represented as the dierence of two concave functions. Denote by DC 1 the class of DC-functions on M which are C 1 on the nonsingular set of M. Since t

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1 SOBOLEV SPACES, LAPLACIAN, AND HEAT KERNEL ON ALEXANDROV SPACES KAUHIRO KUWAE, YOSHIROH MACHIGASHIRA, AND TAKASHI SHIOYA Abstract. We prove the compactness of the imbedding of the Sobolev space W 1;2 () into L 2 () for any relatively compact open subset of an Alexandrov space. As a corollary, the generator induced from the Dirichlet (energy) form has discrete spectrum. The generator can be approximated by the Laplacian induced from the DC-structure on the Alexandrov space. We also prove the existence of the locally Holder continuous Dirichlet heat kernel. 1. Introduction Consider a family M of n-dimensional closed Riemannian manifolds with a uniform lower bound of sectional curvature and a uniform upper bound of diameter for a xed n 2 N. In order to investigate various properties of manifolds in M, it is very useful to study its closure M with respect to the Gromov-Hausdor distance d GH, which is compact by the Gromov compactness theorem [15]. Since the closure M consists of Alexandrov spaces introduced in [2], the study of Alexandrov spaces is nowadays an important topic in Riemannian geometry. This paper is the rst study on the Sobolev spaces, the Laplacian, and the heat equation on Alexandrov spaces. According to an earlier work [25] due to Otsu and the third named author, there is a natural C -Riemannian (and C 1 -dierentiable) structure on the set of nonsingular points of an Alexandrov space M, which enables us to de- ne L p - and W 1;p -functions on M, p 1, and their norms kk L p and kk p W 1;p := kk p L p + krk p L p. We here remark that the nonsingular set of M is not necessarily a Riemannian manifold (in fact, the singular set may be dense in M), and also that M is not a topological manifold in general. Perelman [27] proved that the dierentiable structure stated above extends to a DC-structure on M n S for small >, where S denotes the -singular set of M (see for the denition x2.4), and DC-structure means that any component of coordinate transformations is DC, i.e., Date: March 2, Mathematics Subject Classication. Primary 53C7, 58G11, 58G25; Secondary 31C15, 31C25, 35K5, 53C2, 53C23. Key words and phrases. Alexandrov space, Sobolev spaces, Dirichlet form, Laplacian, DC-structure, heat equation, heat kernel. 1

2 represented as the dierence of two concave functions. Denote by DC 1 the class of DC-functions on M which are C 1 on the nonsingular set of M. Since the second partial derivatives of DC-functions are determined as signed Radon measures, we can do some tensor calculus on M n S. In particular, we can dene the Laplacian, say the DC-Laplacian DC u of any DC 1 -function u on M n S as a signed Radon measure (see x5 for the precise denition). Now that there are no smooth functions on M, we consider the set of DC 1 -functions as a core of the L 2 -space. To state our main theorems, we need some denitions. Let M be an n-dimensional Alexandrov space, n 2, and > a xed number with 1=n. (Note that one-dimensional Alexandrov spaces are Riemannian manifolds.) If M has no boundary, we set M := M n S. If M has nonempty boundary, refer to x2.4 for the denition of M. Let M be an open subset and := \ M, where we note that can be taken to be M itself. Denote by L p () the set of L p - functions on and by W 1;p () the set of W 1;p -functions on. Note that L p () and W 1;p () are complete normed spaces with respect to kk L p and kk W p respectively. Let DC( 1 ) be the set of DC 1 -functions with compact support in, and W 1;p () the W 1;p -closure of the set of W 1;p -functions with compact support in. One of our main theorems is stated as follows. Theorem 1.1. The two imbeddings are both dense. DC 1 ( ) W 1;2 () and DC 1 ( ) L p (); p 1; This theorem implies that the imbedding W 1;2 () L p (), p 2, is dense. The same idea of the proof of Theorem 1.1 shows that the 1-capacity of S is zero (Theorem 3.1). See x2.8 for the denition of the 1-capacity. Denote by H 1;p () the W 1;p -closure of the set of DC 1 -functions with compact support in. Then, Theorem 1.1 implies W 1;2 () = H 1;2 (). The following theorem plays an important role in analysis on Alexandrov spaces. Theorem 1.2. Assume that is relatively compact. Then, the imbedding H 1;p () L p () for any p 1 is compact. In particular, the imbedding W 1;2 () L 2 () is compact. Dene a (canonical) Dirichlet form E : W 1;2 (M) W 1;2 (M)! R of M by E(u; v) := M hru; rvi dh n for u; v 2 W 1;2 (M), where h; i denotes the inner product induced from the Riemannian structure on M and H k the k-dimensional Hausdor measure, k. Note here that the volume measure induced from the Riemannian 2

3 structure coincides with H n (see x2.7). By (E; F) we mean to restrict the domain of E to be F W 1;2 (M). Note that (E; W 1;2 ()) is a strongly local regular Dirichlet form (Proposition 4.1). Let : W 1;2 () D( )! L 2 () denote the generator induced from (E; W 1;2 ()), which is a densely de- ned selfadjoint operator satisfying E(u; v) = ( u; v) L 2. A standard discussion using Theorem 1.2 (see for instance III.21 of [3]) leads to the following: Corollary 1.1. Assume that is relatively compact. Then, the generator has discrete spectrum consisting of nonnegative eigenvalues 1 < 2 < % 1. Every eigenspace E i has at most nite multiplicity L and is L 2 -orthogonal to each other. The total eigenspace 1 E i=1 i is dense in W 1;2 () and in L 2 (). We obtain that 1 = if and only if the 1-capacity of M n is zero (Theorem 7.3). Remark that this assertion is never trivial; in fact we need to establish the Dirichlet heat kernel, Theorem 1.5 below, to prove that any constant function in W 1;2 () is zero provided the 1-capacity of M n is nonzero. We also obtain that if M is compact, the second eigenvalue of M is greater than some positive constant depending only on the dimension, a lower bound of curvature, an upper bound of the diameter, and a lower bound of H n -volume of M (Corollary 7.1). Let us next consider the relation between the generator and the DC-Laplacian DC. Remark that we do not know if D( ) and DC( 1 ) have a dense intersection in W 1;2 (), though they are both dense in W 1;2 (). Theorem 1.3. For a given u 2 D( ), if a sequence u i 2 DC 1 ( ) converges to u in the W 1;2 -topology, then the DC-Laplacian DC u i of u i converges to u dh n in the vague topology, provided = ;. By Theorem 1.1, the above u i always exists. Remark that, if 6= ;, this theorem does not hold even in the case where M is a Riemannian manifold with smooth boundary. Sturm [31, 33, 34] (cf. [32]) proved that on a locally compact separable intrinsic metric space with a strongly local regular Dirichlet form, the parabolic Harnack inequality and the existence of the Dirichlet heat kernel are both derived from the three following conditions: The distance function induced from the Dirichlet form coincides with the original one. The doubling property holds. The weak Poincare inequality holds. We prove all of them for Alexandrov spaces (Theorems 7.1 and 7.2). Besides, we extend Sturm's work to the case where the base space can 3

