Sharp capacitary estimates for rings in metric spaces

SHARP CAP A CIT AR Y ESTIMA TES F OR RINGS IN METRIC SP A CES NICOLA GAR OF ALO AND NIKO MAR O L A Abstract . W e establish sharp estimates for the p -capacit y of metric rings with unrelated radii in metric measure spaces equipp ed with a doubling mea- sure and sup p orting a P oincar´ e inequ alit y . These estimates pla y an essential role in th e study of the local b eh a vior of p -harmon ic Green’s functions. 1. Intr oduction In this pa p er w e establish sharp capacitar y estimates for the metric rings with unrelated radii in a lo cally doubling metric measure space supp orting a lo cal ( 1 , p )-P oincar ` e inequalit y . A motiv ation for pursuing these estimates come s from the s tudy of the asymptotic b ehav ior of p - harmonic Green’s functions in this g eometric setting. Similarly to the classical case (for the la tter the reader should see [30], [36] and [37]), capacitary estimates play a crucial role in studying the lo cal b eha vior of suc h singular functions. F or this asp ect w e refer t he reader to the forthcoming pa p er b y D anielli and the authors [1 1]. P erhaps the most impo rtan t mo del of a metric space with a ric h non-Euclidean geometry is the Heisen b erg group H n , whose underlying manifold is C n × R with the g roup law ( z , t ) ◦ ( z ′ , t ′ ) = ( z + z ′ , t + t ′ − 1 2 I m ( z z ′ )). Kor´ an yi and Reimann [29] first computed explicitly the Q -capacit y o f a metric ring in H n . Here Q = 2 n + 2 indicates the homogeneous dimension of H n attac hed to t he non-isotropic g roup dilations δ λ ( z , t ) = ( λz , λ 2 t ). Their metho d mak es use of a suitable c hoice of “ p olar” co ordinates in the g roup. The Heisen b erg gr oup is the prototype of a g eneral class of nilp oten t stratified Lie groups, no w adays kno wn as Carnot groups. In this more general contex t, Heinonen and Holopainen [18] pro v ed sharp estimates for the Q -capacity o f a ring. Again, here Q indicates the homog eneous dimension att ac hed to the non-isotropic dilations asso ciated with the grading of the Lie a lgebra. In the pap er [6] Cap ogna, D anielli and the first named author es- tablished sharp p -capacitary estimates, for the range 1 < p < ∞ , for 2000 Mathematics S ubje ct Classific ation. Primary: 31B15 , 31C45 ; Seco ndary: 31C15. Key wor ds and phr ases. Capacit y , doubling measur e, Green function, Newtonian space, p -har mo nic, Poincar´ e inequality , singular function, So b o le v space. First author supp orted in part by NSF Grant DMS-0701001 . Second author supp orted by the Academy of Finland and E mil Aalto sen s ¨ a¨ ati¨ o. 1 2 NICOLA GAR OF ALO AND NIKO MAR O L A Carnot–Carath ´ eo dory rings asso ciated with a system of v ector fields of H¨ ormander t yp e. In part icular they prov ed that for a ring cen tered at a p oint x the p -capacity of the ring itself c hanges drastically dep ending on whether 1 < p < Q ( x ), p = Q ( x ) or p > Q ( x ). Here, Q ( x ) is the p oin t wise dimension at x , and suc h n umber in general differs fr om the so-called lo cal homogeneous dimension associated with a fixed compact set containing x . This unsettling phenomenon is no t presen t, for ex- ample, in the analysis of Carnot groups since in that case Q ( x ) ≡ Q , where Q is the a b o v e mentioned homogeneous dimension o f t he group. In [26],[27] Kinnunen and Martio dev elop ed a capacity theory based on the definition of Sob olev f unctions on metric spaces. They a lso pro vided sharp upp er b ounds for the capacit y of a ball. The results in the presen t pap er encompass all previous ones and extend them. F or the relev an t geometric setting o f this pap er w e refer the reader to Section 2. Here, w e confine ourselv es to men tion that a fundamen ta l example of the spaces included in this pap er is obtained b y endo wing a connected Riemannian manifo ld M with the Carat h ´ eo dory metric d asso ciated with a give n s ubbundle of the tangen t bundle, see [7]. If suc h subbundle generates the tangent space at ev ery p oint, then thanks to t he theorem of Cho w [10] and Rashevsky [33] ( M , d ) is a metric space. Suc h metric spaces are kno wn as sub-Riemannian or Carnot-Carath´ eo do ry (CC) spaces. By the fundamental w orks of Rothsc hild and Stein [34 ], Nagel, Stein and W ainger [3 2], and of Jerison [23], ev ery CC space is lo cally doubling, and it lo cally satisfies a ( p, p )-P oincar ` e inequalit y for an y 1 ≤ p < ∞ . Another basic example is provided b y a Riemannian manifold ( M n , g ) with nonnegativ e Ricci tensor. In suc h case thanks to the Bishop comparison theorem the doubling condition holds globally , see e.g. [8], whereas t he (1 , 1)-Poincar ` e inequalit y was pro v ed b y Buser [5]. An in teresting example to whic h our results apply and that do es not fa ll in any of the t wo previously men tioned categories is the space of tw o infinite closed cones X = { ( x 1 , ... , x n ) ∈ R n : x 2 1 + ... + x 2 n − 1 ≤ x 2 n } equipp ed with the Euclidean metric o f R n and with the Leb esgue measure. This space is Ahlfors regular, and it is shown in Ha j lasz– Kosk ela [15, Example 4.2] that a (1 , p ) -P o incar ´ e inequality holds in X if a nd only if p > n . Another example is o btained b y g luing t w o copies of closed n -balls { x ∈ R n : | x | ≤ 1 } , n ≥ 3, along a line segmen t. In this w ay one obtains an Ahlfors regular space that supp orts a (1 , p )- P oincare inequality for p > n − 1. A thorough ov erview of a nalysis on metric spaces can b e fo und in Heinonen [17]. One should also consult Semmes [35] and David and Semmes [12]. The pr esen t note is or ganized as follow s. In Section 2 w e list o ur main assumptions and gather the necessary back ground material. In Section 3 we establish sharp capacitary estimates for spherical r ings SHARP CAP ACIT AR Y ESTIMA TE S F OR RINGS IN METRIC SP A C E S 3 with unrelated radii. Section 4 closes the pap er with a small remark on the existence of p -harmonic Green’s functions. In the setting of metric measure spaces Holopainen and Shanm ugalingam [22] constructed a p -harmonic G reen’s function, called a singular function there, having most of the ch aracteristics o f the fundamen tal solution of the La place op erator. See also Holo painen [21]. Ac kno wledgemen ts. This pap er was completed while the second au- thor was visiting Purdue Univ ersit y in 2007–20 08. He wishes to thank the Departmen t of Mathematics for the hospitalit y and sev eral of its facult y for fr uitful conv ersations. 2. Preliminaries W e b egin b y introducing our main assumptions on the metric space X and on the measure µ . 2.1. General assumptions. Throughout the pap er X = ( X , d, µ ) is a lo cally compact metric space endow ed with a metric d and a p ositiv e Borel r egular measure µ suc h that 0 < µ ( B ( x, r )) < ∞ f or all balls B ( x, r ) := { y ∈ X : d ( y , x ) < r } in X . W e assume that for ev ery compact set K ⊂ X there exist constan ts C K ≥ 1, R K > 0 and τ K ≥ 1, suc h that fo r an y x ∈ K and eve ry 0 < 2 r ≤ R K , one has: (i) the closed balls B ( x, r ) = { y ∈ X : d ( y , x ) ≤ r } are compact; (ii) (lo cal doubling condition) µ ( B ( x, 2 r )) ≤ C K µ ( B ( x, r )); (iii) (lo cal w eak ( 1 , p 0 )-P oincar ´ e inequalit y) there exists 1 < p 0 < ∞ suc h that f or all u ∈ N 1 ,p 0 ( B ( x, τ K r )) a nd all w eak upp er gradien t s g u of u Z B ( x,r ) | u − u B ( x,r ) | dµ ≤ C K r  Z B ( x,τ K r ) g p 0 u dµ  1 /p 0 , where u B ( x,r ) := R B ( x,r ) u dµ := R B ( x,r ) u dµ/µ ( B ( x, r )). Giv en an op en set Ω ⊆ X , and 1 < p < ∞ , the notation N 1 ,p (Ω) indicates the p -Newtonian space on Ω defined b elo w. Hereafter, the constan ts C K , R K and τ K will b e referred to as the lo c al p ar ame ters of K . W e also say tha t a constant C dep ends o n the lo cal doubling constan t of K if C dep ends on C K . The ab ov e assumptions encompass, e.g., all Riemannian manifolds with Ric ≥ 0, but they also include all Carnot–Carath´ eo dory spaces, and therefore, in particular, a ll Carnot groups. F or a detailed discussion of t hese f acts we refer the reader to the pap er by Garofa lo–Nhieu [13]. In the case of Carnot–Carath´ eo dory spaces, recall that if the Lie algebra generating vec tor fields grow at infinity faster than linearly , then the compactness of metric balls of large radii may fa il in general. Consider for instance in R the smo ot h v ector field of H¨ ormander t yp e X 1 = (1 + x 2 ) d dx . Some direct calculations prov e that the distance relativ e 4 NICOLA GAR OF ALO AND NIKO MAR O L A to X 1 is giv en by d ( x, y ) = | arctan( x ) − arctan( y ) | , a nd therefore, if r ≥ π / 2, w e hav e B (0 , r ) = R . 