γ ∈ Γ . Consider the function h ( x ) = ( k ρ k L ∞ if k ρ k L ∞ ≥ ρ ( x ) , ∞ if ρ ( x ) > k ρ k L ∞ . 20 E. DURAND AND J.A. JARAMILLO Notice that k ρ k L ∞ = k h k L ∞ , and since R γ ρ = + ∞ for every γ ∈ Γ 1 and ρ ≤ h , we hav e that R γ h = + ∞ for e very γ ∈ Γ 1 . Now, we deﬁne the function  = h − k h k L ∞ which ha s k  k L ∞ = 0 and Z γ  = Z γ h − k h k L ∞ ℓ ( γ ) = + ∞ for every γ ∈ Γ 1 .  Now we are rea dy to deﬁne the notion of ∞− we ak upp er gr adient . Deﬁnition 5.8. A non-nega tive Borel function g on X is an ∞− we ak upp er gr a- dient of an extended rea l-v alued function f on X, if | f ( γ ( a )) − f ( γ ( b )) | ≤ Z γ g for ∞− a .e. cur ve e very cur ve γ ∈ Υ. Let e N 1 , ∞ ( X, d, µ ), be the class of a ll functions f ∈ L ∞ ( X ) Bo rel for which there exists an ∞− weak upper gra dient in L ∞ . F o r f ∈ e N 1 , ∞ ( X, d, µ ) we deﬁne k u k e N 1 , ∞ = k u k L ∞ + inf g k g k L ∞ , where the inﬁmum is taken ov er a ll ∞− weak upp er gr adients g of u . Deﬁnition 5 .9. (Newtonian space for p = ∞ ) W e deﬁne a n equiv alence re lation in e N 1 , ∞ by u ∼ v if a nd only if k u − v k e N 1 , ∞ = 0. Then the space N 1 , ∞ ( X, d, µ ) is deﬁned as the q uotient e N 1 , ∞ ( X, d, µ ) / ∼ a nd it is equipp ed with the norm k u k N 1 , ∞ = k u k e N 1 , ∞ . Note that if u ∈ e N 1 , ∞ and v = u µ − a.e., then it is not necessarily true that v ∈ e N 1 , ∞ . Indee d, let ( X = [ − 1 , 1] , d, λ ) wher e d denotes the Euclidean distance and λ the Leb esg ue measure. Let u : X → R b e the function u = 1 and v : X → R given by v = 1 if x 6 = 0 a nd v ( x ) = ∞ if x = 0. In this ca se we hav e that u = v µ − a.e. , u ∈ e N 1 , ∞ but v / ∈ e N 1 , ∞ . It ca n b e shown that if u, v ∈ e N 1 , ∞ , and v = u µ − a.e., then k u − v k e N 1 , ∞ = 0. In additio n, N 1 , ∞ ( X ) is a B anach space. Both results can b e chec ked adapting pro p erly the resp ective pro ofs for the case p < ∞ . F o r further details s ee [Sh2]. Lemma 5.10. If f ∈ D ( X ) then Lip( f ) is an u pp er gr adient of f . Pr o of. Let γ : [ a , b ] → X be a rectiﬁable curve para metrized by arc- length which connects x and y . It can be c heck ed that γ is 1 − L ipschitz (see for instance Theo rem 3 . 2 in [Ha1]). The function f ◦ γ is an inﬁnitesimally Lipschitz function and by Stepanov’s diﬀerentiabilit y theore m (see [BRZ]), it is diﬀerent iable a.e. Note that | ( f ◦ γ ) ′ ( t ) | ≤ Lip f ( γ ( t )) at every po int of [ a, b ] wher e ( f ◦ γ ) is diﬀerent iable. Now, we deduce that | f ( x ) − f ( y ) | ≤    Z b a ( f ◦ γ ) ′ ( t ) dt    ≤ Z b a Lip( f ( γ ( t ))) dt as wan ted.  