4 be replaced with any open subset in the full generality (Theorem 8.2 and Corollary 8.2). As a result, we have the following theorems. Theorem 1.4. (1) Any local solution u of the heat equation ( ) u = is locally Holder continuous on (2) Any eigenfunction of is locally Holder continuous on. Here, we see x8 for the meaning of local solution of the heat equation. The following theorem shows the existence of the Dirichlet heat kernel. Theorem 1.5. For any open subset M, there exists a unique, measurable, and locally Holder continuous function ( ; 1 ) 3 (t; x; y) 7?! p t (x; y) 2 [ ; 1 ) satisfying the following (1){(3). (2a) (2b) (2c) (2d) (1) For any t >, u 2 L 2 (), and x 2, we have e?t u(x) = y2 p t (x; y) u(y) dh n : (2) For any s; t > and x; y 2, we have p s+t(x; y) = p t (x; y) = p t (y; x); y2 z2 p s (x; z) p t (z; y) dh n ; p t (x; y) dh n 1; p t (x; y) > if and only if x and y are contained in a common connected component of. (3) Assume that is relatively compact. Denote by 1 2 : : : all the eigenvalues of with multiplicity and by fu i g 1 i=1 the sequence of associated eigenfunctions which is a complete orthonormal basis of L 2 (). Then we have p t (x; y) = 1X i=1 e? it u i (x)u i (y) for any t > and x; y 2, where the convergence is uniform on any compact subset of ( ; 1 ). Remark 1.1. (1) For Lipschitz manifolds, all the above theorems are already obtained. Refer to [19, 6, 31, 21]. Our results are essentially dierent from them because of the existence of singularity where the space is not locally Euclidean topologically. For instance, the quotient of a closed Riemannian manifold by a (not necessarily discrete nor free) isometric group action is an Alexandrov space. 4

5 (2) Sturm [34] constructed a strongly local regular Dirichlet form for metric spaces with some condition, called the strong Measure Contraction Property (strong MCP for short) with an exceptional closed subset, and he proved that the parabolic Harnack inequality and the existence of the heat kernel both hold on out of the exceptional set. We observe that Alexandrov spaces do not necessarily admit the strong MCP, even with any exceptional set of measure zero. It is not dicult to show that for Alexandrov spaces, his characterization of Dirichlet form remains valid and coincides with the canonical Dirichlet form induced from the Riemannian metric. (3) For a xed n 2 N, let M be a family of n-dimensional closed Riemannian manifolds with uniformly bounded sectional curvature and diameter, and M its Gromov-Hausdor closure. Fukaya [11] dened a selfadjoint operator P M on L 2 (M) for any M 2 M which coincides with the Laplacian for M 2 M, and proved that the spectrum of P M is continuous in M 2 M. Notice that all M 2 M are Alexandrov spaces. If a space M 2 M is not collapsed (i.e., has dimension n), the operator P M coincides with the generator M. We conjecture the above result to be true even if M is a family of Alexandrov spaces with uniform lower bound of curvature, upper bound of diameter, and upper bound of dimension. Theorems 1.1{1.3 are proved by Machigashira and Shioya, and Theorems 1.4 and 1.5 are proved by Kuwae, Machigashira, and Shioya. Acknowledgment. The authors would like to thank Prof. Y. Otsu for stimulating discussions. 2. Preliminaries 2.1. Basics for Alexandrov spaces. In this subsection, we dene some notations and describe basic facts for Alexandrov space established mainly in [5]. A metric space is called an intrinsic metric space if the distance between any two points is realized as the inmum of the lengths of continuous curves joining them. In this case, if the space is complete and locally compact, there is a curve of minimal length, called a minimal geodesic, joining the two points. Let M be a complete locally compact intrinsic metric space with distance function d. For x; y 2 M, let xy : [ ; 1 ]! M denote a minimal geodesic joining x to y with parameter proportional to the arc-length. Note that xy is not necessarily unique for xed x; y 2 M. We mean a triangle 4x 1 x 2 x 3, x i 2 M, to be the triple of three minimal geodesics x1 x 2, x2 x 3, and x3 x 1. We say that M is of curvature bounded below by 2 R if the following holds: 5

6 Alexandrov convexity: For any triangle 4x 1 x 2 x 3 in M, there exists a triangle 4x ~ 1 x 2 x 3 = 4~x 1 ~x 2 ~x 3, called a comparison triangle, in a simply connected space form of constant curvature such that d(x i ; x j ) = d(~x i ; ~x j ) for i; j = 1; 2; 3 and that d( x1 x 2 (s); x1 x 3 (t)) d( ~x1 ~x 2 (s); ~x1 ~x 3 (t)) for any s; t 2 [ ; 1 ]. An Alexandrov space is dened to be a complete locally compact intrinsic metric space of curvature bounded below and of nite Hausdor dimension. Assume from now on that M is an Alexandrov space of curvature and n its Hausdor dimension. Then, n is an integer and coincides with the Hausdor dimension of any open subsets of M. For a triangle 4x 1 x 2 x 3 in M, we denote by \x 1 x 2 x 3 the angle between x2 x 1 and x2 x 3 at x 2 and set ~ \x 1 x 2 x 3 := \~x 1 ~x 2 ~x 3 for its comparison triangle ~ 4x 1 x 2 x 3 = 4~x 1 ~x 2 ~x 3. We have \x 1 x 2 x 3 ~ \x 1 x 2 x 3 ; which is analogous to the Toponogov comparison theorem. Recall that the space of directions x at any x 2 M is an (n? 1)- dimensional Alexandrov space of curvature 1 and the tangent cone K x at any x 2 M an n-dimensional Alexandrov space of curvature. Since K x is the Euclidean cone K( x ) over x, there is a natural identication K x ' x [ ; +1 )=( x ); we indicate by t 2 K x the corresponding element of (; t) 2 x [ ; +1 )=( x ). There is a 1-expanding map from x to the standard unit (n? 1)-sphere S n?1, where a map f : (X; d X )! (Y; d Y ) between two metric spaces is said to be L-expanding, L >, if d Y (f(x); f(y)) L d X (x; y) for any x; y 2 X. For Alexandrov spaces, the Bishop-Gromov inequality holds as well as for Riemannian manifolds, i.e., for any R > r > and x 2 M we have H n (B(x; r)) H n (B(x; R)) b n;(r) b n; (R) (see Proposition A.4 of [36]), where B(x; r) denotes the open metric r-ball centered at x and b n; (r) the volume of an r-ball in the n- dimensional simply connected space form of curvature Some general notations. Denote by `const ;;::: ' the symbol expressing some nonnegative constant depending only on ; ; : : :. In particular, `const' means some universal nonnegative constant. Let O ;;::: (); ;;::: () be the symbols which express some functions depending only on ; ; : : : such that lim sup t! jo ;;::: (t)=tj < +1 and lim t! ;;::: (t) = respectively. For x; y 2 R we dene x _ y := maxfx; yg, x ^ y := minfx; yg, and (x) + := x _. 6

