Aspects of area formulas by way of Luzin, Rado, and Reichelderfer on metric measure spaces

ASPECTS OF AREA FORMULAS BY W A Y OF LUZIN, RAD ´ O, AND REICHELDERFER ON METRIC MEASURE S P A CES NIKO MAR OLA AND WILLIAM P . ZIEMER A B S T R AC T . W e c onsider some measu re-theoretic properties of fu nc- tions belongin g to a Sobolev-type cla s s on metric measure spaces th at admit a Poincar ´ e inequality and are equipped with a doublin g measure. The properties we ha ve selected to study are those that are related to area formu las . Mathematic s Subjec t Classific ation (2010) : Primary: 46E35, 46E40; Secondary : 30L99, 28A99 . K e y wor ds and phrases : Area formula, condi tion N, doubling m e asure, L u zin’ s condit ion, metric space, N e wto nian space, Poincar ´ e inequality , Sobolev space, up- per grad ient. 1 . I N T RO D U C T I O N W e in v estigate s ome m easure- theoretic properties of functions belonging to the Banach or vector space-va lued Newtonian space N 1 ,p ( X ) a nd com- pare t hese properties in th e more g ene ral setting with the classical Euclidean ones. Ne wtonian space is a metri c space analogue of the classi ca l Sobolev space W 1 ,p ( R n ) and w as first int roduced and st udied by S hanmugalingam in [29]; here X refers to a comp lete metric measure space with a measure µ th at satisfies a v olume doubling con dition and the space is assumed to support a Poincar ´ e inequalit y . Under these rather standard conditions on the space, we give a metric space version of L uzin’ s condition for the graph mapping similar to one in Mal ´ y et al. [27], we study absolute continuity as defined by Mal ´ y [23] for functions in the Newtonian class, and we als o discuss the condition due to Rad ´ o and Reichelderfe r [28]. W e p ro vide a version of t he area formula for Newtonian functions. In par- ticular , we e xtend the Euclidean results of Hajłasz [10] and Mal ´ y e t al. [27] to Ne wton–Sobole v functions in t he aforemention ed setting of gener al met- ric spaces. W e provide another vi e w to a recent result by Magnani [22] which is related to the area formula in general metric measure spaces. Under rather general assumption s on X (see Section 2) the follo wing area formula will be sh o wn to be valid for the graph m a pping ¯ u of u ∈ 1 2 NIKO MAR OLA AND WILLIAM P . ZIEMER N 1 ,p lo c ( X ; R m ) , where p > m or p ≥ m = 1 , H Q ( ¯ u ( A )) = Z A J ¯ u dµ, whenev er A is a µ -measurable subset and J ¯ u denotes the generalized J a- cobian of ¯ u . In p ar ticular , H Q ( ¯ u ( A )) = 0 whenever µ ( A ) = 0 . Here the exponent Q serves as a subst itute for the dim e nsion of X , and it is associ- ated with the doubl ing constant of the un der lying measure µ (see Section 2). Althought the proofs for these formulas are ra ther standard, ou r g e neral setting causes some unexpected difficulties. T o overc ome these, we care- fully consi der some local properties of so-called generalized Jacobi a n o f a function and couple them with the a forementioned measure-theoretic prop- erties of Ne wt on–Sobole v functions. There is a rich supply of examples of complete metric spaces with a vol- ume doubl ing measure th at sup port a Poincar ´ e inequality and where our results are applicable. T o nam e but a few , we list Carnot–Carath ´ eodory spaces, thus i ncluding the Heisenberg group and more general Carnot groups, as well as Riemannian manifolds with non-negati ve Ricci curvature. In out line, the p a per is or gani ze d as follows: In Section 2 we introduce the necessary background material such as the doubli ng conditi on for the measure, upper gradients, Poincar ´ e inequalit y , Ne wtonian spaces, and ca- pacity . In Section 3 we establish a general criterion for a v ersio n of Luzi n’ s condition in the spi rit of Rad ´ o and Reichelderfer [28, V .3.6], see also Mal ´ y et al. [27]. Then we close Section 3 by proving, wi th the aid of estimates between the capacity and t he Hausdorff content, that the graph mapping of a vector -valued Newtonian function satisfies a v ersi on of the Lu z in conditi on. In Section 4 we deal wi th the are a f ormula. In Se ction 5 we study the Rad ´ o– Reichelderfer condi tion and abso lute continui ty of Newtonian functions in the spirit of Mal ´ y [23]. Ac knowledgements. W e would l ik e to thank N a geswari Shanmugalingam for detail e d com ments and suggestions on sev eral draft versions of th e pa- per . 