Now s uppo se that µ ( B ) > 0 for every o pen ba ll B ⊂ X . It is clea r b y Lemma 5.10 that D ∞ ( X ) ⊂ e N 1 , ∞ ( X ) and tha t the map φ : D ∞ ( X ) − → N 1 , ∞ ( X ) f − → [ f ] . INFINITESIMALL Y LIPSCHITZ FUNCTIONS ON METRIC SP ACES 21 is an inclusion. Indeed, if f , g ∈ D ∞ ( X ) with 0 = [ f − g ] ∈ N 1 , ∞ ( X ), we have f − g = 0 µ − a.e. Thus f = g in a dense subset and since f , g are contin uous we obtain that f = g . Therefore we hav e the following c hain of inclusions: LIP ∞ ( X ) = M 1 , ∞ ( X ) ⊂ D ∞ ( X ) ⊂ N 1 , ∞ ( X ) , ( ∗ ) and k · k N 1 , ∞ ≤ k · k D ∞ ≤ k · k LIP ∞ ≤ 2 k · k M 1 , ∞ . The next exa mple shows that in general D ∞ ( X ) 6 = N 1 , ∞ ( X ). Example 5.11. C onsider the metric space ( X = { B n } n , d e ), where d e is the re- striction to X of the Euclidean metric of R 2 and { B n } n is a s equence of o pe n balls with ra dius conv ergent to zero , as shows the picture b elow: ✲ ✻ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W e deﬁne on X a function in the following way: f ( x, y ) = ( 1 if ( x, y ) ∈ B i i = 2 k + 1 k ∈ Z , 0 if ( x, y ) ∈ B i i = 2 k k ∈ Z . The c onstant function g = 0 is clearly and upp er gradient of f , and so f ∈ N 1 , ∞ ( X ). But, there is no co nt inuous repres ent ative for the function f . Thus, in par ticular, f do es not a dmit a representative in D ∞ . In the following, w e will lo ok for co nditions under which the Sobo lev s paces M 1 , ∞ ( X ) and N 1 , ∞ ( X ) coincide. In particula r, this will giv e us the equalit y of all the spaces in the chain ( ∗ ) ab ove. F or that, we need some preliminary terminology and results. Deﬁnition 5.12. W e say that a measure µ on X is doubling if ther e is a p o sitive constant C µ such that 0 < µ ( B ( x, 2 r )) ≤ C µ µ ( B ( x, r )) < ∞ , for each x ∈ X and r > 0. Here B ( x, r ) denotes the op en ball of center x and radius r > 0 . Deﬁnition 5.13. Let 1 ≤ p < ∞ . W e say that ( X , d, µ ) supp orts a we ak p - Poinc ar´ e ine quality if ther e exis t c onstants C p > 0 and λ ≥ 1 such that for every Borel measura ble function u : X → R and every upper gradient g : X → [0 , ∞ ] o f u , the pair ( u, g ) satisﬁes the inequality Z B ( x,r ) | u − u B ( x,r ) | dµ ≤ C p r  Z B ( x,λr ) g p dµ  1 /p for each B ( x, r ) ⊂ X . Here for arbitrar y A ⊂ X with 0 < µ ( A ) < ∞ we w rite Z A f = 1 µ ( A ) Z A f dµ. The Poincar´ e inequality cr eates a link b etw een the measure, the metric and the gradient and it pro vides a w a y to pass fro m the inﬁnitesimal information which g ives the g radient to larg er scales. Metric spaces with doubling measure and Poincar´ e 22 E. DURAND AND J.A. JARAMILLO inequality a dmit ﬁrst o rder diﬀerential calculus a kin to that in E uclidean spac es. See [Am], [He1] or [He2] for further information ab out thes e topics. The pro of of the next re sult is s trongly inspir ed in P rop osition 3 . 2 in [JJRRS]. How ever, w e include all the details b ecause of the technical diﬀerences , whic h at certain p oints b ecome quite subtle. Theorem 5.14. L et X b e a c omplete metric sp ac e that su pp orts a doubling Bor el me asur e µ which is non-trivial and ﬁnite on b al ls and supp ose that X supp orts a we ak p -Poinc ar ´ e ine quality for some 1 ≤ p < ∞ . L et ρ ∈ L ∞ ( X ) su ch that 0 ≤ ρ . Then, ther e exists a set F ⊂ X of me asur e 0 and a c onst ant K > 0 (dep ending only on X ) su ch that for al l x, y ∈ X \ F ther e exist a r e ctiﬁable curve γ s u ch that R γ ρ < + ∞ and ℓ ( γ ) ≤ K d ( x, y ) . Pr o of. W e may a ssume that 0 < k ρ k L ∞ ≤ 1 . Indeed, in o ther case, w e could take e ρ = ρ/ (1 + k ρ k L ∞ ). Let E = { x ∈ X : ρ ( x ) > k ρ k L ∞ } , which is a set o f measur e zero. B y Theor em 2 . 