7 For a subset A of a topological space, we denote by A and A the interior and the closure of A respectively Cut locus and the exponential map. Let C x for x 2 M be the cut locus to x, i.e., the set of points y 2 M such that minimal geodesics xy do not extend beyond y. Then, C x for any x 2 M is of H n -measure zero (see Proposition 3.1 in [25]). We note that for any y 2 M n C x, the minimal geodesic xy is unique, because geodesics in M do not branch. Dene a map exp?1 x : M n C x! K x for x 2 M by exp?1 x y := d(x; y) y x for y 2 M n C x, where yx 2 x is the direction of xy. The inverse map exp x : U x := exp?1 x (M n C x )! M of exp?1 is called the exponential map at x. The Alexandrov convexity implies that exp?1 x j B(x;r)nCx for any r > is a (1? (r))-expanding map, or equivalently exp x : B(o x ; r)\u x! M is a Lipschitz map with Lipschitz constant (1 + (r)), where o x denotes the vertex of K x Strainer and singularity. We say that a point x 2 M is (m; )- strained, m 2 N, >, if there is a sequence fp i g i=1;:::;m, called a strainer at x, such that for any i; j, ~\p i xp j > ( =2? if i 6= j,? if i =?j. We here call min i d(x; p i ) the length of the strainer fp i g at x. Denote by S m; the set of non-(m; )-strained points in M. It is a closed subset of Hausdor dimension m?1 (1.6 of [5]). More strongly, it is shown in the proof of 1.6 of [5] that any compact subset C S m; has an -net (i.e., a discrete subset of C whose -neighborhood contains C) for any > of at most O C; ( 1?m ) elements. The -singular set S of M is dened to be S n;, and each point in S is called a -singular point. Assume that M has nonempty boundary (see [5] for the denition of the boundary). Then, the doubling theorem (x5 of [26]; 13.2 of [5]) states that the gluing of two copies of M along their boundaries, called the double dbl(m) of M, is an Alexandrov space without boundary. Consider M as a subset of dbl(m) and denote by ^S the set of - singular points of dbl(m) contained in M. When M has no boundary, put ^S := S. In either case, we dene M := M n ^S. Note that S ^S S n?1;. A point x 2 M is said to be singular if K x is not isometric to R n (or equivalently x is not isometric to S n?1 ). We denote by S M the set of singular points of M and call it the singular set of M. It follows that S M = S > S, so that the Hausdor dimension of S M is n? 1 ([25, 5]). Remark that the singular set S M is possibly dense in M (see x of [25]). 7

8 We x a with < 1=n. Then, if a point x 2 X is an (n; )- strained point with a strainer fp i g i=1;:::;n of length `, then the two maps ' := (d(p i ; )) i=1;:::;n and ~' := 1 H n (B(p i ; r=`)) y2b(p i ;r=`) d(y; ) dh n i=1;:::;n on B(x; r) for small r > are both ( n ()+ n; (r=`))-almost isometries ([5, 25]), where a map f : (X; d X )! (Y; d Y ) between metric spaces is said to be -almost isometric (or a -almost isometry) if it is a bi- Lipschitz homeomorphism satisfying d Y (f(x); f(y))? 1 d X (x; y) In particular, M n S is a Lipschitz manifold. < for any x; y 2 X DC-functions. Let U R n be any open subset. A Lipschitz function f : U! R is said to be DC if there are two concave functions g; h : U! R such that f = g? h. Denote by BV the class of (not necessarily continuous) bounded functions f : U! R of bounded variation (cf. [37]). It is well known that the rst i of any DC-(resp. BV -) function f on U is a.e. determined as BV - functions (resp. determined as signed Radon measures), and that 2 f second j of any DC-function f on U are determined as signed Radon measures and 2 = i : Also, a classical theorem due to Alexandrov [1] states that any DCfunction is a.e. twice dierentiable in the sense of Stolz, i.e. the second Taylor expansion converges a.e. A Lipschitz map f = (f 1 ; : : : ; f m ) : U! R m is said to be DC if each f i is a DC-function. If f : U! V and g : V! R` are DC-maps, where V R m is an open subset, then g f is DC. Let f be a Lipschitz function (locally) dened on an Alexandrov space M. The function f is said to be concave if the restriction of the function on any geodesic is concave as a function on an interval of R. We say that f is DC if it is represented as the dierence of two concave functions Structures on topological spaces. We formulate some general concept of structure on topological spaces F-structure. Let us rst present classes of maps between subsets of R n. For an integer n, we consider a family F = f F(U; A) j U R n is an open subset and A U a subset g such that 8

9 (i) each F(U; A) is a class of maps from U to R n, (ii) if A B, then F(U; A) F(U; B), (iii) if f 2 F(U; A), g 2 F(V ; B), and f(u) V, then g f 2 F(U; A \ f?1 (B)): The following are examples of F = ff(u; A)g. Class C 1 : Let C 1 (U; A) be the class of maps from U to R n which are C 1 on A, i.e., they are dierentiable on A and their derivatives are continuous on A. Class DC: Let DC(U; A) be the class of maps from U to R n which are DC on some open subset O R n with A O U. Let X be a paracompact Hausdor space, Y X a subset, and F as above. We call a pair (U; ') a local chart of X if U is an open subset of X and if ' is a homeomorphism from U to an open subset of R n. A family A = f(u; ')g of local charts of X is called an F-atlas on Y X if the following (i) and (ii) hold: (i) We have Y S (U;')2A U. (ii) If two local charts (U; '); (V; ) 2 A satisfy U \ V 6= ;, then '?1 2 F('(U \ V ); '(U \ V \ Y )): Two F-atlases A and A on Y X are said to be equivalent if A [ A is also an F-atlas on Y X. We call each equivalent class of F-atlases on Y X an F-structure on Y X. Assume Y = X. Then, an F-structure on Y X is simply called an F-structure on X. If there is an F-structure on X, it is a topological manifold. We call a space equipped with an F-structure an F-manifold. Notice that F-manifolds for F := C 1 are nothing more than C 1 -dierentiable manifolds in the usual sense C 1 -structure. Assume now that a C 1 -structure on Y X is given. Consider triples (U; '; ), where (U; ')'s are local charts with respect to the C 1 -structure and 's are vectors in R n. Two such (U; '; ) and (V; ; ) are said to be equivalent at a point x 2 Y if x 2 U \ V and if nx i = j for any j = 1; : : : ; n, where (x 1 ; : : : ; x n ) := ', (y 1 ; : : : ; y n ) :=, ( 1 ; : : : ; n ) :=, and ( 1 ; : : : ; n ) :=. The tangent space T x X at a point x 2 Y is de- ned to be the set of all the equivalence classes [(U; '; )] x at x. The tangent spaces have the structure of n-dimensional linear space over R. We dene the topology on the tangent bundle T X := `x2y T xx in the usual manner. The notion of the dierentiability of a function f : X! R at a point x 2 Y and its derivative df x : T x X! R at x 2 Y 9