2.2. Lo c al doubling property . W e note that assumption (ii) im- plies that for ev ery compact set K ⊂ X with lo cal parameters C K and R K , for any x ∈ K and ev ery 0 < r ≤ R K , one has for 1 ≤ λ ≤ R K /r , (2.1) µ ( B ( x, λr )) ≤ C λ Q µ ( B ( x, r )) , where Q = log 2 C K , and the constan t C dep ends only on the lo cal doubling constant C K . The exp onent Q serv es as a lo cal dimension of the doubling measure µ restricted to the compact set K . In addition to suc h lo cal dimension, for x ∈ X we define the p ointwise dimension Q ( x ) b y Q ( x ) = sup { q > 0 : ∃ C > 0 suc h that λ q µ ( B ( x, r )) ≤ C µ ( B ( x, λr )) , for all λ ≥ 1 , 0 < r < ∞} . The inequalit y (2.1) readily implies that Q ( x ) ≤ Q for ev ery x ∈ K . Moreo ver, it f ollo ws that (2.2) λ Q ( x ) µ ( B ( x, r )) ≤ C µ ( B ( x, λr )) for any x ∈ K , 0 < r ≤ R K and 1 ≤ λ ≤ R K /r , and the constant C dep ends on the lo cal doubling constant C K . F ur thermore, for a ll 0 < r ≤ R K and x ∈ K (2.3) C 1 r Q ≤ µ ( B ( x, r )) µ ( B ( x, R K )) ≤ C 2 r Q ( x ) , where C 1 = C ( K, C K ) and C 2 = C ( x, K, C K ). F or more on doubling measures, see, e.g. Heinone n [17] and the references therein. 2.3. Upp er gradien ts. A path is a con tinuous mapping from a com- pact interv al, and w e sa y that a nonnegative Borel function g on X is an upp er g r adient of a n extended real v alued function f on X if for all rectifiable paths γ jo ining p oints x and y in X we ha v e (2.4) | f ( x ) − f ( y ) | ≤ Z γ g ds. whenev er bo th f ( x ) and f ( y ) are finite, and R γ g ds = ∞ otherwise. See Cheeger [9] and Shanm ugaling am [38] for a detailed discussion of upp er gra dien ts. If g is a nonnegat iv e measurable function on X and if (2.4) ho lds f or p -almost ev ery path, then g is a w e ak upp er gr adie n t o f f . By saying that (2 .4) holds for p -almost eve ry path w e mean that it fails only for a path family with zero p - mo dulus (see, for example, [38 ]). SHARP CAP ACIT AR Y ESTIMA TE S F OR RINGS IN METRIC SP A C E S 5 A function f : X → R is Lipschitz , denoted b y f ∈ Lip( X ), if there exists a constan t L ≥ 0 suc h that | f ( x ) − f ( y ) | ≤ Ld ( x, y ) for ev ery x, y ∈ X . The upp er p o intwise Lipschitz c onstant o f f at x defined b y Lip f ( x ) = lim sup r → 0 sup y ∈ B ( x,r ) | f ( y ) − f ( x ) | r is an upp er gradien t of f . W e note that for c ∈ R , Lip f ( x ) = 0 f or µ -a.e. x ∈ { y ∈ X : f ( y ) = c } . If f has an upp er gradien t in L p ( X ), then it has a m i n imal we ak upp er gr adient g f ∈ L p ( X ) in the sense that for ev ery w eak upp er g radien t g ∈ L p ( X ) of f , g f ≤ g µ -almost every where (a.e.), see Corollary 3.7 in Shanm uga lingam [39], and L emma 2.3 in J. Bj¨ orn [3] for the p oint wise c hara cterization o f g f . Thanks to the results in Cheeger [9], if X satisfie s assumptions ( ii) and (iii), then for f ∈ Lip( X ) one has g f ( x ) = Lip f ( x ) f or µ -a.e. x ∈ X . W e recall the followin g v ersion of the c hain rule. Lemma 2.1. L et u ∈ Lip( X ) a nd f : R → R b e absolutely c ontinuous and differ entiable. Then g f ◦ u ( x ) ≤ | ( f ′ ◦ u )( x ) | Lip u ( x ) for µ -al m ost every x ∈ X . 