2 . M E T R I C M E A S U R E S PAC E S : D O U B L I N G A N D P O I N C A R ´ E W e b riefl y recall the basic definition s and collect some well-known re- sults needed later . For a thorou gh treatment we refer t he reader to a mono - graph by A. and J. Bj ¨ orn [3] and Heino nen [13]. Throughout the paper , if not ot herwise st a ted, X := ( X , d, µ ) i s a com - plete m etric space endowed with a metric d and a posi ti ve complete Borel regular measure µ such th at 0 < µ ( B ( x, r )) < ∞ for all balls B ( x, r ) := { y ∈ X : d ( x, y ) < r } ; and if B = B ( x, r ) , then we denote τ B = ASPECTS OF AREA FORMULAS ON METRIC MEASURE SP A CES 3 B ( x, τ r ) for each τ > 0 . W e also denote the metric ball B ( x, r ) by B X ( x, r ) if necessary . Also throughout the paper , if not otherwise st a ted, let Y := ( Y , ˜ d, ν ) be a complete s epar able metric measure space with a pos- itive complete Borel regular measure ν . A function f : X → Y is called L -Lipschitz if for all x, y ∈ X , ˜ d ( f ( x ) , f ( y )) ≤ Ld ( x, y ) . W e let Lip( f ) be the infimum of such L . In our treatment, it is natu ra l to assume some connection b e tween the measure and the metric. Also by dim e nsion we mean s ome quantity which relates t he measure of a m e tric ball to its radius. W e shall clarify th e se concepts bel o w . Our s tanding assumptions on th e metric space X are as follows. (D) The measure µ is dou bling, i.e., there exists a constant C µ ≥ 1 , called the doubling constant of µ , such that µ ( B ( x, 2 r )) ≤ C µ µ ( B ( x, r )) . for all x ∈ X and r > 0 . (PI) The sp ac e X sup ports a weak (1 , p ) -Poin c ar ´ e inequality for so me p ≥ 1 (see below). W e not e the doubling condition (D) implies that for every x ∈ X and r > 0 , we hav e for λ ≥ 1 (2.1) µ ( B ( x, λr )) ≤ C λ Q µ ( B ( x, r )) , where Q = log 2 C µ , and the constant depends only on C µ . The e x ponent Q serves as a dim ension of the dou bling measure µ ; we em phasize that it need not be an integer . When it is nece ssary to emphasize the r elat ion- ship between Q and X , we will use the notation X Q . Complete metric spaces verifying condition (D) are precisely those t hat hav e finite Assouad dimension [ 13]. Th is notio n of dimension, ho wever , need not to be uniform in space. In wh at follows, we assume further that t her e exists a constant C > 0 , depending onl y on C µ , such that the measure µ satisfies th e l o wer mass bound (2.2) C r Q ≤ µ ( B ( x, r )) for all x ∈ X and 0 < r < diam( X ) . It follo ws from ( D) that µ satis fi es the following local v ersion of (2.2): For a fi xed x 0 ∈ X and a scale r D > 0 we ha ve (2.3) ˜ C r Q ≤ µ ( B ( x, r )) for all balls B ( x , r ) ⊂ X with x ∈ B ( x 0 , r D ) and 0 < r < r D , wh e re ˜ C = C r − Q D µ ( B ( x 0 , r D )) and C is from (2.1). 4 NIKO MAR OLA AND WILLIAM P . ZIEMER Let s ≥ 0 . W e define the (spherical) H a usdorff s -measure in X as i n Federer [8, 2.10.2] (see also [13]) a nd will denote it by H s . W e also denote by H s ∞ the Hausdorff s -content in X defined as H s ∞ ( E ) = inf  ∞ X i =1 r s i : E ⊂ ∞ [ i =1 B ( x i , r i ) , x i ∈ E  , where the in fi mum is taken over all countable covers of E by b a lls B ( x i , r i ) . W e note h e re that if X is a p r oper , i.e. b oundedly com pact, m etric s pace , then Hausdorf f content is inner regular in the follo w ing s ense H s ∞ ( E ) = sup {H s ∞ ( K ) : K ⊂ E , K com pact } whenev er E ⊂ X i s a Borel set. See Federer [8, Corollary 2.10.23]. W e shall also need the concept of the Hausdorff measure of codimension s of E ⊂ X which w e define by app lying the Carath ´ eodory construction to the function h ( B ( x, r )) = µ ( B ( x, r )) r s . Above, we use t he con vention h ( B ( x, 0)) := h ( ∅ ) = 0 . W e thus d e fine t he restricted Hausdorf f content of codimension s as follows e H s R ( E ) = inf  ∞ X i =1 h ( B ( x i , r i )) : E ⊂ ∞ [ i =1 B ( x i , r i ) , x i ∈ E , r i ≤ R  , where 0 < R < ∞ . When R = ∞ , we have the corresponding H a usdorff content of E and denot e it by e H s ∞ ( E ) . Finally , the Hausdorf f measure of codimension s is defined as e H s ( E ) = lim R → 0 e H s R ( E ) . W e remark t hat if the m ea sure µ is Q -regular , i.e., µ ( B ( x, r )) ≈ r Q , for some Q ≥ 1 , e H s ( E ) ≈ H Q − s ( E ) . Let us mention th a t the lower m ass bound (2.2) for t he measure µ implies that H Q is absolutely continuous with respect to µ and that H Q − s ( E ) ≤ C e H s ( E ) . The upper s -density of a finite Borel re gu lar measure ζ at x is defined by Θ ∗ s ( ν, x ) = lim sup r → 0 + ζ ( B ( x, r )) ω s r s , where ω s is the Lebesgue measure of the unit ball in R s when s i s a positive integer , and ω s = Γ(1 / 2) s / Γ( s/ 2 + 1) otherwise. W e record that if for all x in a Borel set E ⊂ X , Θ ∗ s ( ζ , x ) ≥ α , 0 < α < ∞ , then ζ ≥ α C H s E , ASPECTS OF AREA FORMULAS ON METRIC MEASURE SP A CES 5 where the pos iti ve cons tant C depends on ly on s . On t he ot her hand, if Θ ∗ s ( ζ , x ) ≤ α w e obtain ζ E ≤ αC H s E , where a positive constant C depends only on s . See Federe r [8, 2.10.1 9]. Recall that the following g e neral c overing theorem is valid in our setting. From a giv en f amil y of balls B with sup { diam B : B ∈ B } < ∞ covering a set E ⊂ X we can select a pairwise disjoint subfamily B ′ of balls su c h that E ⊂ S B ∈B ′ 5 B , see [8, Coroll a ry 2.8.5]. If X is separable, then B ′ is countable and B ′ = { B i } i ≥ 1 . In this no te, a curve γ in X is a continuous m apping from a compact interval [0 , L ] to X . W e recall that each curve c an be parametrized by 1- Lipschitz map ˜ γ : [0 , L ] → X . A nonnegativ e Borel functi on g on X is an upper gradient o f a function f : X → Y if for all rectifiable curves γ , we hav e (2.4) ˜ d ( f ( γ ( L )) , f ( γ (0))) ≤ Z γ g ds. See C heeger [5] and Shanmugalingam [ 29] for a discussion on upper gradi- ents. If g is a nonnegativ e measurable functi on on X and if (2.4) holds for p -alm ost e very curve, p ≥ 1 , then g is a weak u pper gradient of f . By say- ing that (2.4 ) holds for p -almost e very curv e we mean that it f ail s o nly for a curve f amily with zero p -modul us (see, e.g., [29]). If u has an up per gra di- ent in L p ( X ) , then it has a minimal weak upper gr adient g f ∈ L p ( X ) in the sense that for ev ery weak upper gradient g ∈ L p ( X ) of f , g f ≤ g µ -almost e verywhere (a.e.), see Corollary 3.7 in Shanmugalingam [30]. While the results in [29] and [30] are formulated for real-v alued functions and their upper gradients, they are applicable for m etric space v alu e d functions and their upper gradients; the proofs of t hese results require only the mani pula- tion of upper gradients, which are alw ays real-valued. W e define Sobolev spaces on metric s pac es foll o wing Shanmugalin gam [29]. Let Ω ⊆ X be nonempty and open. Whene ver u ∈ L p (Ω) and p ≥ 1 , let (2.5) k u k N 1 ,p (Ω) := k u k 1 ,p :=  Z Ω | u | p dµ + Z Ω g p u dµ  1 /p . The Ne wton ian space on Ω is t he quotient space N 1 ,p (Ω) = { u : k u k N 1 ,p (Ω) < ∞} / ∼ , where u ∼ v i f and only if k u − v k N 1 ,p (Ω) = 0 . The space N 1 ,p (Ω) is a Banach sp a ce and a lattice. If Ω ⊂ R n is o pen, then N 1 ,p (Ω) = W 1 ,p (Ω) 6 NIKO MAR OLA AND WILLIAM P . ZIEMER as Banach spaces. For these and ot her p roperties of Ne wtonian spaces we refer to [29 ]. The class N 1 ,p (Ω; R m ) consists of those mappings u : Ω → R m whose component functions each belong to N 1 ,p (Ω) = N 1 ,p (Ω; R ) . Qualitative properties li k e Lebesgue points, densi ty of Li pschitz functions, quasicontinui ty , etc. may be in vestigated component wise. A function belon gs to the local Newtonian s pace N 1 ,p lo c (Ω) if u ∈ N 1 ,p ( V ) for all bounded open sets V with ¯ V ⊂ Ω , the latter space bein g defined by cons idering V as a m etric space wi th the m etric d and the m easure µ restricted to it. Ne wtonian spaces share many properties of the classical Sobolev spaces. For example, if u, v ∈ N 1 ,p lo c (Ω) , then g u = g v µ -a.e. i n { x ∈ Ω : u ( x ) = v ( x ) } , furthermore, g min { u,c } = g u χ { u 6 = c } for c ∈ R . W e shall also n e ed a New tonian space with zero boundary values . For a measurable set E ⊂ Ω , let N 1 ,p 0 ( E ) = { f | E : f ∈ N 1 ,p (Ω) and f = 0 on Ω \ E } . This space equipped with the norm inherited from N 1 ,p (Ω) is a Banach space. W e say that X supports a w eak (1 , p ) -P oincar ´ e inequal ity if there e xist constants C > 0 and τ ≥ 1 such t hat for all balls B ( z , r ) ⊂ X , all measur- able functions f on X and for all weak upper gradients g f of f , (2.6) Z B ( z,r ) | f − f B ( z,r ) | dµ ≤ C r  Z B ( z,τ r ) g p f dµ  1 /p , where f B ( z,r ) := R B ( z,r ) f dµ := R B ( z,r ) f dµ/µ ( B ( z , r )) . It is well known that the embedding N 1 ,p ( X ) → L p ( X ) is not su rjec tiv e if and only if there exists a curve family in X with a pos iti ve p -modu lus. Moreover , th e validity of a Poincar ´ e i nequality can sometim es be stated in terms of p -modulus . More precisely , to requi re that (2.6) holds in X is to require that the p -mod ulus of curves between e very pair of distinct points of the space is suffi ciently lar ge, see Theorem 2 in Keith [15]. It is not e worthy that by a result of K eith and Zhong [16] in a compl e te metric space equip ped with a d oubling measure and supporting a weak (1 , p ) -Poincar ´ e inequal ity there exists ε 0 > 0 s uch that the space admits a weak (1 , p ′ ) -Poincar ´ e in e quality for each p ′ > p − ε 0 . The following Luzin-type approx imation theorem shall be of use later in the paper . W e refer to Shanmugalingam [29, Theorem 4.1] for the proof which, in t urn, is a modification of an idea due to S. Semm es. See also Hajłasz [9, Theorem 5]. Theor em 2.1. Suppo se X sati sfies (D) and (PI) for some 1 < p < ∞ . Let u ∈ N 1 ,p ( X ) . Then for every ε > 0 ther e is a Lipschitz fu nction f ε : X → ASPECTS OF AREA FORMULAS ON METRIC MEASURE S P ACES 7 R such that µ ( { x ∈ X : u ( x ) 6 = f ε ( x ) } ) < ε and k u − f ε k 1 ,p < ε . In other wor ds, with F ε := { x ∈ X : u ( x ) 6 = f ε ( x ) } , we have u | X \ F ε is Lipschitz. Capacity. There are se veral equi valent definitions for capacities, and t he following are the ones we find most sui table for our purposes. Let 1 ≤ p < ∞ and Ω ⊂ X bou nded. • The variational p -capacity of a set E ⊂ X is the number cap p ( E ) = inf k g u k p L p ( X ) , where the i nfi mum is t a ken over all u ∈ N 1 ,p ( X ) such that u ≥ 1 on E ; recall that g u is the minimal p -weak upp er gradient of u . • The relative p -capacity o f E ⊂ Ω is the num ber Cap p ( E , Ω) = inf k g u k p L p (Ω) , where the i nfimum is taken over all u ∈ N 1 ,p 0 (Ω) such that u ≥ 1 on E . • The Sobolev p -capacity of E ⊂ X is the numb e r C p ( E ) = inf k u k p N 1 ,p ( X ) , where the i nfi mum is t a ken over all u ∈ N 1 ,p ( X ) such that u ≥ 1 on E . Observe that if µ ( X ) < ∞ the constant function will