10 are dened in the same way as for dierentiable manifold. The tangent bundle T X (and the cotangent bundle T X) induces the notion of (continuous) vector and tensor elds and forms on Y (compare x5) Structures on Alexandrov spaces. We again consider an n- dimensional Alexandrov space M. Let > be a xed number with 1=n. Recall that around any x 2 M n S we have a local chart (U; ~'), U := B(x; r), as in x2.4. The family of all the (U; ~')'s on M induces: a C 1 -structure on M n S M M ([25]), a DC-structure on M n S M ([27]). (Remark here that for each local chart (U; ~') as above, there is an ane isomorphism T on R n such that T ~' is regular on U in the sense of [27].) In particular, M n S is a DC-manifold. We have the structure which is DC on M n S and C 1 on M n S M. Call such the structure the DC 1 -structure with singular set S M (cf. x5). We also mean by DC 1 the class of DC-functions on M which is C 1 on M n S M. Remark that for any DC-local chart (U; ') of M, a function f : U! R is DC if and only if f '?1 : '(U)! R is DC (see x3 of [27]). In [25], a natural C -Riemannian metric g on M n S M is introduced. We have the following: The distance function on M n S M induced from g coincides with the original one of M ([25]). There is a natural identication (or isometry) between the tangent cone K x and the tangent space T x M for any x 2 M n S M (Lemma 3.6(1) of [25]). The integral R MnS M f! M of any Borel measurable function f on M with respect to the volume form! M induced from g coincides with the integral R M f dhn with respect to the n-dimensional Hausdor measure H n (x7.1 of [25]). The metric g is BV (x4.2 of [27]). Associated with the Riemannian metric g, we can dene the gradient ru on M n S M in the pointwise sense and also in the weak sense as a L 1 loc-vector eld Basics for abstract Dirichlet form. Let X be a locally compact separable metric space and m a positive Radon measure with full topological support, namely m(g) = implies G = ; for any open set G of X. Denote by L 2 (X; m) the totality of m-measurable functions on X which is square integrable with respect to R m. L 2 (X; m) is a real Hilbert space with inner product (u; v) m := uv dm; u; v 2 X L 2 (X; m). We consider a symmetric nonnegative denite bilinear form E with densely dened domain F on L 2 (X; m). For > we set E (u; v) := E(u; v) + (u; v) m ; u; v 2 F. The pair (E; F) is said to be a Dirichlet form on L 2 (X; m) if F is complete with respect to 1

11 E 1=2 1 -norm and if for any u 2 F we have u ] := _ u ^ 1 2 F and E(u ] ; u ] ) E(u; u). Let C (X) be the totality of continuous functions on X with compact support. A Dirichlet form (E; F) on L 2 (X; m) is said to be regular if F \C (X) is dense in F with respect to E 1=2 1 -norm and dense in C (X) with respect to the uniform norm. A subset C of F \C (X) is said to be a special standard core if (C.1) C is dense in F with respect to E 1=2 1 -norm and C is a dense subalgebra of C (X), (C.2) for any compact set K X and relatively compact open set G X with K G, the set C contains a nonnegative element u with u = 1 on K and u = on X n G, (C.3) for any " > there exists a smooth function " : R! [?"; "+1 ] such that " (t) = t for any t 2 [ ; 1 ], " (t)? " (s) t? s for any s < t, and that " u 2 C for any u 2 C. A Dirichlet form (E; F) on L 2 (X; m) is said to be strongly local if E(u; v) = for any u; v 2 F such that u = const m-a.e. on a neighborhood of supp[jvjm], the support of the measure jvjm. The 1-capacity Cap() over X with respect to a Dirichlet form (E; F) is dened as follows: for any open subset G of X, Cap(G) := inff E 1 (u; u) j u 2 F; u 1 m-a.e. on G g; and for any subset B of X, Cap(B) := inff Cap(G) j B G and G is open g: It follows that Cap(B) m(b) for any m-measurable B X. We know (see Lemma 2.2.7(ii) of [12]) that for any compact subset K of X, Cap(K) = inff E 1 (u; u) j u 2 C, u 1 on K g; where C is a special standard core. A statement P (x) with respect to x 2 X is said to hold quasi everywhere, q.e. in short, if the set f x 2 X j P (x) fails g is of zero 1-capacity. We refer to [12] for more details of the theory of Dirichlet form. 3. Construction of Cut-off Functions In this section, we shall prove Theorem 1.1. We follow [8, 18] for the primitive idea of the proof, where Chavel-Feldman [8] treated a domain outside a submanifold of codimension two and Li-Tian [18] considered algebraic varieties. Dierent from their cases, we have the diculty to estimate the area of the boundary of the r-neighborhood of a -singular set for all small r >, because of no regularity of the -singular set. Our idea is to consider the r-neighborhood of an -net of a -singular set. Since the number of points in the -net varies for smaller, we need some delicate discussion. 11

12 Throughout this and the subsequent sections, for n 2 N, 2 R let M be an n-dimensional Alexandrov space of curvature. We take a compact subset C M with the property: () for any > there is an -net N of C with at most O C ( 2?n ) points. Notice that any compact subset of ^S for any > possesses this property (see x2.4). Let k 2 N be any xed number. For each i k, we nd a 3?(i+1) -net N i of C with #N i O C (3 (i+1)(n?2) ) and set A i :=B(N i ; 3?i ) n B(N i ; 2 3?(i+1) ); A i :=B(N i ; 2 3?(i+1) ) n B(N i+1 ; 3?(i+1) ): Then, since B(N i ; 2 3?(i+1) ) B(N i+1 ; 3?(i+1) ), all A i and A i are disjoint to each other and S ik (A i [ A i) = B(N k ; 3?k ) n C. Lemma 3.1. For any r > and i k with 2 3?(i+1) r 3?i, we have H n?1 (@B(N i ; r)) O C; (r) and H n (B(N i ; r)) O C; (r 2 ): Proof. It follows from the Alexandrov convexity that H n?1 (@B(x; r)) O (r n?1 ) for any x 2 M and r >. Hence, for any r > and i k with 2 3?(i+1) r 3?i, we have H n?1 (@B(N i ; r)) X x2n i H n?1 (@B(x; r)) #N i O (r n?1 ) O C; (r): The other inequality is proved in the same way. Lemma 3.2. Let u: M! R be a continuous nonnegative function. Then, for any i k we have 3?i u dh n const u dh A i n?1 dr:?(i+1) i ;r) Proof. In this proof, we always assume that minimal geodesics are parameterized by arc-length. Fix a number i k. Denote by? x, x 2 M, the set of minimal geodesics from N i to x (i.e., they have length equal to d(n i ; x)). Taking a geodesic x in? x for each x 2 M, we set ' r (x) := x (r), r >. Since the cut locus at any point in M is of H n -measure zero, ' r is a H n -a.e. continuous map on M. For any r 1 ; r 2 with 2 3?(i+1) r 1 < r 2 3?i, we dene an H n -a.e. continuous map f r1 ;r 2 : A(N i ; r 1 ; r 2 i ; r 1 ) [ r 1 ; r 2 ] by f r1 ;r 2 (x) := (' r1 (x); d(x; N i )) for x 2 A(N i ; r 1 ; r 2 ), 12