2 in [He1], there exists a co nstant C dep ending only on the doubling consta nt C µ of X such that for each f ∈ L 1 ( X ) and fo r all t > 0 µ ( { M ( f ) > t } ) ≤ C t Z X | f | dµ Recall that M ( f )( x ) = sup r > 0 { R B ( x,r ) | f | dµ } . F o r e ach n ≥ 1 we c an choose V n be an op en set s uch that E ⊂ V n and µ ( V n ) ≤  1 n 2 n  p (see Theor em 1 . 10 in [M]). Note that E ⊆ T n ≥ 1 V n = E 0 and µ ( E 0 ) = µ ( E ) = 0. Next, cons ider the family of functions ρ n = k ρ k L ∞ + X m ≥ n χ V m and the function ρ 0 given b y the fo rmula ρ 0 ( x ) = ( k ρ k L ∞ if x ∈ X \ E 0 , + ∞ otherwise. W e hav e the following prop er ties: (i) ρ n | X \ V n ≡ k ρ k L ∞ . (ii) ρ ≤ ρ 0 ≤ ρ m ≤ ρ n if n ≤ m . (iii) ρ n | E 0 ≡ + ∞ . (iv) ρ n ∈ L p ( X ) is lower se micontin uous; in fact k ρ n − k ρ k L ∞ k L p ≤ 1 n . Indeed, since each o f the sets V m are op en then the functions χ V m are low er semicontin uo us (see P rop osition 7 . 1 1 in [F ]) a nd so o nce we chec k that k ρ n − k ρ k L ∞ k L p ≤ 1 n , w e will b e done. F or that, is is enough to prov e that P m ≥ n k χ V m k L p ≤ 1 n , which follows from the formula X m ≥ n k χ V m k L p = X m ≥ n ( µ ( V n )) 1 /p = X m ≥ n 1 m 2 m ≤ 1 n X m ≥ n 1 2 m ≤ 1 n . (v) µ ( { M (( ρ n − k ρ k L ∞ ) p ) > 1 } ) ≤ C n p . Indeed, a s we ha ve seen ab ov e µ ( { M (( ρ n − k ρ k L ∞ ) p ) > 1 } ) ≤ C 1 Z X | ρ n − k ρ k L ∞ | p = C k ρ n − k ρ k L ∞ k p L p < C 1 n p . INFINITESIMALL Y LIPSCHITZ FUNCTIONS ON METRIC SP ACES 23 F o r each n ≥ 1 consider the set S n = { x ∈ X : M (( ρ n − k ρ k L ∞ ) p )( x ) ≤ 1 } W e claim tha t: S n ⊂ S m if n ≤ m and F = X \ S n ≥ 1 S n has me asur e 0 . Indeed, if n ≤ m , we hav e that 0 ≤ ρ m − k ρ k L ∞ ≤ ρ n − k ρ k L ∞ and s o 0 ≤ ( ρ m − k ρ k L ∞ ) p ≤ ( ρ n − k ρ k L ∞ ) p ; hence S n ⊂ S m . On the other hand by (v) ab ov e, w e hav e µ ( X \ S n ) ≤ C n p . Thus, 0 ≤ µ ( F ) = µ  X \ [ n ≥ 1 S n  = µ  \ n ≥ 1 ( X \ S n )  = lim n →∞ µ ( X \ S n ) ≤ lim n →∞ C n p = 0 . After all this prepar atory work, o ur aim is to prov e that there exis ts a constant K > 0 dep ending o nly o n X such that for a ll x, y ∈ X \ F there exist a rectiﬁable curve γ such that R γ ρ < + ∞ and ℓ ( γ ) ≤ K d ( x, y ). The co nstant K will be constructed along the remainder of the pro o f. In what follo ws let m 0 be the smallest int eger for which S m 0 6 = ∅ . Fix n ≥ m 0 and a p oint x 0 ∈ S n ⊂ X \ F . As one can chec k str aightforw ardly , it is enough to prove tha t for each x ∈ S n there exists a rectiﬁable curve γ s uch that R γ ρ < + ∞ and ℓ ( γ ) ≤ K d ( x, y ), wher e the cons tant K depends only o n X and not o n x 0 or n . F o r our pur po ses, we deﬁne the set Γ xy as the set of all the r ectiﬁable curves connecting x and y . Since a complete metric s pace X supp orting a doubling measure and a weak p − Poincar´ e inequality is q uasi-conv ex (see Theo rem 17 . 