d o as a test function, thus all sets are of zero variational p -capacity . Howe ver , t his is not true for the relati ve p -capacity whene ver X \ Ω is “large” , say , C p ( X \ Ω) > 0 . Under our assum ptions, these capacities enjoy the standard properties of capacities. F or in stance, wh en p > 1 they are Choquet capacities, i.e., the capacity of a Borel set can be obtained by approximating with comp ac t sets from insid e and open sets from outside. It i s notew orth y , howe ver , that the Choquet property f ail s for p = 1 in the general metric setting. This does not cause any problem s for us as we mainl y deal with comp a ct set s in this note. In a recent paper by Kinnunen–Hakkarainen [12] the BV -capacity was proved to be a Choq uet capac ity . See, e.g., Ki nnunen–Martio [ 18], [19] for a discussion on capacities on metric spaces. The Sobolev capacity is the correct g a uge for disti nguishing b e tween Ne wtonian functions: if u ∈ N 1 ,p ( X ) , then u ∼ v if and on ly if u = v p -qu asie verywhere, i.e., outside a set of zero Sobolev p -capacity . More- over , by Shanmugalingam [29] if u, v ∈ N 1 ,p ( X ) and u = v µ -a.e., t hen u ∼ v . A function u ∈ N 1 ,p ( X ) is said to be quas icontinuous , if there exists an open set G ⊂ X with arbitrarily small Sobole v p -capacity such that the restriction of u to X \ G is continuous. A mapping in N 1 ,p ( X ; R m ) 8 NIKO MAR OLA AND WILLIAM P . ZIEMER is said to be quasicontinuous if each o f its component functions is quasi- continuous. Recall that all functions in N 1 ,p ( X ) are q uasicontinuous, s e e Bj ¨ orn et al. [4]. Since Ne wtonian functions h a ve Lebesgue points outside a set o f zero Sobole v capacity , in what foll o ws we m ay as sume that e very Ne wtonian function is precisely represented. 3 . G R A P H S O F N E W T O N I A N F U N C T I O N S : L U Z I N ’ S C O N D I T I O N Let Q > 0 . Recall t hat a m apping f : X → Y is said to satisfy Luzin ’ s condition ( N Q ) if H Q ( f ( E )) = 0 whenev er E ⊂ X satisfies µ ( E ) = 0 . By way of mot i vation, the v alidit y of Luzin ’ s condition i mplies certain change of v ariable form ulas, thus it is of independent interest in analysis. Let E ⊂ X . W e denote by ¯ f : X → X × Y the graph mapping of f ¯ f ( x ) = ( x, f ( x )) , x ∈ X , and G f ( E ) is the graph of f ove r E defined by G f ( E ) = { ( x, f ( x )) : x ∈ E } ⊂ X × Y . It is well known th a t if th e mapping f is Borel measurable, then th e graph G f ( X ) is Borel measurable as w e ll, see, e.g., [10, Lem ma 18]. W e, fur - thermore, denote by pr X : X × Y → X the projectio n pr X ( x, y ) = x , and by pr Y : X × Y → Y the projecti on pr Y ( x, y ) = y . Observe that Lip(pr X ) = Lip(pr Y ) = 1 . Als o it is well-known that if f : X → Y is continuous, then G f ( X ) i s homeomorphic to X . Lemma 3.1. Let f : X → R m , m ≥ 1 , b e me asurable. Then pr X ( G f ( X ) ∩ E ) is measurable for e very Bor el measurable subset E ⊂ X × R m . Pr oof. Let f ∗ and f ∗ be Borel measurable representatives o f f ; Borel regu- larity of the m ea sure µ imp lies that i f f is measurable, then there exist Borel measurable functions f ∗ , f ∗ such that f ∗ ≤ f ≤ f ∗ and f ∗ ( x ) = f ∗ ( x ) for µ -a.e. x ∈ X . Thus the g ra ph G f ∗ ( X ) of f ∗ and the graph G f ∗ ( X ) of f ∗ are Borel subsets of X × R m . Then Kuratowski [20, Theorem 2, p. 385] implies that t he projections pr X ( G f ∗ ( X ) ∩ E ) and pr X ( G f ∗ ( X ) ∩ E ) are Borel measurable for e very Borel m easurable set E ⊂ X × R m . Since f ∗ and f ∗ agree up to a set of µ -measure zero, so do sets pr X ( G f ∗ ( X ) ∩ E ) and pr X ( G f ∗ ( X ) ∩ E ) , implying that pr X ( G f ( X ) ∩ E ) is µ -measurable.  W e now state a general criterion for the condition ( N Q ) similar to that of Rad ´ o and Reichelderfer , s e e [28, V .3.6] and Mal ´ y [ 23]. In Euclidean spaces this result was obtained by Mal ´ y et al. [27]. In what follows, we suppose that 1 ≤ m < Q , where m is related to R m . ASPECTS OF AREA FORMULAS ON METRIC MEASURE S P ACES 9 Theor em 3.2. Suppose X sat isfies condition (D) a nd t he lower mass bound (2.2) is satisfied. Let f : X Q → R m be a measurable function. Denote Ξ z ,r = G f ( X Q ) ∩ B ( z , r ) , wher e z ∈ X Q × R m and 0 < r < diam( X Q ) . Suppos e that ther e exists a weight Φ ∈ L 1 lo c ( X Q ) such that (3.1) H Q − m ∞ (pr X (Ξ z ,r )) ≤ 1 diam(Ξ z ,r ) m Z pr X (Ξ z , 4 r ) Φ dµ for all z ∈ X Q × R m and all 0 < r < diam ( X Q ) / 4 . Then ther e exists a positive constant C < ∞ , depending on C µ and m , such that (3.2) H Q ( ¯ f ( E )) ≤ C Z E Φ dµ for each Bor el measurable set E ⊂ X Q . In particular , ¯ f satisfies L uzin’ s condition ( N Q ) . Pr oof. Define a set function σ on the Cartesian product X Q × R m by σ ( E ) = Z pr X ( G f ( X Q ) ∩ E ) Φ dµ, E ⊂ X Q × R m . By a V itali-type covering theorem there is a pairwise disjo int count able subfamily of ball s { B i } := { B ( x i , r i ) } such that we may cover pr X (Ξ z ,r ) as follo ws pr X (Ξ z ,r ) ⊂ [ i B ( x i , 5 r i ) =: [ i 5 B i . For each i let M i denote the greatest integer satisfying ( M i − 1) r i < diam(Ξ z ,r ) . Since Ξ z ,r ∩ pr − 1 X (5 B i ) is bou nded in X Q × R m , it can be contained in a lar ge enough cylinder of th e form B ( x i , 5 r i ) × R i , where R i is a cube in R m with sid e- length dia m(Ξ z ,r ) . Since M i r i ≥ diam Ξ z ,r , R i may be covered by M m i cubes {R j i } with side r i . W e hence obtain H Q ∞ (Ξ z ,r ∩ pr − 1 X (5 B i )) ≤ C M m i r Q i ≤ C ( M i r i ) m r Q − m i ≤ C (dia m(Ξ z ,r ) + r i ) m µ (5 B i )(5 r i ) − m . As r i ≈ diam(5 B i ) ≤ diam pr X (Ξ z ,r ) ≤ diam(Ξ z ,r ) su mming over i shows that H Q ∞ (Ξ z ,r ) ≤ C diam(Ξ z ,r ) m ∞ X i =1 µ (5 B i ) (5 r i ) m . 