13 where A(N i ; r 1 ; r 2 ) := B(N i ; r 2 ) n B(N i ; r 1 ). Since minimal geodesics in M do not branch, it is easily veried that f r1 ;r 2 is injective. Let us now prove: Sublemma 3.1. The restriction f r1 ;r 2 j A(Ni ;r 1 ;r 2 ) is locally L-expanding for some constant L > depending only on 3 2k. Proof. Fix an arbitrary x 2 A(N i ; r 1 ; r 2 ) and put x := d( x (); N i n f x ()g). Let y 2 A(N i ; r 1 ; r 2 ) be any point close enough to x compared with a given 2 ( ; x ). Then, there is a 2? x with sup d((t); y (t)) < : t It follows from d((); y ()) < x that () = y (). If x () 6= y (), then d( x (); ()) = d( x (); y ()) x, and hence, by the Alexandrov convexity, d( x (r 1 ); (r 1 )) c x, where c x > is a constant depending on x but independent of y. Assuming < c x =2, we have d(f r1 ;r 2 (x); f r1 ;r 2 (y)) d( x (r 1 ); y (r 1 )) c x =2: If x () = y (), applying the Alexandrov convexity to 4xy x () yields d(f r1 ;r 2 (x); f r1 ;r 2 (y)) L d(x; y) for some constant L > depending only on 3 2k. This completes the proof. By the uniform continuity of u on a compact subset of M, for any > there is a > such that if a nite sequence 2 3?(i+1) = r 1 < < r m = 3?i satises max j (r j+1? r j ), then for any j, u? u 'rj < on A(Ni ; r j ; r j+1 ): The above sublemma implies that dh n A(N i ;r j ;r j+1 ) (x) L?n dh i ;r j )[ r j ;r j+1 ](y) under the change of variable y = f rj ;r j+1 (x) (where dhx n is the Hausdor measure on a metric space X), so that A i u dh n = X j which tends to 3?i L?n L?n X j i ;r) This completes the proof. A(N i ;r j ;r j+1 i ;r j ) u dh n (u + )(r j+1? r j ) dh n?1 ; (u + ) dh n?1 dr as!. 13

14 Following [8, 18], we dene a function ^ k : [ ; 1 )! [ ; 1 ] by the following: ^ k (r) := := 3?k ; := e??2 ; 8 >< >: 1 if r,? r? 2 if 2 r,? r? 1 if r 2, if r. Consider the function k : M! [ ; 1 ] dened by k (x) := 8 >< >: if x 2 C, ^ k ( 3?i ) if x 2 2 A i, ^ k (d(x; N i )? 3?i ) if x 2 A 6 i, 1 if x 2 M n B(N k ; 3?k ). Clearly, k is DC. Since the derivative ^ k is monotone nondecreasing, if 2 3?(i+1) r 3?i, then ^ k r? 3?i 6 ^ k and hence, by Lemmas 3.1 and 3.2, 1X 3?i kr k k 2 L = ^ 2 k r? 3?i 6 i=k 23?(i+1) ^ k 3 4 r 2 O C; (r) dr; 3 4 r ; 2 H n?1 (@B(N i ; r)) dr which tends to zero as k! 1. Besides, it follows from Lemma 3.1 that k1? k k L p! as k! 1 for any xed p 1. Consequently, the following lemma is obtained: Lemma 3.3. For a relatively compact open subset M, the following (1), (2), and (3) hold. (1) For any bounded function u 2 W 1;2 (), we have lim kr( ku)? ruk k!1 L 2 = : (2) For any p 1 and u 2 L p (), we have (3) In particular, lim k!1 k ku? uk L p = : W 1;2 ( n C) = W 1;2 () and L p ( n C) = L p (): Proof. (1) follows from kr( k u)? ruk L 2 k(1? k )ruk L 2 + kur k k L 2 : (2) and (3) are straight forward consequences. 14

15 Proof of Theorem 1.1. It suces to prove the theorem in the case where is relatively compact. Assume rst that = ;. It is clear that DC( 1 ) W 1;2 (). Recall that the set C := \ ^S has the property () (see x2.4). Let N i be as in the above. Then, M ;k := M n B(N k ; 3?k ) is a DC 1 -manifold with singular set M ;k \S M. Since \M ;k is covered by nitely many DC 1 -local charts, a standard mollier discussion yields W 1;2 ( \ M ;k ) DC 1 ( ) W 1;2 for any k, where A W 1;2 denotes the W 1;2 -closure of a set A of functions. Therefore, by Lemma 3.3(3), W 1;2 () = W 1;2 ( ) = [ k W 1;2 ( \ M ;k ) W 1;2 DC 1 ( ) W 1;2 : If 6= ;, applying the above to the double images dbl( ) dbl() dbl(m) instead of M yields DC 1 (dbl( )) W 1;2 = W 1;2 (dbl()): Remarking that a small perturbation of the double of any u 2 DC 1 ( ) is an element of DC 1 (dbl( )), we have DC 1 ( ) W 1;2 = W 1;2 (). In the same way, we obtain DC 1 ( ) Lp = L p (). The following is important for the potential theory on Alexandrov spaces. Theorem 3.1. The 1-capacity of ^S is zero. In particular, the 1- capacity of S is also zero. Proof. Remark that Proposition 4.1 in the next section implies that the set DC (M) of DC-functions on M with compact support is a special standard core of (E; W 1;2 (M)). Therefore, any compact subset C M with () satises Cap(C) = inff kuk 2 W 1;2 lim k!1 k1? k k 2 W 1;2 = ; j u 2 DC (M); uj C 1 g where k is as above. In particular, the 1-capacity of any compact subset of ^S is zero. By (2.1.6) of [12], this completes the proof. 4. Compactness Result The main purpose of this section is to prove Theorem 1.2. For we need to mollify any function to be a DC 1 -function with uniform Lipschitz constant. Here, a diculty is caused by the existence of singularities where the space is not locally Euclidean. Our idea is to introduce a new concept of mollier which is dened even on such singularities. The uniform estimate of the L p -closeness between functions and their molliers (Theorem 4.1) strongly relies on the Alexandrov convexity. 15

