arXiv:1107.3996v1 [math.AP] 20 Jul 2011 Two characterization of BV functions on Carnot groups via the heat semigroup Marco Bramanti∗ Michele Miranda Jr† Diego Pallara‡ July 21, 2011 Abstract In this paper we provide two different characterizations of sets with finite perimeter and functions of bounded variation in Carnot groups, analogous to those which hold in Euclidean spaces, in terms of the short-time behaviour of the heat semigroup. The second one holds under the hypothesis that the reduced boundary of a set of finite perimeter is rectifiable, a result that presently is known in Step 2 Carnot groups. MSC (2000): 22E30, 28A75, 35K05, 49Q15. Keywords: Functions of bounded variation, perimeters, Carnot groups, Heat semigroup. 1 Introduction There are various definitions of variation of a function, and the related classes BV of functions of bounded variation, that make sense in different contexts and are known to be equivalent in wide generality. In the Euclidean framework, i.e., Rn endowed with the Euclidean metric and the Lebesgue measure, the variation of f ∈ L1 (Rn ) can be defined as Z  |Df | (Rn ) = sup f div g dx : g ∈ C01 (Rn , Rn ), kgkL∞ (Rn ) ≤ 1 , (1) Rn and |Df |(Rn ) < +∞ is equivalent to saying that f ∈ BV (Rn ), i.e., the distributional gradient of f is a (Rn -valued) finite Radon measure. This approach can be generalized to ambients where a measure and a coherent differential structure, which allows to define a divergence operator, are defined. Alternatively, the variation of f can be defined through a relaxation procedure,   Z n n 1 n |Df | (R ) = inf lim inf |∇fk |dx : fk ∈ Lip(R ), fk → f in L (R ) , (2) k→∞ Rn and the space BV can be defined, accordingly, as the finiteness domain of the relaxed functional in L1 (Rn ). To generalize (2), no differential structure is needed, apart from ∗ Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy. e-mail: marco.bramanti@polimi.it † Dipartimento di Matematica, Università di Ferrara, via Machiavelli, 35, 44100, Ferrara, Italy. e-mail: michele.miranda@unife.it ‡ Dipartimento di Matematica “Ennio De Giorgi”, Università del Salento, C.P.193, 73100, Lecce, Italy. e-mail: diego.pallara@unisalento.it 1 Lipschitz functions and a suitable substitute of the modulus of the gradient. Definitions (1) and (2) have been extended to several contexts, such as manifolds and metric measure spaces (see e.g. [22], [2], [20]), and in particular Carnot-Carathéodory spaces and Carnot groups, see [10], [6], [17]. However, we point out that the original definition of variation of a function, and in particular of set of finite perimeter, has been given by E. De Giorgi in [13] by a regularization procedure based on the heat kernel. He proved that Z n |Df |(R ) = lim |∇Tt f |dx, (3) t→0 Rn (where ∇ denotes the gradient with respect to the space variables x ∈ Rn and Tt f (x) = R h(t, x − y)f (y)dy is the heat semigroup given by the Gauss-Weierstrass kernel h) and Rn that if the above quantity is finite then f ∈ BV (Rn ). Equality (3) has been recently proved for Riemannian manifolds M in [11], with the only restriction that the Ricci curvature of M is bounded from below, thus generalizing the result in [23], where further bounds on the geometry of M were assumed. In this paper the framework is that of Carnot groups, where it is known that formulae (1) and (2) (where of course the intrinsic differential structure is used) give equivalent definitions of |Df |, see [17]; we show here that (3) holds in a weaker sense, all the quantities involved being again the intrinsic ones. Namely, the two sides of (3) are equivalent, although we don’t know whether they are equal (see Theorem 11 for the precise statement). We think that it is interesting to comment on the proof of (3) in the various situations. Thinking of the left hand side as defined by (1), the ≤ inequality follows easily by lower semicontinuity of the total variation and the L1 convergence of Tt f to f , so that the more difficult part is the ≥ inequality. In Rn it follows from the (trivial) commutation relation Di Tt f = Tt Di f between the heat semigroup and the partial derivatives, whereas in the 2 Riemannian case, as treated in [11], it follows from the estimate kdTt f kL1 (M) ≤ eK t |df |(M ), where d denotes the exterior differential on M , which replaces the gradient in (3), and −K 2 is a lower bound for the Ricci curvature of M . In the subriemannian case we are dealing with, the Euclidean commutation does not hold, and we are not able to prove that equality (3) still holds, but only that the semigroup approach identifies the BV class. The proof relies on algebraic properties and Gaussian estimates on the heat kernel