10 NIKO MAR OLA AND WILLIAM P . ZIEMER Hence by t a king the infimum over all co verings we ha ve obtained the fol- lowing estimate H Q ∞ (Ξ z ,r ) ≤ C diam(Ξ z ,r ) m e H m ∞ (pr X (Ξ z ,r )) , where the constant C depends only on C µ and m . Assumption (3.1) together with this estimate gi ves for e ach z ∈ X × R m and 0 < r < diam ( X Q ) / 4 H Q ∞ (Ξ z ,r ) ≤ C diam(Ξ z ,r ) m e H m ∞ (pr X (Ξ z ,r )) (3.3) ≤ C Z pr X (Ξ z , 4 r ) Φ dµ ≤ C σ ( B ( z , 4 r )) . Since for H Q -almost e very z ∈ G f ( X Q ) , see Federer [7, Lemma 10.1], (3.4) lim sup r → 0 + H Q ∞ (Ξ z ,r ) ω Q r Q ≥ C , it follows from (3.3) that lim sup r → 0 + σ ( B ( z , r )) ω Q r Q ≥ C for H Q -almost ev ery z ∈ G f ( X Q ) . Lemma 3.1 i mplies that σ is a measure on th e Borel sigma algebra of X Q × R m , and i t m ay be extended to a regular Borel outer measure σ ∗ on all of X Q × R m in the usual way σ ∗ ( A ) := inf { σ ( E ) : A ⊂ E , E is a Borel set } . Since Φ ∈ L 1 lo c ( X Q ) it follows t hat σ ∗ is a R adon measure on X Q × R m . Therefore, by (3.4) H Q ( E ) ≤ C σ ∗ ( E ) for all E ⊂ G f ( X Q ) . Finally , give n a µ m ea surable set E ⊂ X Q , choose a Borel set G with E ⊂ G . Then ¯ f ( E ) ⊂ G × R m , G × R m is a Borel set, and H Q ( ¯ f ( E )) ≤ C σ ∗ ( ¯ f ( E )) ≤ C σ ( G × R m ) = C Z G Φ dµ. The proof is completed by taking th e infimum ov er all such G . If E ⊂ X Q such that µ ( E ) = 0 then i t readily follows t hat H Q ( f ( E )) = 0 . Th is completes the proof.  In (3.1) we may replace the Hausdorf f content H Q − m ∞ (pr X (Ξ z ,r )) with an inequality in volving e H m ∞ (pr X (Ξ z ,r )) on t he left hand side. W e shall show , as an application of Theorem 3.2, that the graph mapping of a N e wtonian fun c tion satisfies a version of Luzin’ s con dition ( N Q ) . W e start with a fe w auxiliary esti mates. W e sh all need the fol lo wing relation be- tween the p -capactity and the Hausdorf f content when p ≥ 1 . For the proof ASPECTS OF AREA FORMULAS ON METRIC MEASURE S P ACES 11 of the next lemma the reader should consult Costea [6, Thoerem 4.4] and Kinnunen et al. i n [17, Theorem 3.5] for the case (I) and (II), respectively . Lemma 3.3. Suppos e X sat isfies condit ions (D) a nd (PI), and th e lower mass bound (2.2 ) is satisfied. (I) Let 1 < p ≤ Q and E ⊂ X and suppose Q − p < t ≤ Q . Then H t ∞ ( E ∩ B ( x, r )) ≤ C r t − Q + p Cap p ( E ∩ B ( x, r ) , B ( x, 2 r )) , wher e x ∈ X , r > 0 , and C depends on C µ , p , t , and the constant s in the weak (1 , p ) -P oincar ´ e inequality . (II) Let p = 1 and E ⊂ X compact. Then e H 1 ∞ ( E ) ≤ C cap 1 ( E ) , wher e the constant C depends only on the doubl ing constant C µ and the constants in the weak (1 , 1) -P oincar ´ e inequality . Remark 3.4. If u ∈ N 1 ,p 0 ( B ( x, 2 r ); R m ) such that u ≥ 1 on E ∩ B ( x, r ) , g u is a mini mal p -weak upp e r gradient of u , and m , where 1 ≤ m < min { p, Q } , we ob tain H Q − m ∞ ( E ∩ B ( x, r )) ≤ C r p − m Z B ( x, 2 r ) g p u dµ, where the constant C is as in Lemma 3.3 (I). If u ∈ N 1 , 1 ( X ; R ) such that u ≥ 1 on E and g u is a minimal 1-weak upper gradient of u , Lemma 3.3 (II) implies that e H 1 ∞ ( E ) ≤ C Z X g u dµ, where the constant C is from Lemma 3.3 (II). The preceding estimates im ply the fol lo wing. Observe also that the graph mapping is always one-to-one. Theor em 3.5. Suppose that X sa tisfies conditio ns (D) and (PI) with some 1 ≤ p ≤ Q , and the lo w er mass bound (2.2) is sati sfied. Let u ∈ N 1 ,p ( X Q ; R m ) , wher e either p > m or p ≥ m = 1 . Then the graph mappi ng u satisfies Luzin’ s condition ( N Q ) . The assumption t hat p > m or p ≥ m = 1 is necessary already i n the Euclidean case. W e refer to a discuss ion in Mal ´ y et al. [27]. Pr oof of Theor em 3.5. It is su f ficient to verify the h ypothesis of Th eore m 3 .2 with some locally integrable function Φ on X Q . Assume first p > m and, to th is end, fix a po int z = ( ˜ x, ˜ y ) ∈ X Q × R m and r > 0 . W e observe the following Ξ z ,r = G u ( X Q ) ∩ B ( z , r ) ⊂ ( G u ( X Q ) ∩ ( B X ( ˜ x, r ) × B ( ˜ y , r ))) . 