16 Lemma 4.1. Let r > be a number and f a nonnegative function on ( ; r ). Then, for any x 2 M, we have y2b(x;r) f(d(x; y)) dh n (1 + n; (r)) jxj<r f(jxj) dx; where jxj is the Euclidean norm of x 2 R n and dx the Lebesgue measure over R n. Proof. Recall that exp?1 j B(x;r)nCx is (1? n; (r))-expanding and there is a 1-expanding map ' from x to S n?1 R n. Dene ~' : B(x; r)nc x! R n by ~'(y) := d(x; y)'(y x), where y x is the direction at x of xy. Then, ~' is (1? n; (r))-expanding. Changing the variable as x := ~'(y), we have dx (1? n; (r)) dh n (y). This completes the proof. Lemma 4.2. Let x 2 M be a point, r >, and f a nonnegative function on B(x; r). Then, for any t 2 ( ; 1 ), we have y2b(x;r) f( xy (t)) dh n 1 + n;(r) f(z) dh t z2b(x;tr) n : n Proof. Fix any x 2 M, r > and t 2 ( ; 1 ). For y 2 B(x; r) n C x, we put z := z(y) := xy (t). Then, by the Alexandrov convexity, we see that the map y 7! z is (1? (r)) t-expanding and hence dh n (z) (1? n; (r)) t n dh n (y). This completes the proof. Putting f 1 in Lemmas 4.1 and 4.2 leads to: Corollary 4.1. For any x 2 M and < r < R, we have (1) H n (B(x; r)) (1 + n; (r))! n?1 r n ; (2) H n (B(x; R)) H n (B(x; r)) (1 + n;(r)) Rn r ; n where! n?1 is the volume of S n?1. Remark that Corollary 4.1 is also directly obtained by the Alexandrov convexity and the Bishop-Gromov inequality. Let n be a xed smooth nonnegative function on [ ; 1 ) such that n = const n around, n (r) = for r 1, and R n n (jxj)dx = 1: Note that for h >, jxj n R n h y2m 16 dx = h n : For < h < 1, we dene a DC-function ^1 h on M by ^1 h (x) := 1 d(x; y) h n n h dh n :

17 It follows that ^1 h becomes DC 1, because of the same proof of that the function M 3 x 7! R y2b(z;r) d(x; y) dhn is C 1 on M n S M (see Lemma 5.1 of [25]). Lemma 4.1 implies that ^1 h (x) 1 + n; (h). Let M be a relatively compact open subset. Lemma 4.3. We have inf <h<1 x2 ^1 h (x) > : Proof. Let h 1 > h 2 >. Applying Lemma 4.2 for t := h 2 =h 1 and f(z) := n (d(x; z)=h 2 ) yields ^1 h1 (x) = 1 h n 1 = 1 h n 1 y2b(x;h 1 ) t d(x; y) n h 2 dh n d(x; xy(t)) n y2b(x;h 1 )nc x h n;(h 1 ) h n 2 z2b(x;h 2 ) = (1 + n; (h 1 ))^1 h2 (x): n d(x; z) h 2 dh n dh n Thus, it suces to prove inf x2 ^1 h (x) > for a xed h >, which is in fact follows from the continuity of ^1 h (x) in x 2 M. Let us introduce the mollier of functions on M. For a number < h < 1 and a function u 2 L 1 (B(; h)), we dene two DC 1 - functions ^u h and u h on by ^u h (x) := 1 h n y2m u h (x) := ^u h (x)=^1 h (x): d(x; y) n u(y) dh n ; h Lemma 4.4. For any u 2 L 1 (B(; h)) and x; y 2, we have ju h (x)j const n;; h n kuk L 1 (B(;h)) ; (1) juh(x)? uh(y)j const n;; (2) Proof. Set A := A() := inf 1>h> x2 It follows that for any x 2, h n+1 kuk L 1 (B(;h)) d(x; y): ^1 h (x); B := sup n ; and C := sup j nj: (4.4.1) and j^u h (x)j B h n kuk L 1 (B(x;h)) ; ju h (x)j B Ah n kuk L 1 (B(;h)) : 17

18 To prove (2), we rst verify the Lipschitz continuity of ^u h. In fact, for any u 2 L 1 (B(; h)) and x; y 2, we see (4.4.2) j^u h (x)? ^u h (y)j C jd(x; z)? d(y; z)j ju(z)j dh h z2b(x;h)[b(y;h) n n+1 C h d(x; y) ju(z)j dh z2b(x;h)[b(y;h) n : n+1 It follows that ju h (x)? u h (y)j ^1 h (y) j^u h (x)? ^u h (y)j + j^u h (y)j ^1 h (y)? ^1 h (x) ; ^1 h (x) ^1 h (y) which together with Corollary 4.1(1), (4.4.1), and (4.4.2) completes the proof. Note that using the above mollier u h, we can prove that for p 1, H 1;p () := DC 1 () W 1;p is dense in L p (), while it is also derived from Theorem 1.1. The following theorem is essential for the proof of Theorem 1.2. Theorem 4.1. Assume n 2. u 2 H 1;p (B(; h)), we have For any < h 1, p 1, and ku? u h k L p () const n;;p; h kruk L p (B(;h)) : Proof. It suces to prove the theorem for any DC 1 -function u. Let u 2 DC 1 (B(; h)) be a given function. We then dene, even at a singular point x 2 B(; h), the norm of the gradient jru(x)j to be the supremum of the directionally derivatives (u) of u at x with respect to all directions 2 x. Of-course, for x 2 B(; h) n S M, the gradient ru(x) is determined and its norm is compatible with that dened here. For any x 2 and y 2 B(x; h), since u xy is DC, we have and hence ju(x)? u(y)j h ju(x)? u h (x)j B 1 Ah n y2b(x;h) B Ah n? jru( xy (t))j dt ju(x)? u(y)j dh n y2b(x;h) jru( xy (t))j dh n dt;

19 where A and B are as in the proof of Lemma 4.4. inequality, ju(x)? u h (x)j p B p A p h p(n?1) Hn (B(x; h)) p?1 1 const n;;p; h n?p 1 y2b(x;h) y2b(x;h) jru( xy (t))j p dh n dt: By the Holder jru( xy (t))j p dh n dt Thus, it follows from Lemma 4.2 that for any x 2 and p 1, Therefore, ju(x)? u h (x)j p const n;;p; h n?p 1 = const n;;p; h n?p z2b(x;h) const n;;p; h p?1 x2 1 jru(z)j t z2b(x;th) p dh n dt n jru(z)j p 1 z2b(x;h) ju(x)? u h (x)j p dh n const n;;p; h p?1 z2b(;h) jru(z)j p Since Lemma 4.1 implies that x2b(z;h) x2b(z;h) d(x;z) h dh n d(x; z) n?1 (1 + n;(h)) 1 dt dhn tn jru(z)j p 1 d(x; z) n?1 dhn : const n; h for any z 2 M, the proof is completed. 1 d(x; z) n?1 dhn dh n : jxj<h dx jxj n?1 Proof of Theorem 1.2. We prove the theorem in a standard way by using Theorem 4.1 (see, for instance, the proof of Theorem 7.22 of [13]). Let us give here a brief outline of the proof. Take a W 1;p -bounded subset U of H 1;p () and put U h := fu h j u 2 Ug for any h >. Then, Theorem 4.1 implies that U is contained in ch-neighborhood of U h with respect to the L p -norm, where c > is a constant. We apply the Ascoli- Arzela Theorem to U h to prove the precompactness of U h. This shows the precompactness of U. Let M be a (not necessarily relatively compact) open subset. Since for any u 2 C () the mollier u h is a function in DC 1 () for small enough h >, we directly have: 19