1 in [Ch]), it is clear that Γ xy is nonempty . W e deﬁne the function u n ( x ) = inf n ℓ ( γ ) + Z γ ρ n : γ ∈ Γ x 0 x o . Note that u n ( x 0 ) = 0. W e will prov e that in S n the function u n is bo unded by a Lips chit z function v n with a cons tant K 0 which dep ends only on X a nd k ρ k L ∞ (and no t on x 0 nor n ) such that v n ( x 0 ) = 0. Assume this for a moment. W e have 0 ≤ u n ( x ) = u n ( x ) − u n ( x 0 ) ≤ v n ( x ) − v n ( x 0 ) ≤ K 0 d ( x, x 0 ) < ( K 0 + 1) d ( x, x 0 ) . Thu s, there exists a rectiﬁa ble curve γ ∈ Γ x 0 x such that ℓ ( γ ) + Z γ ρ ≤ ℓ ( γ ) + Z γ ρ n ≤ ( K 0 + 1) d ( x, x 0 ) . Hence, tak ing K = K 0 + 1, we will hav e ℓ ( γ ) ≤ K d ( x, x 0 ) and Z γ ρ < + ∞ , as we wan ted. Therefore, conside r the functions u n,k : X → R g iven by u n,k = inf n ℓ ( γ ) + Z γ ρ n,k : γ ∈ Γ x 0 x o where ρ n,k = min { ρ n , k } which is a lower semicontin uous function. Let us see that the functions u n,k are Lipschitz for each k ≥ 1 (and in particular contin uous) and that ρ n,k + 1 ≤ ρ n + 1 a re upp er gr adients fo r u n,k . Since X is qua si-conv ex, it follows that u n,k ( x ) < + ∞ for all x ∈ X . Indeed, let y , z ∈ X , C q the constant o f quasi-co nv ex it y for X and ε > 0. W e may assume that u n,k ( z ) ≥ u n,k ( y ). Le t γ y ∈ Γ x 0 y be such that u n,k ( y ) ≥ ℓ ( γ y ) + Z γ y ρ n,k − ε. 24 E. DURAND AND J.A. JARAMILLO On the other hand, for each rectiﬁable curve γ y z ∈ Γ y z , we hav e u n,k ( z ) ≤ ℓ ( γ y ∪ γ y z ) + Z γ y ∪ γ yz ρ n,k , and so | u n,k ( z ) − u n,k ( y ) | = u n,k ( z ) − u n,k ( y ) ≤ ℓ ( γ y z ) + Z γ yz ρ n,k = Z γ yz ( ρ n,k + 1) . Thu s, ρ n,k + 1 is a n upp er gradient for u n,k . In particular, if ℓ ( γ z y ) ≤ C q d ( z , y ), we deduce that | u n,k ( z ) − u n,k ( y ) | ≤ ( k + 1 ) ℓ ( γ z y ) ≤ C q ( k + 1 ) d ( z , y ) and so u n,k is a C q ( k + 1)-Lipschit z function. Our purpos e now is to prov e that the restriction to S n of each function u n,k is a Lipsc hitz function on S n with re sp ect to a co nstant K 0 which depends only o n X . Fix y , z ∈ S n . F or each i ∈ Z , deﬁne B i = B ( z , 2 − i , d ( z , y )) if i ≥ 1, B 0 = B ( z , 2 d ( z , y )), and B i = B ( y , 2 i d ( z , y )) if i ≤ − 1. T o simplify notation we wr ite λB ( x, r ) = B ( x, λr ). In the ﬁrst inequality of the follo wing estimation w e use the fact that, since u n,k is contin uous , a ll po in ts of X are Leb esg ue p oints o f u n,k . Using the weak p -Poincar´ e ineq uality a nd the doubling condition we g et the third inequality . F ro m the Minkowski ineq uality we deduce the ﬁfth while the last one follows from the deﬁnition o f S n : | u n,k ( z ) − u n,k ( y ) | ≤ X i ∈ Z    Z B i u n,k dµ − Z B i +1 u n,k dµ    ( ∗ ) ≤ X i ∈ Z 1 µ ( B i ) Z B i    u n,k − Z B i +1 u n,k dµ    dµ ≤ C µ C p d ( z , y ) X i ∈ Z 2 −| i |  1 µ ( λB i ) Z λB i ( ρ n,k + 1) p  1 /p ≤ C µ C p d ( z , y ) X i ∈ Z 2 −| i |  1 µ ( λB i ) Z λB i (( ρ n,k − k ρ k L ∞ ) + k ρ k L ∞ + 1) p  1 /p ≤ C µ C p d ( z , y ) X i ∈ Z 2 −| i |  k ρ k L ∞ + 1 +  1 µ ( λB i ) Z λB i ( ρ n,k − k ρ k L ∞ ) p  1 /p  ≤ 3 C µ C p d ( z , y ) X i ∈ Z 2 −| i | ≤ K 0 d ( z , y ) where K 0 = 9 C µ C p is a constant that depends o nly on X . Recall that C µ is the doubling cons tant and C p is the co nstant which a ppe ars in the weak p − Poincar´ e inequality . Let us see with more detail inequality ( ∗ ). If i > 0 , we hav e that    Z B i u n,k dµ − Z B i +1 u n,k dµ    ≤ 1 µ ( B i +1 )    Z B i +1  u n,k − Z B i u n,k dµ  dµ    ≤ µ ( B i ) µ ( B i ) 1 µ ( B i +1 )    Z B i  u n,k − Z B i u n,k dµ  dµ    ≤ C µ µ ( B i )    Z B i  u n,k − Z B i u n,k dµ  dµ    . W e have use d that B i +1 ⊂ B i for i > 0 a nd that µ is a doubling mea sure and so µ (2 B i +1 ) = µ ( B i ) ≤ C µ µ ( B i +1 ). The ca ses i < 0 and i = 0 ar e similar. Thu s, the restriction of u n,k to S n is a K 0 -Lipschitz function for all k ≥ 1 . Note that u n,k ≤ u n,k +1 and ther efore we may deﬁne v n ( x ) = sup k { u n,k ( x ) } = lim k →∞ u n,k ( x ) . INFINITESIMALL Y LIPSCHITZ FUNCTIONS ON METRIC SP ACES 25 Whence v n is a K 0 -Lipschitz function on S n . Since v ( x 0 ) = 0 a nd x 0 ∈ S m when m ≥ m 0 we hav e that v ( x ) < ∞ and so , it is enough to chec k that u n ( x ) ≤ v n ( x ) for x ∈ S n . Now, ﬁx x ∈ S n . F o r each k ≥ 1 there is γ k ∈ Γ x 0 x such that ℓ ( γ k ) + Z γ k ρ n,k ≤ u n,k ( x ) + 1 k ≤ K 0 d ( x, x 0 ) + 1 k . In particular , ℓ ( γ k ) ≤ K 0 d ( x, x 0 ) + 1 := M for ev e ry k ≥ 1 and so, b y repa rametriza- tion, we may assume tha t γ k is an M - Lipschitz f unction and γ k : [0 , 1] → B ( x 0 , M ) for a ll k ≥ 1. Since X is complete and do ubling, a nd ther efore closed balls ar e com- pact, we are in a po sition to use the Ascoli- Arzela theorem to o btain a s ubsequence { γ k } k (whic h w e denote again by { γ k } k to s implify nota tion) a nd γ : [0 , 1] → X such that γ k → γ uniformly . F or each k 0 , the function 1 + ρ n,k 0 is low er s emicon- tin uous, and therefore by Lemma 2 . 2 in [JJRRS] a nd the fact that { ρ n,k } is an increasing seq uence of functions, we hav e ℓ ( γ ) + Z γ ρ n,k 0 = Z γ (1 + ρ n,k 0 ) ≤ lim inf k →∞ Z γ k (1 + ρ n,k 0 ) ≤ lim inf k →∞ Z γ k (1 + ρ n,k ) . Using the monotone conv er gence theor em o n the le ft ha nd side a nd letting k 0 tend to inﬁnity yields ℓ ( γ ) + Z γ ρ n ≤ lim inf k →∞ Z γ k (1 + ρ n,k ) . Since γ ∈ Γ x 0 x we hav e u n ( x ) ≤ ℓ ( γ ) + Z γ ρ n ≤ lim inf k →∞ Z γ k (1 + ρ n,k ) ≤ lim inf k →∞  u n,k ( x ) + 1 k  ≤ v n ( x ) , and that completes the pro of.  Remark 5. 15. In Theorem 5 .14 we can c hange the h ypo thesis of completeness fo r the space X by lo ca l compactness. The pro of is analogo us to the one of Theorem 1 . 