12 NIKO MAR OLA AND WILLIAM P . ZIEMER Hence we ha ve that pr X (Ξ z ,r ) ⊂ ( B X ( ˜ x, r ) ∩ u − 1 ( B ( ˜ y , r ))) , moreover u ( x ) ∈ B ( ˜ y , r ) for µ -a.e. x ∈ B X ( ˜ x, r ) ∩ u − 1 ( B ( ˜ y , r )) . Let us define the function v : X Q → R by v ( x ) = max  2 − | u ( x ) − u ( ˜ x ) | r , 0  , and consi der an open subset O ⊂ X Q such th at { x ∈ X Q : v ( x ) > 0 } ⊂ O . Then ( g u /r ) χ O is a p -weak upper gradient of v [2 9 , Lem ma 4.3], wher e g u is a mi nimal p -weak u pper gradient of u . Let η : X Q → R be a L ipschitz cut-off functio n so that η = 1 on B X ( ˜ x, r ) , η = 0 in X Q \ B X ( ˜ x, 2 r ) , 0 ≤ η ≤ 1 , and g η ≤ 2 /r . Th e n v η ≥ 1 on B X ( ˜ x, r ) ∩ u − 1 ( B ( ˜ y , r )) , and v η ∈ N 1 ,p 0 ( B ( ˜ x, 2 r )) . Moreover , the product rule for upper gradients giv es us the f ollowing g vη ≤ g v + 2 v /r µ -a.e. Thus v η is admissible for the relativ e p -capacity and L e mma 3.3 (I) implies that H Q − m ∞ (pr X (Ξ z ,r )) ≤ H Q − m ∞ ( B X ( ˜ x, r ) ∩ u − 1 ( B ( ˜ y , r ))) ≤ C r p − m Z B X ( ˜ x , 2 r ) ∩ O g p vη dµ ≤ C r p − m Z B X ( ˜ x , 2 r ) ∩ O  v p r p + g p v  dµ ≤ C r − m Z B X ( ˜ x , 2 r ) ∩ u − 1 ( B ( ˜ y , 2 r )) (1 + g p u ) d µ. Since B X ( ˜ x, 2 r ) ∩ u − 1 ( B ( ˜ y , 2 r )) ⊂ pr X (Ξ z , 4 r ) , above reasoning gives us that H Q − m ∞ (pr X (Ξ z ,r )) ≤ C r m Z pr X (Ξ z , 4 r ) (1 + g p u ) d µ. This verifies the assumpti ons of Theorem 3.2 wi th Φ = C (1 + g p u ) , and thus concludes the p r oof wh en p > m . Th e case p ≥ m = 1 is dealt with by a similar ar gum ent together with the estimate in Lemma 3.3 (II).  4 . A S P E C T S O F A R E A F O R M U L A S F O R N E W T O N I A N F U N C T I O N S In this section we shall prove versions of the area formu la for Newto- nian function s. In t he metric measure space setti ng t hese formulas have been studied pre viou sly by Ambrosio –Kirchheim [1], Magnani [21, 22], and Mal ´ y [24, 25], to name but a fe w . In particular , in [24] coarea prop- erties and coarea formula, which is considered as dual to the area form ula, ASPECTS OF AREA FORMULAS ON METRIC MEASURE S P ACES 13 are thoroughly stud ied in metric spaces. W e also refer to Hajłasz [10] for a very nice discussion on the topic in Eucli dea n spaces. W e define the generalized Jacobian of a continuous map f : X → Y at x as follo ws J f ( x ) := lim sup r → 0 ν ( f ( B ( x, r ))) µ ( B ( x, r )) , where, we recall, ν m easures Y . It follows from [8, 2.2.13] applied to th e pull-back measure ν f ( E ) := ν ( f ( E ) ) , that f ( E ) is ν -measurable for ev ery Borel set E ⊂ X . Moreover , for µ -a.e. x , the generalized Jacobian J f ( x ) is finite, see Federer [8, 2.9]. It is also easy to see that if g : X → Y is another cont inuous m ap s uch t hat g = f on an open subs e t A ⊂ X , then J f ( x ) = J g ( x ) for µ -a .e. x ∈ A . An alt er nativ e, but maybe less tractable, way t o define a generalized Ja- cobian of f at x could be as follows. Set e J f ( x ) := lim sup r → 0 f ∗ ν ( B ( x, r )) µ ( B ( x, r )) , where f ∗ ν is a measure which results by Carath ´ eodory’ s constructio n from ζ ( A ) = ν ( f ( A )) , A ⊂ X , on the family of all Borel su bsets of X , see [8, 2.10.1]. Hence if A is a Borel sub set of X , then f ∗ ν ( A ) = sup ( X B ∈H ζ ( B ) : H is a Borel partition of A ) cf. [8, Theorem 2.10.8]; for any Borel set A ⊂ X the following identity will be satisfied [8, Theorem 2.10.10] f ∗ ν ( A ) = Z Y N ( f | A , y ) dν ( y ) , where the mul tiplicity function of f relativ e to a subset A is written as N ( f | A , y ) = #( A ∩ f − 1 ( y )) for each y ∈ Y . T o compare these two notions, we ha ve that J f ( x ) = e J f ( x ) = J f | D ( x ) for µ -a.e. x ∈ D , where D ⊂ X is closed and f | D is assumed to be one-to- one. Here we denote J f | D ( x ) := lim sup r → 0 ν ( f ( B ( x, r ) ∩ D )) µ ( B ( x, r )) . Let us clarify this. Clearly , J f | D ( x ) ≤ J f ( x ) ≤ e J f ( x ) for µ -a.e. x ∈ D . On the other hand, since f i s on e- to-one on D we ha ve that ζ ( A ) := ν ( f ( A )) is , in fact, a measure on D , and that ζ ( A ) = f ∗ ν ( A ) for every 14 NIKO MAR OLA AND WILLIAM P . ZIEMER Borel subset of D . Thus we obtain as in Magnani [ 22, proof of Theorem 2] for e very ( density point) x ∈ D e J f ( x ) ≤ lim sup r → 0 ν ( f ( B ( x, r ) ∩ D )) µ ( B ( x, r )) + lim sup r → 0 f ∗ ν ( B ( x, r ) \ D ) µ ( B ( x, r )) = J f | D ( x ) , where the last equali ty foll o ws form [8, Corollary 2.9.9] applied to e J f ( x ) χ D , where χ D is the characteristic function of the set D . Magnani [22] h a s rece ntly presented a unified approach to the a rea for- mula for merely continuous m appings between m etric s pac es, and thus without any notion of differentiability . W e remark th at in the present pa- per a funct ion in N 1 ,p lo c ( X Q ; R m ) alth ough having some “diff erentiability” properties, need not to be e ven continu ous as all Newtonian functi ons are, a priori, only quasicontinuous. Let u s state the following area formula. Theor em 4.1. Suppose X satisfies conditions (D) an d (PI) with s ome 1 ≤ p ≤ Q , and t he lower mass bound (2.2) is satis fied. Let u ∈ N 1 ,p lo c ( X Q ; R m ) , wher e p > m or p ≥ m = 1 . Then the following ar ea f ormula is valid (4.1) H Q ( ¯ u ( A )) = Z A J ¯ u ( x ) dµ ( x ) , whenever A ⊂ X is µ -measurable. Pr oof. By Theorem 3.5 the graph mapping ¯ u satisfies Luzin’ s condi tion ( N Q ) and is, moreover , one-to-on e on X . Thus the pull-back measure H Q ( ¯ u ( A )) , A ⊂ X Q arbitrary µ -measurable subset, is absolute continu- ous with respect to the doubling measure µ . Let { f i } i ≥ 1 , f i : X Q → R m , be a sequence of Lipschit z functions and E 1 ⊂ E 2 ⊂ . . . ⊂ X Q associated closed sets such that u i := u | E i = f i | E i and µ ( X Q \ S i E i ) = 0 . The existence of such sets and funct ions follows from Theorem 2.1. Th en the following identity is v alid by the area formula obtained in [22] (4.2) Z E i J ¯ f i ( x ) dµ ( x ) = H Q ( ¯ f i ( E i )) . Since u i ( x ) = f i ( x ) for x ∈ E i , E i closed, i t follows that J ¯ u i ( x ) = J ¯ f i ( x ) for µ -a.e. x ∈ E i . The equality (4.2) remains true f or measurable A ⊂ E ∞ , where E ∞ = S ∞ i =1 E i , and moreover , (4.2) will also be valid whenev er µ ( A ) = 0 . Thus (4.1) holds for all µ -measurable set A ⊂ X Q .  