20 Lemma 4.5. The imbedding DC 1 () C () is dense with respect to the uniform norm. Using Theorem 1.1 and the above lemma, we can easily prove the following, where the proof is omitted. Proposition 4.1. The set DC 1 () is a special standard core of the Dirichlet form (E; W 1;2 ()). In particular, (E; W 1;2 ()) is a strongly local regular Dirichlet form. 5. Tensor Calculus under DC-Structure In this section, we review the DC tensor calculus introduced by Perelman [27] and give some extensions of it. We present the denition of DC-Laplacian and see that the Green formula holds (Proposition 5.2), which is needed for the proof of Theorem 1.3. A paracompact Hausdor space V is said to be a DC 1 -manifold with a subset S V if V possesses a DC-atlas A on V which is also a C 1 - atlas on V n S V. We say that each (U; ') 2 A is a DC 1 -local chart and S the singular set of V symbolically. Throughout this section, let V be an n-dimensional DC 1 -manifold with singular set S V such that '(U \ S) is of H n?1 -measure zero for any DC 1 -local chart (U; '). Recall that for an n-dimensional Alexandrov space M and for < 1=n, the pair of V := M n S and S := S M satises such the condition. We say that a function f (locally) dened on V is DC 1 if f := f '?1 is everywhere DC and C 1 on '(U n S) for any DC 1 -local chart (U; '). Denote by BV the class of functions f (locally) dened on V such that f is everywhere BV and continuous on '(U n S) for any DC 1 - local chart (U; '). We see that the partial derivatives of DC 1 -functions with respect to local charts are BV -functions. Lemma 5.1 (x4. of [27]). For any f; h 2 BV we have the following. (1) The product fh is BV. (2) If f does not change its sign and is bounded away from zero, then 1=f is BV. (3) The partial derivatives of BV -functions on R n are determined as signed Radon measures ( f h) f h + h ; i where the upper bar means the push forward through a DC 1 -local chart (U; '). Note that to obtain (3) of the lemma, it is necessary that '(U \ S) is of H n?1 -measure zero. A distributional tensor eld on V is dened to be a correction of distributional tensor elds T (U;') on '(U) R n for all DC 1 -local charts 2

21 (U; ') on V which satisfy the compatibility condition under chart transformation: T (U ; ) = ( '?1 ) T (U;') for any two local charts (U; ') and (U ; ) with U \ U 6= ;, where ( '?1 ) is the pull back. Compare [35]. In what follows, we assume all tensor elds to be BV, DC 1, or Radon measures. Looking at the denition Xf := i i of the P derivative of a function f on V with respect to a vector eld X = i, we observe that this is dened if f is DC 1 and i are Radon measures, or if f and X are both BV. We here remark that we cannot multiply Radon measure by Radon measure. We can carefully verify that the Lie bracket, the dierential of form, and the Lie derivative are all dened on V in the usual way. Note that the dierential d! of a BV form! on V is a Radon measured form and the Lie derivative L X! of a BV form! with respect to a BV vector eld X is a Radon measured form. We say that a subset N V is an m-dimensional DC 1 -submanifold of V if for any x 2 N there are a DC 1 -local chart (U; ') of V around x and an m-dimensional plane L R n such that '(N \ U) = L \ U. Clearly, a DC 1 -structure on any DC 1 -submanifold of V is naturally induced. However, the codimension of the singular set of the submanifold may be smaller than the required one to obtain Lemma 5.1(3), and we remark that some tensor calculus does not work on the submanifold. For an (n? 1)-form! on V, the (n? 1)-form! on a DC 1 -hypersurface N (i.e., an (n? 1)-dimensional DC 1 -submanifold) is dened by!(x 1 ; : : : ; X n?1 ) :=!(X 1 ; : : : ; X n?1 ) for vector elds X 1 ; : : : ; X n?1 tangent to N. The Stokes formula is proved in the same way as for C 1 -manifolds. Proposition 5.1 (The Stokes formula). Let D V be an orientable compact subset bounded by a DC 1 -hypersurface. Then, for any BV (n? 1)-form! we have D d! Assume from now on that V has a BV -Riemannian metric g = (g ij ) (i.e., a symmetric positive denite BV (2; )-tensor). Recall ([27]) that the canonical Riemannian metric of any Alexandrov space is BV. The volume form! V := p G dx 1^ ^dx n is dened as a BV n-form, where G := det(g ij ). We can also consider! V to be a nonnegative Radon measure on V, so that it is dened even when V is non-orientable. For Alexandrov spaces,! V is identied with dh n. P We dene the DC-divergence divdc X of a BV -vector eld X = on V by i div DC X := L X! V = X pg i dx 1 ^ ^ dx n i 21

22 which is considered as a signed Radon measure on V as well as the volume form! V. Notice that the ordinary divergence div X of a smooth vector eld X on a Riemannian manifold satises div DC X = (div X)! V. The DC-Laplacian DC u of a DC 1 -function u on V is dened as a signed Radon measure by DC u :=? div DC ru: Let D V be a compact set bounded by a DC 1 -hypersurface. Then, the inward unit normal vector eld is a.e. determined and the BV -Riemannian metric is induced from g. Proposition 5.2 (The Green formula). For any BV -vector eld X on D we have D div DC X hx; ; where h; i denotes the inner product induced from the metric g. Proof. The proof is a quite standard discussion. Assume rst that V is orientable. A direct calculation shows that div DC X = d(i X! V ), where i X is the interior product operator. Moreover, hx; = hx; i (i! V ) = (i X! V ): The Stokes formula leads to the Green formula. If V is non-orientable, the Green formula for the orientable double cover ^D ^V of D V proves the proposition. 6. DC-Approximation of the Generator The main purpose of this section is to prove Theorem 1.3. First we prove the following proposition, which not only is useful for the proof of Theorem 1.3, but also says the signicant claim that the DC 1 -structure on an Alexandrov space extends to a C 1 -dierentiable structure outside a neighborhood of the -singular set. Proposition 6.1. Let M be a relatively compact open subset. For any given two numbers ; > with 1=n, there is a n; ()-almost isometric bi-dc 1 -homeomorphism f ; from a relatively compact open subset ; M n S with n B(S ; ) ; to a C 1 -Riemannian manifold N ;. In particular, every compact n-dimensional Alexandrov space of curvature and without -singularity is n; ()-almost isometric and bi-dc 1 -homeomorphic to a closed C 1 -Riemannian manifold, provided < 1=n. Proof. We prove the theorem by the same idea in [23]. Note that in [23], Otsu treated only Riemannian manifolds of small excess, but the same discussion can be applied to Alexandrov spaces without essential 22