2.4. Capacity. Let Ω ⊂ X be op en and E ⊂ Ω a Borel set. The r elative p -c ap aci ty of E with resp ect to Ω is the nu m b er Cap p ( E , Ω) = inf Z Ω g p u dµ, where the infim um is take n o v er all functions u ∈ N 1 ,p ( X ) suc h that u = 1 o n E and u = 0 on X \ Ω. If suc h a function do not exist, w e set Cap p ( K , Ω) = ∞ . When Ω = X w e simply write Cap p ( E ). Observ e that if E ⊂ Ω is compact the infim um ab ov e could b e tak en o v er all functions u ∈ Lip 0 (Ω) = { f ∈ Lip( X ) : f = 0 on X \ Ω } suc h that u = 1 on E . Supp ose that Ω ⊂ X is o p en. A c ondenser is a triple ( E , F ; Ω ), where E , F ⊂ Ω are disjoin t non-empt y compact sets. F or 1 ≤ p < ∞ the p - c ap acity of a c on denser is the nu m b er cap p ( E , F ; Ω) = inf Z Ω g p dµ, where the infim um is take n o ve r all p -w eak upp er gra dien ts g of all functions u in Ω suc h that u = 0 o n E , u = 1 on F , and 0 ≤ u ≤ 1. F or other prop erties as well as equiv alen t definitions of the capac- it y we refer to Kilp el¨ ainen et al. [25], Kinnu nen–Martio [26, 27], and Kallunki–Shanm uga lingam [24 ]. See also Gol’dshtein and T ro yano v [14]. 6 NICOLA GAR OF ALO AND NIKO MAR O L A 2.5. Newt onian spa ces. W e define Sob olev spaces on the metric space follo wing Shanm ugalingam [3 8]. Let Ω ⊆ X b e nonempt y and op en. Whenev er u ∈ L p (Ω), let k u k N 1 ,p (Ω) =  Z Ω | u | p dµ + inf g Z Ω g p dµ  1 /p , where the infimum is take n ov er all w eak upp er gradien ts of u . The Newtonian sp ac e on Ω is the quotien t space N 1 ,p (Ω) = { u : k u k N 1 ,p (Ω) < ∞} / ∼ , where u ∼ v if and only if k u − v k N 1 ,p (Ω) = 0. The Newtonian space is a Banac h space a nd a lattice, moreo v er, Lipschitz functions are dense; for the prop erties of Newtonian spaces w e refer to [38] and Bj¨ orn et al. [1]. T o b e able to compare the b oundary v alues of Newtonian functions w e need a Newtonian space with zero b oundary v alues. Let E b e a measurable subset of X . The Newtonian sp ac e with zer o b oundary values is the space N 1 ,p 0 ( E ) = { u | E : u ∈ N 1 ,p ( X ) and u = 0 on X \ E } . The space N 1 ,p 0 ( E ) equipped with t he norm inherited from N 1 ,p ( X ) is a Banac h space, see Theorem 4.4 in Shanm ugalingam [39]. W e say that u b elongs to the lo c al Newtonian s p a c e N 1 ,p lo c (Ω) if u ∈ N 1 ,p (Ω ′ ) for ev ery op en Ω ′ ⋐ Ω ( or equiv alently that u ∈ N 1 ,p ( E ) for ev ery measurable E ⋐ Ω). 3. Cap acit ar y est ima tes The aim of this section is to establish sharp capacity estimates for metric rings with unrelated radii. W e emphasize an in teresting feature of Theorems 3.2 a nd 3.4 that cannot b e observ ed, for example, in the setting of Carnot groups. That is the dep endence of the estimates o n the cen ter of the ring. This is a consequenc e of the fa ct that in this generalit y Q ( x 0 ) 6 = Q where x 0 ∈ X , see Section 2. The results in this section will play an essen tial role in the subseque n t dev elopments , see the forthcoming pap er by Danielli and the author s [11]. F or now on, let 0 < r < 1 10 diam( X ) and fix a ba ll B ( x 0 , r ) ⊂ X . W e ha ve the fo llo wing estimate. Lemma 3.1. L et u ∈ Lip( X ) such that u = 0 in X \ B ( x 0 , r ) . Then (3.1) | u ( x ) | ≤ C  r p 0 − 1 Z B ( x 0 ,r ) (Lip u ) p 0 ( y ) d ( x, y ) µ ( B ( x, d ( x, y ))) dµ ( y )  1 /p 0 , for al l x ∈ B ( x 0 , r ) . F or the pro of see, e.g., M¨ ak el¨ ainen [31], Theorem 3.2 and Remark 3.3. SHARP CAP ACIT AR Y ESTIMA TE S F OR RINGS IN METRIC SP A C E S 7 W e are ready to pro v e sharp capacitary estimates for metric rings with unrelated radii. Theorem 3.2. (Estimates fr om b elow ) L et Ω ⊂ X b e a b ound e d op en set, x 0 ∈ Ω , and Q ( x 0 ) b e the p ointwise dimen s ion at x 0 . Then ther e exists R 0 = R 0 (Ω) > 0 such that fo r any 0 < r < R < R 0 we have Cap p 0 ( B ( x 0 , r ) , B ( x 0 , R )) ≥              C 1 (1 − r R ) p 0 ( p 0 − 1) µ ( B ( x 0 ,r )) r p 0 , if 1 < p 0 < Q ( x 0 ) , C 2 (1 − r R ) Q ( x 0 )( Q ( x 0 ) − 1)  log R r  1 − Q ( x 0 ) , if p 0 = Q ( x 0 ) , C 3 (1 − r R ) p 0 ( p 0 − 1)     (2 R ) p 0 − Q ( x 0 ) p 0 − 1 − r p 0 − Q ( x 0 ) p 0 − 1     1 − p 0 , if p 0 > Q ( x 0 ) , wher e C 1 = C  1 − 1 2 Q ( x 0 ) − p 0 p 0 − 1  p 0 − 1 , C 2 = C µ ( B ( x 0 , r )) r Q ( x 0 ) , C 3 = C µ ( B ( x 0 , r )) r Q ( x 0 )  2 p 0 − Q ( x 0 ) p 0 − 1 − 1  p 0 − 1 , with C > 0 dep ending only on p 0 and the do ubli n g c onstant of Ω . Pr o of. Let u ∈ Lip ( X ) such that u = 0 on X \ B ( x 0 , R ), u = 1 in B ( x 0 , r ), and 0 ≤ u ≤ 1. Then by Lemma 3.1 1 = | u ( x 0 ) | ≤ C  R p 0 − 1 Z B ( x 0 ,R ) (Lip u ) p 0 ( y ) d ( x 0 , y ) µ ( B ( x 0 , d ( x 0 , y ))) dµ ( y )  1 /p 0 ≤ C  R R − r  p 0 − 1 Z B ( x 0 ,R ) (Lip u )( y ) d ( x 0 , y ) µ ( B ( x 0 , d ( x 0 , y ))) dµ ( y )  1 /p 0 ≤ C  R R − r  1 − 1 /p 0  Z B ( x 0 ,R ) (Lip u ) p 0 ( y ) d µ ( y )  1 /p 2 0 ·  Z B ( x 0 ,R ) \ B ( x 0 ,r ) d ( x 0 , y ) p ′ 0 µ ( B ( x 0 , d ( x 0 , y ))) p ′ 0 dµ ( y )  1 /p ′ 0 p 0 , where p ′ 0 = p 0 / ( p 0 − 1). W e choose k 0 ∈ N so that 2 k 0 r ≤ R < 2 k 0 +1 r . Then we get Z B ( x 0 ,R ) \ B ( x 0 ,r ) d ( x 0 , y ) µ ( B ( x 0 , d ( x 0 , y ))) dµ ( y ) ≤ C k 0 X k =0 Z B ( x 0 , 2 k +1 r ) \ B ( x 0 , 2 k r ) d ( x 0 , y ) p ′ 0 µ ( B ( x 0 , d ( x 0 , y ))) p ′ 0 dµ ( y ) 8 NICOLA GAR OF ALO AND NIKO MAR O L A ≤ C k 0 X k =0 (2 k r ) p ′ 0 µ ( B ( x 0 , 2 k r )) p ′ 0 − 1 ≤ C r p ′ 0 µ ( B ( x 0 , r )) p ′ 0 − 1 k 0 X k =0 2 k ( p ′ 0 − Q ( x 0 )( p ′ 0 − 1)) . If 1 < p 0 < Q ( x 0 ), then p ′ 0 − Q ( x 0 )( p ′ 0 − 1) < 0, and w e obtain (3.2) 1 ≤ C − 1 1  R R − r  p 0 ( p 0 − 1) r p 0 µ ( B ( x 0 , r )) Z B ( x 0 ,R ) (Lip u ) p 0 dµ. If p 0 = Q ( x 0 ), then p ′ 0 − Q ( x 0 )( p ′ 0 − 1) = 0, and w e find (3.3) 1 ≤ C − 1 2  R R − r  Q ( x 0 )( p 0 − 1) r Q ( x 0 ) µ ( B ( x 0 , r )) k p 0 − 1 0 Z B ( x 0 ,R ) (Lip u ) p 0 dµ. Finally , if p 0 > Q ( x 0 ), then p ′ 0 − Q ( x 0 )( p ′ 0 − 1) > 0, and w e hav e (3.4) 1 ≤ C − 1 3  R R − r  p 0 ( p 0 − 1) r Q ( x 0 ) µ ( B ( x 0 , r )) ·     (2 R ) p 0 − Q ( x 0 ) p 0 − 1 − r p 0 − Q ( x 0 ) p 0 − 1     p 0 − 1 Z B ( x 0 ,R ) (Lip u ) p 0 dµ. T aking the infimum ov er all comp eting u ’s in (3.2)– (3.4) we reac h the desired conclusion.  Remark 3.3. Observ e that if X supp orts the we ak (1 , 1)-Poincar ´ e in- equalit y , i.e. p 0 = 1, these estimates reduce to the capacitary estimates, e.g., in Cap ogna et al. [6, Theorem 4.1]. Theorem 3.4. (Estimates fr om ab ove ) L et Ω , x 0 , and Q ( x 0 ) b e a s in The or em 3.2. Then ther e e xists R 0 = R 0 (Ω) > 0 such that for any 0 < r < R < R 0 we have Cap p 0 ( B ( x 0 , r ) , B ( x 0 , R )) ≤              C 4 µ ( B ( x 0 ,r )) r p 0 , if 1 < p 0 < Q ( x 0 ) , C 5  log R r  1 − Q ( x 0 ) , if p 0 = Q ( x 0 ) , C 6     (2 R ) p 0 − Q ( x 0 ) p 0 − 1 − r p 0 − Q ( x 0 ) p 0 − 1     1 − p 0 , if p 0 > Q ( x 0 ) , wher e C 4 is a p ositive c onstant dep en d ing only on p 0 and the lo c al doubling c onstant of Ω , wher e as C 5 = C µ ( B ( x 0 , r )) r Q ( x 0 ) , SHARP CAP ACIT AR Y ESTIMA TE S F OR RINGS IN METRIC SP A C E S 9 with C > 0 dep ending only on p 0 and the lo c a l doubling c onstant of Ω . Final ly, C 6 = C  2 p 0 − Q ( x 0 ) p 0 − 1 − 1  − 1 , with C > 0 dep en ding o n p 0 , the lo c al p ar ameters of Ω , a nd µ ( B ( x 0 , R 0 )) . Pr o of. F or i = 0 , 1 and p 6 = Q ( x 0 ), we define h ( t ) =        1 , if 0 ≤ t ≤ r , t p 0 − Q i p 0 − 1 − R p 0 − Q i p 0 − 1 r p 0 − Q i p 0 − 1 − R p 0 − Q i p 0 − 1 , if r ≤ t ≤ R, 0 , if t ≥ R , where Q 0 = Q and Q 1 = Q ( x 0 ). Note t hat h ∈ L ∞ ( R ), supp( h ′ ) ⊂ [ r , R ], and that h ′ ∈ L ∞ ( R ), thus h is a Lipsc hitz f unction. Let u = h ◦ d ( x 0 , y ). By the chain r ule, see Lemma 2.1, w e obtain for µ -a .e. g p 0 u ≤ ( | ( h ′ ◦ d ( x 0 , y )) | Lip d ( x 0 , y )) p 0 =     p 0 − Q i p 0 − 1     p 0 d ( x 0 , y ) (1 − Q i ) p 0 p 0 − 1    r p 0 − Q i p 0 − 1 − R p 0 − Q i p 0 − 1    p 0 . F urthermore, we ha ve tha t Cap p 0 ( B ( x 0 , r ) , B ( x 0 , R )) ≤ Z B ( x 0 ,R ) \ B ( x 0 ,r ) g p 0 u dµ ≤ k 0 X k =0 Z B ( x 0 , 2 k +1 r ) \ B ( x 0 , 2 k r ) g p 0 u dµ ≤ C     r p 0 − Q i p 0 − 1 − R p 0 − Q i p 0 − 1     − p 0 k 0 X k =0 (2 k r ) (1 − Q i ) p 0 p 0 − 1 µ ( B ( x 0 , 2 k r )) , where k 0 ∈ N is c hosen so t hat 2 k 0 r ≤ R < 2 k 0 +1 r . A t this p oint w e need t o ma k e a distinction. If 1 < p 0 < Q ( x 0 ) ≤ Q , then w e select i = 0, and we ha ve b y the do ubling prop erty that Cap p 0 ( B ( x 0 , r ) , B ( x 0 , R )) ≤ C     r p 0 − Q p 0 − 1 − R p 0 − Q p 0 − 1     − p 0 µ ( B ( x 0 , r )) r (1 − Q ) p 0 p 0 − 1 k 0 X k =0 2 k ( p 0 − Q ) p 0 − 1 ≤ C     1 −  R r  p 0 − Q p 0 − 1     − p 0 µ ( B ( x 0 , r )) r p 