Let us discuss an alternati ve formulati on of the area formula which can be obtained by using Theorem 2 in Magnani [22]. Assume X satisfies con- ditions (D) and (PI) wit h som e 1 ≤ p < ∞ , and assume further that there exist disj oint µ -measurable sets { A j } j ≥ 1 such that they occupy µ -a.e. o f X , ASPECTS OF AREA FORMULAS ON METRIC MEASURE S P ACES 15 i.e. µ ( X \ S j A j ) = 0 . Let u ∈ N 1 ,p lo c ( X Q ; R N ) , where Q ≤ N . Assum e further that u satisfies Luzin’ s con dition ( N Q ) and u | A j is o ne- to-one for each i = 1 , 2 , . . . . Then the following area formula is v alid Z A θ ( x ) J u ( x ) dµ ( x ) = Z R N X x ∈ u − 1 ( y ) θ ( x ) d H N ( y ) , whenev er A ⊂ X is µ -measurable and θ : A → [0 , ∞ ] is a m ea surable function. In particul ar , Z A J u ( x ) dµ ( x ) = Z R N N ( u | A , y ) d H N ( y ) is v alid whene ver A ⊂ X is µ -m ea surable. 5 . N E W T O N I A N F U N C T I O N S : A B S O L U T E C O N T I N U I T Y , R A D ´ O , R E I C H E L D E R F E R , A N D M A L ´ Y Absolutely continuous functi ons on the real line sati sfy Luzin’ s condi- tion, are continuous, and differentiable almo st e verywhere. It i s well-known that these p r operties for the Sobolev class W 1 ,p ( R m ) depend on p . For in- stance, functions in W 1 ,m ( R m ) may be no where d if ferentiable and no where continuos whereas functions in W 1 ,p ( R m ) , p > m , hav e H ¨ o lder continu- ous representatives and are differe ntiable almo st ev erywhere. W e consider Luzin’ s conditio n, absol ute conti nuity , and differentiability for the Banach space va lued Newtonian space N 1 ,p ( X Q ; V ) , when p ≥ Q , and t hus e xtend some related results studied in Heinonen et al. [14]. Here V := ( V , k · k V ) is an arbitrary Banach space of positive dim ension. W e refer the reader to [14 ] for a detailed d iscussion on the Banach space valued Newtonian functions. Suppose X satisfies conditi ons (D) and (PI) with som e 1 ≤ p < ∞ ; the following i s kno wn: • Let p > Q . In this case each function u ∈ N 1 ,p ( X Q ; R ) is locall y (1 − Q/p ) -H ¨ older continuou s (Shanmug a lingam [29]), moreove r u is differentiable µ -a.e. with respect to the st rong measurable dif fer- entiable structure (see Cheeger [5]). For the latter result we refer to Balogh et al. [2]. • Let p = Q . Then ev ery cont inuous pseudomonot one m a pping in N 1 ,Q lo c ( X Q ; V ) satisfies Luzin’ s condition ( N Q ) (Heinonen et a l. [14, Theorem 7.2]). It would be interesting to genera lize Calderon’ s differentiability theorem to Banach space v alued Ne wt onian functions. 16 NIKO MAR OLA AND WILLIAM P . ZIEMER Recall that following Mal ´ y–Martio [26], a map f : X → V is pseu- domonotone if there exists a constant C M ≥ 1 and r M > 0 such that diam( f ( B ( x, r ))) ≤ C M diam( f ( ∂ B ( x, r ))) for all x ∈ X and al l 0 < r < r M . Note that we denot e ∂ B ( x, r ) := { y ∈ X : d ( y , x ) = r } . Let Ω be open such that Ω ⊂ X Q . W e s ho w next that u ∈ N 1 ,p (Ω; V ) , p ≥ Q , is abs olutely continuous in the foll o wing sense. Following Mal ´ y [23] we say that a mapping f : Ω → V is Q -absolutely cont inuous if for each ε > 0 there exists δ = δ ( ε ) > 0 such t hat for every pairwise di sjoint finite family { B i } ∞ i =1 of (closed) balls in Ω we hav e t hat ∞ X i =1 diam( f ( B i )) Q < ε, whenev er P ∞ i =1 µ ( B i ) < δ . Furthermore, we say that a mapping f : X → V sati sfie s the Q -Rad ´ o–Reichelderfer condi tion , condition (RR) for short, if there exists a non-ne gative contro l function Φ f ∈ L 1 lo c ( X ) s uch that (5.1) diam( f ( B ( x, r ))) Q ≤ Z B ( x,r ) Φ f dµ for e very b a ll B ( x, r ) ⊂ X with 0 < r < R . A condi tion simi lar to this was used by Rad ´ o and Reichelderfer in [28, V .3.6] as a sufficient condition for the mapping s wi th the condition (RR) to b e dif ferentiable a.e. and to satisfy Luzin’ s condition, see also Mal ´ y [23]. A function f i s said to satisfy condition (RR) weakly if (5.1) holds true with a di lated ball B ( x, α r ) , α > 1 , on the right-hand side of the equation. It re adily f ollows that condition (R R) implies (local) Q -absolute continu- ity of f . Indeed, let ε > 0 and { B ( x i , r x i ) } , 0 < r x i < R , a pairwise disjoint finite family of balls in Ω such that E = S i B ( x i , r x i ) , and µ ( E ) < δ . Then condition (RR) and pairwise disjointness of { B ( x i , r x