6 in [JJRRS], a nd we do not include the details. Corollary 5.1 6. L et X b e a c omplete m etric sp ac e that supp orts a doubling Bor el me asur e µ which is non-trivial and ﬁnite on b al ls. If X supp orts a we ak p -Poinc ar´ e ine quality for 1 ≤ p < ∞ , then L IP ∞ ( X ) = M 1 , ∞ ( X ) = N 1 , ∞ ( X ) with e qu ivalent norms. Pr o of. If f ∈ N 1 , ∞ ( X ), then there exists a n ∞− weak upp er g radient g ∈ L ∞ ( X ) of f . W e deno te Γ 1 the fa mily of curves for whic h g is no t an upper gr adient for f . Note that Mo d ∞ Γ 1 =0. By Lemma 5 .7 there exists a Borel measura ble function 0 ≤  ∈ L ∞ ( X ) s uch that, R γ  = + ∞ for every γ ∈ Γ 1 and k  k L ∞ = 0. Consider ρ 0 = g +  ∈ L ∞ ( X ) which is an upp er gradient of f and satisﬁes that k ρ 0 k L ∞ = k g k L ∞ . No te that R γ ρ 0 = + ∞ fo r all γ ∈ Γ 1 and that by Lemma 5 .7 the family of curves Γ 2 = { γ ∈ Υ : R γ ρ 0 = + ∞} has ∞− mo dulus zero . Fina lly , consider the set { x ∈ X : g ( x ) +  ( x ) ≥ k ρ 0 k L ∞ } and deﬁne ρ ( x ) = ( k ρ 0 k L ∞ if x ∈ X \ E , + ∞ if x ∈ E . Then ρ is a n upp er gr adient of f and it satisﬁes that k ρ k L ∞ = k ρ 0 k L ∞ = k g k L ∞ . Note that if R γ ρ < + ∞ , then the set γ − 1 (+ ∞ ) has measure zer o in the domain o f γ (b eca use o therwise R γ ρ = + ∞ ). Thus, if R γ ρ < + ∞ , we hav e in particular that R γ ρ = k ρ k L ∞ ℓ ( γ ). By Theor em 5 .14 there exists a set F ⊂ X of mea sure 0 and a 26 E. DURAND AND J.A. JARAMILLO constant K > 0 (depe nding o nly on X ) such that for all x, y ∈ X \ F there ex ist a rectiﬁable curve γ such that R γ ρ < + ∞ and ℓ ( γ ) ≤ K d ( x, y ). Let now x, y ∈ X \ F and γ be a r ectiﬁable curve satisfying the precedent conditions. Then | f ( x ) − f ( y ) | ≤ Z γ ρ ( ∗ ) = k ρ k L ∞ ℓ ( γ ) ≤ k ρ k L ∞ K d ( x, y ) . Then f is k ρ k L ∞ K − Lips chit z a.e. Thus, LIP ∞ ( X ) = M 1 , ∞ ( X ) = N 1 , ∞ ( X ).  Remark 5. 17. Note that if we would hav e chosen a s upper gra dient ρ 0 instead of ρ , the inequality ( ∗ ) might not be necessary true. T o see this, it is enough to deﬁne a function which is zero a.e. and c onstant but ﬁnite o n a set of zero measur e. Our purp ose now is to see under which conditions the spa ces D ∞ ( X ) and N 1 , ∞ ( X ) c oincide. F or that, we need ﬁr st to use the lo ca l version of the weak p -Poincar´ e inequality (se e for ex ample Deﬁnition 4 . 2 . 1 7 in [Sh1]). Deﬁnition 5.18. Let 1 ≤ p < ∞ . W e say that ( X, d, µ ) supp or ts a lo c al we ak p - Poinc ar´ e ine quality with constant C p if for ev er y x ∈ X , there exists a neigh bor ho o d U x of x and λ ≥ 1 such that whenever B is a ba ll in X such that λB is co n tained in U x , and u is an integrable function on λB with g as its upp er gradient in λB , then Z B ( x,r ) | u − u B ( x,r ) | dµ ≤ C p r  Z B ( x,λr ) g p dµ  1 /p . Corollary 5.1 9. L et X b e a c omplete m etric sp ac e that supp orts a doubling Bor el me asur e µ which is non-trivial and ﬁnit e on b al ls. If X supp orts a lo c al we ak p - Poinc ar´ e ine quality for 1 ≤ p < ∞ . Then N 1 , ∞ ( X ) = D ∞ ( X ) with e qu ivalent norms. Pr o of. If f ∈ N 1 , ∞ ( X ), then there exists a n ∞− weak upp er g radient g ∈ L ∞ ( X ) of