0 . This completes the pro of in the range 1 < p 0 < Q ( x 0 ). When p 0 > Q ( x 0 ), from the second inequalit y in (2.3) it fo llo ws µ ( B ( x 0 , 2 k r )) ≤ C 2 k Q ( x 0 ) r Q ( x 0 ) , 10 NICOLA GAROF ALO AND NIKO MAROLA where the constant C dep ends on p 0 , the lo cal doubling constant o f Ω, Ω, and µ ( B ( x 0 , R 0 )). Then w e set i = 1, a nd obtain Cap p 0 ( B ( x 0 , r ) , B ( x 0 , R )) ≤ C     r p 0 − Q ( x 0 ) p 0 − 1 − R p 0 − Q ( x 0 ) p 0 − 1     − p 0 µ ( B ( x 0 , r )) r Q ( x 0 )+ (1 − Q ( x 0 )) p 0 p 0 − 1 k 0 X k =0 2 k ( p 0 − Q ( x 0 )) p 0 − 1 ≤ C (2 p 0 − Q ( x 0 ) p 0 − 1 − 1) − 1     (2 R ) p 0 − Q ( x 0 ) p 0 − 1 − r p 0 − Q ( x 0 ) p 0 − 1     1 − p 0 . This end the pro o f in the rang e p 0 > Q ( x 0 ). When p 0 = Q ( x 0 ) w e set h ( t ) =      1 , if 0 ≤ t ≤ r,  log R r  − 1 log R t , if r ≤ t ≤ R, 0 , if t ≥ R , As ab ov e, let u = h ◦ d ( x 0 , y ), and Lemma 2.1 implies for µ -a.e. g p 0 u ≤  log R r  − p 0 1 d ( x 0 , y ) p 0 . W e ha v e Cap p 0 ( B ( x 0 , r ) , B ( x 0 , R )) ≤ Z B ( x 0 ,R ) \ B ( x 0 ,r ) g p 0 u dµ ≤ k 0 X k =0 Z B ( x 0 , 2 k +1 r ) \ B ( x 0 , 2 k r ) g p 0 u dµ ≤ C  log R r  − Q ( x 0 ) k 0 X k =0 (2 k r ) − Q ( x 0 ) µ ( B ( x 0 , 2 k r )) ≤ C µ ( B ( x 0 , r )) r Q ( x 0 )  log R r  − Q ( x 0 ) , where the inequ alit y (2.1) w as used and k 0 ∈ N w as c hosen so that 2 k 0 r ≤ R < 2 k 0 +1 r . This completes the pro o f.  W e ha v e the fo llo wing immediate corollary . Corollary 3.5. I f 1 < p 0 ≤ Q ( x 0 ) , then we have Cap p 0 ( { x 0 } , Ω) = 0 . W e close this section b y stating for completeness the follo wing we ll- kno wn estimate fo r the conformal capacit y . In the setting of Carnot groups it w as first prov ed by Heinonen in [16]. F or a discussion in met- ric spaces, see Heinonen–Kosk ela [20, Theorem 3.6] and Heinonen [17, Theorem 9 .19]. In [17] a w eak (1 , 1 )-P oincar ´ e inequality is assumed. By ob vious mo difications, ho w eve r, the pro of carries out in our setting as w ell. W e hence omit the pro of. SHARP CAP A CIT AR Y ESTIMA TES FOR RINGS IN METRIC SP A CE S 11 Theorem 3.6. Supp ose that E an d F ar e c onne cte d close d subsets of X such that F is unb ounde d an d F ∩ ∂ B ( z , r ) 6 = ∅ , and E joi n s z to ∂ B ( z , r ) . Then ther e is a uniform c onstant C > 0 , dep ending only on X, such that cap Q ( E , F ; X ) ≥ C > 0 . 4. A re mark on t he existe nce of singular functions In this s ection w e giv e a remark on the existence of singular functions or p 0 -harmonic Green’s functions on relativ ely compact domains Ω ⊂ X . The existence of singular functions in metric space setting was pro v ed in Holo painen–Shanm ugaling am [22] in Q -regular metric spaces (see b elow) supp o rting a lo cal P oincar´ e inequalit y . W e start off by recalling the definition of p 0 -harmonic funcition on metric spaces. Let Ω ⊂ X b e a domain. A function u ∈ N 1 ,p 0 lo c (Ω) ∩ C (Ω) is p 0 -harmonic in Ω if for all relativ ely compact Ω ′ ⊂ Ω a nd for all v suc h that u − v ∈ N 1 ,p 0 0 (Ω ′ ) Z Ω ′ g p 0 u dµ ≤ Z Ω ′ g p 0 v dµ. It is kno wn that nonnegativ e p 0 -harmonic functions satisfy Har- nac k’s inequalit y and the strong maxim um principle, there are no non-constan t nonnegative p 0 -harmonic functions on all of X , a nd p 0 - harmonic functions ha v e lo cally H¨ older con tinuous repre sen tatives . See Kinn unen–Shanmugalingam [28] (see also [2]). In this section we also assume that X is line arly lo c al ly c onne cte d : there exists a constan t C ≥ 1 suc h that eac h p o in t x ∈ X has a neigh b orho o d U x suc h that fo r ev ery ball B ( x, r ) ⊂ U x and fo r ev ery pair of p oints y , z ∈ B ( x, 2 r ) \ B ( x, r ), t here exists a pa th in B ( x, C r ) \ B ( x, r /C ) joining the p oin ts y and z . The follo wing definition was giv en b y Holopa inen and Shanm ug alingam in [22]. Definition 4.1. Let Ω b e a relatively compact domain in X and x 0 ∈ Ω. An extended real-v alued function G = G ( · , x 0 ) on Ω is said to b e a singular function with singularity at x 0 if 1. G is p 0 -harmonic a nd p ositiv e in Ω \ { x 0 } , 2. G | X \ Ω = 0 and G ∈ N 1 ,p 0 ( X \ B ( x 0 , r )) for all r > 0, 3. x 0 is a singularity , i.e., lim x → x 0 G ( x ) = Cap p 0 ( { x 0 } , Ω) 1 / (1 − p 0 ) , and lim x → x 0 G ( x ) = ∞ if Cap p 0 ( { x 0 } , Ω) 1 / (1 − p 0 ) = 0, 4. whenev er 0 ≤ α < β < sup x ∈ Ω G ( x ), C 1 ( β − α ) 1 − p 0 ≤ Cap p 0 (Ω β , Ω α ) ≤ C 2 ( β − α ) 1 − p 0 , 12 NICOLA GAROF ALO AND NIKO MAROLA where Ω β = { x ∈ Ω : G ( x ) ≥ β } , Ω α = { x ∈ Ω : G ( x ) > α } , and 0 < C 1 , C 2 < ∞ ar e constants dep ending only on p 0 . Note that the singular function is necessarily non-constant, and con- tin uous on Ω \ { x 0 } . W e ha v e the fo llo wing theorem on the existence . Theorem 4.2. L et Ω b e a r elatively c omp act domain in X , x 0 ∈ Ω , and Q ( x 0 ) the p ointwise dim e nsion a t x 0 . Then ther e exis ts a singular function on Ω with singularity at x 0 . Mor e over, i f p 0 ≤ Q ( x 0 ) , then every singular func tion G with singularity at x 0 satisfies the c o n dition lim x → x 0 G ( x ) = ∞ . Essen tially , the pro of follo ws from the Harnac k inequalit y on spheres and Corollary 3.5. In particular, it is in the Harnac k inequalit y on spheres that X is needed to b e linearly lo cally connected. See, e.g., Bj¨ o rn et al. [4, Lemma 5.3]). W e o mit the pro of. Remark 4.3. The theorem w as first prov ed b y Holopainen and Shan- m uga lingam in [22, Theorem 3.4] under the additional as sumption that the measure on X is Q -regular, i.e., for all balls B ( x, r ) a double in- equalit y C − 1 r Q ≤ µ ( B ( x, r )) ≤ C r Q holds. (If µ is Q -regular then X is called a n Ahlfors regular space.) There are, ho w ev er, man y instanses where this is not satisfied. F or example, the weigh ts mo difying the Leb esgue measure in R n , see [19], or systems of v ector fields of H¨ ormander t yp e, see e.g. Cap o gna et al. [6], are, in general, not Q - regular for an y Q > 0. In this sense o ur observ ation seems to generalize sligh t ly the results obtained in [22]. Reference s [1] Bj ¨ orn, A., Bj ¨ orn, J. and Shanmugalingam, N. , Quasicontin uity of Newton–Sob olev functions a nd density of Lipschitz functions on metric spaces, to app ear in H ouston J. Math. [2] Bj ¨ orn, A. and Marola, N., Moser iteration for (quasi)minimizers on metric spaces, Manuscripta Math. 121 (200 6), 339– 3 66 . [3] Bj ¨ orn, J., Boundar y contin uity for quasiminimizers on metric spa ces, Il linois J. Math. 46 (2002 ), 3 83–40 3 . [4] Bj ¨ orn, J. , MacM anus, P. and Shan mugalingam, N. , F at sets and p oint- wise b oundary estimates for p -har monic functions in metric spaces , J. Anal. Math. 85 (2001), 339– 369. [5] Buser, P. , A note on the iso pe rimetric constant, Ann. Sci. ´ Ec ole Norm. Sup. (4) 15 (1982), 213– 230. [6] Capogna, L ., Danielli, D . and G ar of alo, N ., Capacitar y estimates and the lo cal b ehavior of solutions o f nonlinear sub elliptic equations, Amer. J. Math. 118 (1996 ), 115 3 –119 6. [7] Cara th ´ eodor y, C., Un tersuch ung en ¨ uber die Grundlangen der Thermo dy- namik, Math. Ann. 67 (190 9 ), 355–3 86 . SHARP CAP A CIT AR Y ESTIMA TES FOR RINGS IN METRIC SP A CE S 13 [8] Cha v el, I., Eigenvalues in Riemannian Ge ometry, Pur e and Applied Ma th- ematics, vol. 115, Academic Pres s, O rlando, 1984 . [9] Cheeger, J., Differentiabilit y of Lipschitz functions on metric measure spaces, Ge om. F unct. Anal. 9 (1999 ), 42 8–517. [10] Chow, W.L., ¨ Uber System von linearen partiellen Differen tialgleich ung en er- ster Ordnug, Math. Ann. 117 (1939), 98 –105. [11] Danielli, D., Garof alo, N. and Marola, N. , Lo cal b ehavior of p -harmo nic Green’s functions o n metric spaces, Pr eprint , 2008. [12] Da vid , G. and Semmes, S., F r actur e d fr actals and br oken dr e ams. Self-similar ge ometry t hr ough metric and me asu r e, O xford Lectur e Series in Mathematics and its Applications, 7 . The Clare ndon Press , Oxford University Press , New Y ork, 1 997. [13] Garof a lo, N. and Nhieu, D. -M., Isop er imetric a nd Sob olev ineq ualities fo r Carnot-Ca rath´ eo do ry s paces and the existence of minimal sur faces, Comm. Pur e Appl. Math. 49 (199 6 ), 1081– 1144 . [14] Gol ’dshtein, V. a