i ) } im ply X i diam( f ( B ( x i , r x i ))) Q ≤ X i Z B ( x i ,r x i ) Φ f dµ = Z E Φ f dµ < ε. Local absol ute continuity of a fun c tion follows e ven if the functions satis fi es condition (RR) weakly . Condition (RR) also impl ies that the map f h as finite pointwise Lips- chitz constant almost e verywhere, see W il drick–Z ¨ urcher [31, Proposit ion 3.4]. Com bined with a Stepanov-type dif ferentiabilit y th e orem [2], this has implication s for differentiability [5]. W e also refer to a recent paper [32]. For the next prop osition, we recall that t he no nce ntered Hardy–Li ttle wood maximal function restricted t o Ω , denoted M Ω , is defined for an integrable ASPECTS OF AREA FORMULAS ON METRIC MEASURE S P ACES 17 (real-v alu ed) function f on Ω by M Ω f ( x ) := sup B Z B ( x,r ) | f | d µ, where th e supremum is taken over all balls B ⊂ Ω containing x . Consider further the restrained noncentered m aximal function M Ω ,R in which the supremum is t ak en only over ball s in Ω with radius less than R . Then M Ω f = sup R> 0 M Ω ,R f . It i s standard also in the metric space sett ing, we refer to Hei nonen [13], that for 1 < p ≤ ∞ the operator M Ω is bounded on L P , i.e., there exists a const ant C , depending on C µ and p , such that for all f ∈ L p kM f k L p ≤ C k f k L p . W e have the following generalization. Pr oposition 5.1. Suppose X satisfies condition s (D) and (PI) with (I) p = Q . If u ∈ N 1 ,Q lo c ( X Q ; V ) is continuous and pseudomonotone, then u sati sfies condition (RR), and t hus is (locally) Q -absolutely continuous. (II) some p > Q . Then u ∈ N 1 ,p lo c ( X Q ; V ) s atisfies conditio n (RR) weakly , and thus is (locally) Q -absolu tely continuous . Pr oof. Let Ω ⋐ X Q be open, and fix x ∈ Ω . (I) : Let B ( x, r x ) , 0 < r x < min { r D , r M } , be a ball such t hat B ( x, 12 τ r x ) ⊂ Ω ; τ ≥ 1 is the dilatation constant appearing in th e P o incar ´ e inequality . By a Sobole v embedding theorem Hajłasz–K oskela [11, Theorem 7.1] there exists a constant C , depending on C µ and the constants in the weak (1 , Q ) - Poincar ´ e in e quality , a nd a radius r x < r < 2 r x such that (5.2) k u ( z ) − u ( y ) k p V ≤ C d ( z, y ) p/Q r p (1 − 1 /Q ) x Z B ( x, 5 τ r x ) g p u dµ for each z , y ∈ Ω wi th d ( y , x ) = r = d ( z , x ) , where p ∈ ( Q − ε 0 , Q ) . In fact, [11, Theorem 7.1] is stated and proved only for real-valued functions, but the argument is valid also when the tar get is a Banach space as we may make use of t he Lebesgue differentation theorem for Banach s pac e valued maps as in [14, Propositi on 2.10]. Since u is ps e udomonoto ne we obtain from (5.2) diam( u ( B ( x, r x ))) p ≤ C p M diam u ( ∂ B ( x, r ))) p ≤ C r p x Z B ( x, 5 τ r x ) g p u dµ, where C depends on C µ , C M , and t he const a nts in the weak (1 , Q ) -Poincar ´ e inequality . For each y ∈ B ( x , r x ) we hav e Z B ( x, 5 τ r x ) g p u dµ ≤ Z B ( y, 10 τ r x ) g p u dµ ≤ M Ω , 12 τ r x g p u ( y ) . 18 NIKO MAR OLA AND WILLIAM P . ZIEMER Compining the p r eceding two est imates and i nte grati ng over y ∈ B ( x, r x ) we get diam( u ( B ( x, r x ))) p ≤ C r p x Z B ( x,r x ) M Ω , 12 τ r x g p u dµ. Recall that Q − ε 0 < p < Q ; we get diam( u ( B ( x, r x ))) p ≤ C r p x µ ( B ( x, r x )) − p/Q  Z B ( x,r x ) ( M Ω , 12 τ r x g p u ) Q/p dµ  p/Q ≤ C r p x µ ( B ( x, r x )) − p/Q  Z B ( x,r x ) g Q u dµ  p/Q , which implies together with (2.3) that diam( u ( B ( x, r x ))) Q ≤ C ˜ C Z B ( x,r x ) g Q u dµ, where C depends on C µ , C M , and t he const a nts in the weak (1 , Q ) -Poincar ´ e inequality , and ˜ C is from (2.3). As g Q u ∈ L 1 lo c ( X ) this verifies the fact that u satisfies condition (RR), and thus is locally Q -absolutely continuo us. (II) : Let B ( x, r x ) , 0 < r x < r D , be a ball such that B ( x, 5 τ r x ) ⊂ Ω . Theorem 5.1 (3) i n Hajł asz–K oskela [11, Theorem 5.1] implies that there exist a constant C , depending on C µ , p , and t he constant s appearing in the weak (1 , p ) -P o incar ´ e inequality , such that k u ( z ) − u ( y ) k V ≤ C d ( z, y ) 1 − Q/p r Q/p x  Z B ( x, 5 τ r x ) g p u dµ  1 /p for all z , y ∈ B ( x, r x ) . In fact, [11, Theorem 5.1] i s stated and prov ed only for real-valued functions, but the argument is valied also when the target is a Banach space. Y oung’ s inequality ab ≤ a p /p + b p ′ /p ′ and (2.3) imply diam( u ( B ( x, r x ))) Q ≤ C r Q x µ ( B ( x, r x )) Q/p  Z B ( x, 5 τ r x ) g p u dµ  Q/p ≤ C  ˜ C − 1 µ ( B ( x, r x )) + Z B ( x, 5 τ r x ) g p u dµ  ≤ C  Z B ( x,αr x )  ˜ C − 1 + g p u  dµ  . Hence u satisfies condition (RR) weakly with α = 5 τ and with Φ u = C ( ˜ C − 1 + g p u ) , ˜ C is from (2.3).  ASPECTS OF AREA FORMULAS ON METRIC MEASURE S P ACES 19 The fact th a t a c ontinuous pseudomon otone fun c tion u ∈ N 1 ,Q lo c ( X Q ; V ) verifies Luzin’ s condit ion ( N Q ) would easily follow also from Proposi- tion 5.1 (I). R E F E R E N C E S [1] A M B RO S I O , L . and K I R C H H E I M , B . , Rectifiable sets in m etric and Banach spaces, Math. Ann. 318 (200 0), 527–5 55 . [2] B A L O G H , Z . 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