23 changes. We here focus on how to modify the proof in [23] and mention only the outline of the proof. Refer to x5 of [23] for the detailed discussion. For >, the -strain radius at a point x 2 M is dened to be the supremum of such > that there is an (n; )-strainer at x of length at least. It follows that the -strain radius on a compact subset of M n S is bounded away from zero. Denote by ` > its inmum on n B(S ; ). We x an r > with r ; ; `. Take a maximal :3rdiscrete net fp i g of n B(S ; ) and an (n; )-strainer fp i g =1;:::;n at each p i of length at least `. Then, the map d(y; ) dh n f i := from B(p i ; r) to U i 1 H n (B(p i ; r )) y2b(p i ;r ) =1;:::;n := f i (B(p i ; r)) is a n; ()-almost isometric bi- DC 1 -homeomorphism, where < r r. There are isometric maps F ij : U i! R n satisfying that for any i and j, jf ij f i? f j j < n; () r j df ij df i? df j j < n; () on B(p i ; :9r) \ B(p j ; :9r); on H n -a.e. B(p i ; :9r) \ B(p j ; :9r). Since F ij 's do not necessarily satisfy F ki F jk F ij = id, we perturb F ij 's to satisfy this in the following. Let : [ ; 1 )! [ ; 1 ) be a C 1 -function such that = 1 on [ ; 1=2 ], = on [ 1; 1 ), and?4. Set i (x) := (jxj =:8r) for x 2 U i. We set ~F 1j := F 1j and ~F j1 := ~F?1 1j, and dene ~F 2j : U 2! R n for j 2 by ~F 2j (x) := 1 ~ F 21 (x) ~ F 1j ~ F 21 (x) + (1? 1 ~ F 21 (x))f 2j (x) for x 2 U 2. Then, ~F 2j is a dieomorphism between open subsets of R n, and we put ~F j2 := ~F?1 2j. By the denition, ~F 2j = ~F 1j ~F 21 holds on ~F 12 (V 1 ), where V i := f i (B(p i ; :4r)). We can dene other dieomorphisms ~F ij : U i! R n inductively by a similar idea. Let N ; be the quotient space of the disjoint union S i (V i fig) modulo the equivalence relation dened by the following: Two elements (x; i) 2 V i fig and (y; j) 2 V j fjg are said to be equivalent, (x; i) (y; j), if and only if y = ~F ij (x). Denote by ~V i the subset of N ; corresponding to V i fig and by i : ~ V i! V i the projection. Then, N ; is a C 1 -dierentiable manifold with atlas f( ~V i ; i )g i, where ~F ij : V i! V j is the associated coordinate transformation. We can de- ne a C 1 -Riemannian metric on N ; in some natural way. The desired homeomorphism f ; : ; dened to be the inductive limit of f (i), where f (1) (x) := f 1 (x); := S i B(p i; :4r)! N ; is f (2) (x) := 1 f (1) (x) ~F 12 f (1) (x) + (1? 1 f (1) (x)) f 2 (x) : : : 23

24 Remark 6.1. Assume that M is compact and has no -singularity, < 1=n. In the proof above, the map f := f ; and the Riemannian manifold N := N ; both depend on r,, fp i g, etc., so that f and N are not unique. Now, take the map f and the manifold N depending on some other r,, fp ig, etc. Since f f?1 : N! N is an almost isometry, we can deform it to a C 1 -dieomorphism and therefore N is dieomorphic to N. Besides, if M is a Riemannian manifold, then f is a C 1 -dieomorphism. Theorem 1.3 is an immediate consequence of the following: Theorem 6.1. We have M v DC u = E(u; v) for any u 2 DC 1 (M n S ) and v 2 DC 1 (M). Proof. Let u 2 DC 1 (M n S ) and v 2 DC 1 (M) be any xed functions and M a relatively compact open subset containing the support supp[u] of u. Setting := d(supp[u]; S ) >, we apply Proposition 6.1 to. It follows that supp[u] n B(S ; ) ;. Find a C 1 - mollier : [ ; 1 )! N ; of the distance function d(f ; (supp[u]); ) which is close enough to d(f ; (supp[u]); ). By the Sard theorem, we can choose a small regular value r > of such that?1 (r) is a closed C 1 -hypersurface of N ; and that f ; (supp[u])?1 ([ ; r )) f ; (). Then, the set ~ := f?1 ; (?1 ([ ; r ])) contains no -singular points of M and its ~ = f?1 ; (?1 (r)) is a DC 1 -hypersurface of M n S. It is easy to show that v DC u = hru; rvi dh n? div DC (vru): We integrate this on ~ and apply the Green formula (Proposition 5.2) to eliminate the last term. This completes the proof. Remark that Proposition 6.1 is not necessarily needed to obtain Theorem 6.1. In fact, covering supp[u] by nitely many local charts, we can directly nd ~ as above through some tedious discussion. The reason why we present Proposition 6.1 is to clarify the proof and also the signicance of its own. The proposition provides a C 1 -dierentiable structure on any compact Alexandrov space without -singularity, < 1=n. It follows from Remark 6.1 that the dierentiable structure is uniquely determined in the way of the proof of Proposition 6.1, and that if the Alexandrov space is a Riemannian manifold, this dierentiable structure coincides with the original one. The following proposition and its subsequent corollary suggest that the dierentiable structure is quite natural and is considered to be a canonical one. 24

25 Proposition 6.2 (compare [15, 36]). For any xed n 2 N and 2 R, if a sequence fx i g of n-dimensional compact Alexandrov spaces of curvature converges to an n-dimensional compact Alexandrov space X without -singularity, < 1=n, then X i for suciently large i is dieomorphic to X. Proof. It is known that X i for suciently large i is n ()-almost isometric to X (see [36]). This proves the proposition (cf. Remark 6.1). As an immediate consequence of the proposition, we have a dierentiable sphere theorem which generalizes [24]. Corollary 6.1 (compare [24, 5]). There exists an n > depending only on n such that any n-dimensional Alexandrov space of curvature 1 and Hausdor measure! n? n is dieomorphic to S n, where! n is the volume of the unit n-sphere S n. Let W 1;p ;loc and Lp 1;p loc denote the localizations of W and L p respectively. Using Theorem 1.3 we prove the following: Proposition 6.3. There exists a unique linear operator : W 1;2 ;loc (M) D()! L2 loc(m) with the following properties (1){(4). (1) The domain D() is dense in W 1;2 ;loc (M) and in L2 loc (M). (2) We have D() \ W 1;2 () = D( ) and j = D()\W 1;2 () for any open subset M (3) For a given u 2 D(), if a sequence u i 2 DC(M 1 ) converges to u in the W 1;2 loc -topology, then DC u i converges to u dh n in the vague topology. (4) The operator extends to the linear operator G x (D())! G x (L 2 loc(m)) for any x 2 M, where G x (F) is the set of germs around x of functions in a class F. It seems natural to call the above the Laplacian of M. Notice that if M is compact, (2) of Proposition 6.3 implies that = M and D() = D( M ). We need a lemma for the proof of Proposition 6.3. Lemma 6.1. Let U; V M be two relatively compact open subsets. If two functions u 2 D( U ) and v 2 D( V ) coincide on an open subset O U \ V, then U u = V v on O. Proof. By Theorem 1.1, there are two sequences u i 2 DC 1 (U ) and v i 2 DC 1 (V ) respectively tend to u and v in the W 1;2 -topology. Let O be any xed open subset whose closure is contained in O. Since u = v 25