f . W e constr uct in the same w ay a s in Corolla ry 5.16 an upper gradient ρ of f which satisﬁes k ρ k L ∞ = k g k L ∞ , R γ ρ ≥ | f ( γ (0)) − f ( γ ( L )) | for all γ ∈ Υ and R γ ρ = k ρ k L ∞ ℓ ( γ ) for all γ ∈ Υ such that R γ ρ < + ∞ . Fix x ∈ X . Using a lo cal version of Theo rem 5.14 we obtain that there exists a neighborho o d U x and a constant K > 0 (depending only on X ) s uch that for a lmost every z , y ∈ U x , there exist a rectiﬁable curve γ connecting z and y such that R γ ρ < + ∞ and ℓ ( γ ) ≤ K d ( z , y ). Let now y ∈ U x and γ a rectiﬁable curve satisfying the precedent conditions. Then | f ( x ) − f ( y ) | ≤ Z γ ρ = k ρ k L ∞ ℓ ( γ ) ≤ k ρ k L ∞ K d ( x, y ) . Under the hypo thesis of the corollary it can b e easily chec ked that f is con tin uous on X and so, ther e is no obstr uction to take the sup erior limit lim sup y → x | f ( x ) − f ( y ) | d ( x, y ) . Thu s, we de duce that L ip f ( x ) ≤ K k ρ k L ∞ . Since this is tr ue for e ach x ∈ X , we hav e k Lip f k ∞ ≤ K k ρ k L ∞ < + ∞ and we conclude that f ∈ D ∞ ( X ).  Observe that under the hypo thesis o f Coro llary 5.1 9 we hav e that X is a lo cally radially quasiconv ex metric s pace. W e se e thro ughout a very simple example tha t in ge neral there exis t metric spaces X for which the following holds: LIP ∞ ( X ) = M 1 , ∞ ( X ) D ∞ ( X ) = N 1 , ∞ ( X ) . INFINITESIMALL Y LIPSCHITZ FUNCTIONS ON METRIC SP ACES 27 Indeed, cons ider the metric space ( X , d, λ ) whe re X = C \{ Re(z) ≥ 0 , | Im(z) | ≤ 1 / 2 } , d is the metric induced by the Euc lidean one and λ denotes the L ebes gue measure. Since X is a complete metric space that suppo rts a doubling measure and a loca l weak p -Poincar´ e inequality for an y 1 ≤ p < ∞ , by Co rollar y 5.19 , we have that D ∞ ( X ) = N 1 , ∞ ( X ). Let f ( z ) = ar g( z ), for each z ∈ X . One ca n chec k that f ∈ D ∞ ( X ) = N 1 , ∞ ( X ). How ever, f / ∈ LIP ∞ ( X ), and so LIP ∞ ( X ) D ∞ ( X ) = N 1 , ∞ ( X ) . Ackno wledgements It is a great plea sure to thank Profess ors Jos e F. F er nando and M. Isab el Garrido for many v aluable conv ersations concerning this pap er. References [Am] L. Ambrosio, P . Till i: T opics on A nalysis i n Metric Spaces. Oxford Lecture Series in Mathematics and its Applications 25 . Oxfor d University Pr ess, Oxfor d , (2004 ). [BRZ] Z. M. Balogh, K. Rog ovin, T. Z ¨ urch er: The Stepa no v Diﬀerentiabilit y Theorem in Metric Measure Spaces. J. Ge om. Ana l. 14 no. 3, (2004), 405–422. [Ch] J. Cheeger: Diﬀerentiabilit y of Lipschitz F unct ions on metric measure spaces. Ge om. F unct. Anal. 9 (1999), 428–517. [F] G.B. F olland: Real Analysis, Mo dern T ech niques and Their Appli cations. Pure and Applied Mathematics (1999). [F u] B. F uglede : Extremal length and f unctional completion. A c t a. Math. 98 (1957), 171– 219. [GJ1] M. I. Garrido, J. A. 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Infinitesimally Lipschitz functions on metric spaces

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