nd Tro y anov, M., Ca pa cities in metric spaces , Inte gr al Equations Op er ator The ory 44 (2002), 21 2–24 2. [15] Haj lasz , P. and Koskela, P., Sob olev met Poincar´ e, Mem. Amer. Math. So c. 145 (2000 ), 1 -110. [16] Heinonen, J., A ca pa city e stimate o n Carnot g roups, Bul l. Sci. Math. 119 (1995), 475– 484. [17] Heinonen, J. , L e ctur es on Analysis on Metric Sp ac es, Springer-V erlag, New Y ork, 2 001. [18] Heinonen, J. and Holop ainen, I ., Q ua siregular maps on Ca rnot g r oups, J . Ge om. Anal. 7 (19 9 7), 109–14 8 . [19] Heinonen, J., Kilpel ¨ ainen, T. and Mar tio , O., Nonline ar Potential The- ory of De gener ate El liptic Equations, O xford Univ ersity P ress, Oxford, 19 93. [20] Heinonen, J. a nd K oskela, P. , Quasiconfor mal maps in metric spaces with controlled geo metry , A cta Math. 181 (1998), 1–61. [21] Holop ainen, I., Nonlinear p otential theory and q uasiregula r mappings o n Riemannian manifolds, Ann. Ac ad. Sci. F enn. Ser. A I Math. Diss. 74 (1990 ), 1–45. [22] Holop ainen, I. and Shanmugal ingam, N., Singular functions on metric measure spaces, Col le ct. Math. 53 (2002 ), 31 3–332. [23] Jerison, D., The Poincar ´ e inequality for vector fields sa tisfying H¨ ormander’s condition, Duke Math. J. 53 (198 6), 503–5 23 . [24] Kallunki, S. and Shanmugal ingam, N., Modulus and c o nt inuous capacity , Ann. A c ad. Sci. F enn. Math. 26 (2 001), 455–4 64 . [25] Kilpel ¨ ainen, T. , Kinn unen, J. a nd Mar tio, O., Sob ole v spaces with zero bo undary v alues o n metric spaces, Potential Anal. 12 (2 000), 233–2 47. [26] Kinnunen, J. and Mar tio, O., The Sob olev capacity on metric spa ces, Ann. A c ad. Sci. F enn. Math. 21 (1996), 3 6 7–38 2. [27] Kinnunen, J. and Mar tio, O. , Cho quet prop er ty for the Sobo lev capac it y in metric spaces, in Pr o c e e dings on Analysis and Ge ometry (Novosibirsk, Ak adem- goro dok , 199 9), pp. 285– 290, Sob olev Institute Pre ss, No vosibirsk, 200 0. [28] Kinnunen, J. and Shanmugalingam, N., Regularity of quasi-minimizers o n metric spaces, Manuscripta Math. 105 (2001), 401 – 423. [29] Kor ´ anyi, A. a nd Reimann, H . M., Horizontal normal vectors and confor- mal capa cit y of spher ical rings in the Heisenberg gr oup, Bul l. Sci. Math. 111 (1987), 3–21. 14 NICOLA GAROF ALO AND NIKO MAROLA [30] Littman, W., St amp acchia, G. and W einber ger, H.F., Reg ular p oints for elliptic equa tions with disc o nt inuous co e fficien ts, Ann. S cu ola N orm. Sup. Pisa (3) 18 (196 3 ), 4 3–77. [31] M ¨ akel ¨ ainen, T. , Ada ms inequality on metric measure spa ces, pr eprint , 2 0 07. [32] Nagel, A., Stein, E.M. and W ainger, S., B a lls and metrics defined by vector fields I: basic prop erties, Ac ta Math. 155 (1985), 1 0 3–14 7. [33] Rashevsky, P.K., Any tw o p oints of a totally nonholono mic space may be connected by an admissible line, Uch. Zap. Pe d. Inst. im. Liebkne chta, Ser. Phys. Math., (Russian) 2 (19 38), 83–94. [34] Rothschild, L.P. and Stein, E.M., Hyp o elliptic differential oper ators and nilpo ten t gro ups, A cta Math. 137 (1976), 24 7–32 0. [35] Semmes, S. , Metric space s a nd mappings s een at many s cales (App endix B in Gromov, M. , Metric Struct ur es for Riemannian and N on-Riema nnian Sp ac es ), Ed. Pr ogress in Mathematics, Bir kh¨ auser, Boston, 19 99. [36] Serrin, J., Lo cal be havior of solutions of quasilinea r equations, A cta Math. 111 (1964), 243– 302. [37] Serrin, J., Isolated singularities of solutions of quasilinear eq uations, A ct a Math. 113 (1965 ), 219 – 240. [38] Shanmugalingam, N ., Newtonian s paces: An extensio n of Sobo lev spaces to metric measure spa ces, R ev. Mat. Ib er o americ ana 16 (20 0 0), 243–27 9 . [39] Shanmugalingam, N., Har monic functions on metric s pa ces, Il linois J. Math. 45 (2 001), 1021– 1050. Dep ar tment of Ma thema tics, P urdue University, West La f a yette, IN 47907, U SA E-mail addr ess , Nicola Ga rofalo: garof alo@ma th.pu rdue.edu Dep ar tment of Ma thema tics and S ystems Ana l ysis, Helsink i Univer- sity of Technolo gy, P .O. Box 1100 FI-02015 TKK, Finland E-mail addr ess , Niko Marola: niko. marola @tkk.fi