and its derivatives which allow to estimate the commutator [D, Tt ]. Still different arguments are used in the infinite dimensional case of Wiener spaces. Recently, inspired by [21], another connection between the heat semigroup and BV functions in Rn has been pointed out in [24], where the following equality is proved: √ Z Z π |Df |(Rn ) = lim √ h(t, x − y)|f (x) − f (y)| dydx, (4) t→0 2 t Rn Rn which means that f ∈ BV (Rn ) if and only if the right hand side is finite, and equality holds. Equality (4) is first proved for characteristic functions and then extended to the general case using the coarea formula. The proof for characteristic functions of sets of finite perimeter, in turn, is based on a blow-up argument on the points of the reduced boundary and uses the rectifiability of the reduced boundary. In this paper we extend (4) to Carnot groups (see Theorem 14) but, accordingly to the preceding comments, our proof depends upon the rectifiability of the reduced boundary, which is presently known to be true only in groups of Step 2, see [18]. Therefore, our proof would immediately generalize to more general groups, if the rectifiability of the reduced boundary were proved. Notice also that in general the heat kernel doesn’t have the symmetries that the Gaussian kernel has, such 2 as rotation invariance. As a consequence, equation (4) assumes a different form in general Carnot groups of step two (see (29) in Theorem 14 below); however, in the special but important case of groups of Heisenberg type, (29) simplifies to a form very similar to (4) (see Remark 15). We point out that in the Euclidean case both (3) and (4) hold not only in the whole of Rn , but also in a localized form and with the heat semigroup replaced by the semigroup generated by a general uniformly elliptic operator, under suitable boundary conditions, see [4]. Finally, let us mention that the short-time behavior of the heat semigroup in Rn has been shown to be useful also to describe further geometric properties of boundaries, see [3]. Aknowledgments Miranda was partially supported by GNAMPA project “Metodi geometrici per analisi in spazi non Euclidei; spazi metrici doubling, gruppi di Carnot e spazi di Wiener”. The authors would like to thanks L. Capogna, N. Garofalo, G. Leonardi, V. Magnani, R. Monti, R. Serapioni and D. Vittone for many helpful discussions, as well as the anonymous referees for their useful suggestions. 2 Preliminaries and Main Results We collect in the first two subsections the notions on Carnot groups and BV functions needed in the paper. After that, we devote the last subsection to the statement of the main results and some comments. 2.1 Carnot groups and heat kernels Basic definitions. Here we recall some basic known facts about Carnot groups, whose precise definition we shall give after introducing some notation. We refer to [7, § 1.4] for the proofs and further details. Let Rn be equipped with a Lie group structure by the composition law ◦ (which we call translation) such that 0 is the identity and x−1 = −x (i.e., the group inverse is the Euclidean opposite); let Rn be also equipped with a family {D (λ)}λ>0 of group automorphisms of (Rn , ◦) (called dilations) of the following form D (λ) : (x1 , . . . , xn ) 7→ (λω1 x1 , . . . , λωn xn ) where ω1 ≤ ω2 ≤ . . . ≤ ωn are positive integers, with ω1 = ω2 = . . . = ωq = 1, ωi > 1 for i > q for some q < n. If Lx is the left translation operator acting on functions, (Lx f )(y) = f (x◦ y), we say that a differential operator P is left invariant if P (Lx f ) = Lx (P f ) for every smooth function f . Also, we say that a differential operator P is homogeneous of degree δ > 0 if P (f (D(λ)))(x) = λδ (P f )(D(λ)x) for every test function f , λ > 0, x ∈ Rn , and a function f is homogeneous of degree δ ∈ R if f (D(λ)x) = λδ f (x) for every λ > 0, x ∈ Rn . Now, for i = 1, 2, ..., q, let Xi = n X qij (x) ∂xj j=1 3 (5) be the unique left invariant vector field (with respect to ◦) which agrees with ∂xi at the origin, and assume that the Lie algebra generated by X1 , X2 , ..., Xq coincides with the Lie algebra of G. Then we say that G = (Rn , ◦, {D(λ)}λ>0 ) is a (homogeneous) Carnot group or a homogeneous stratified group. More explicitly, setting V1 = span{X1 , X2 , ..., Xq }, Vi+1 = [V1 , Vi ] (the space generated by the commutators [X, Y ], with X ∈ V1 , Y ∈ Vi ), there is k ∈ N such that Rn = V1 ⊕ · · · ⊕ Vk , with Vk 6= {0} and Vi = {0} for i > k. The integer k is called the step of G. Since V1 at 0 can be identified with Rq , by the left invariance of the vector fields Xi we can identify V1 with Rq for every x ∈ Rn . Note that all vector fields in Vj are j-homogeneous and call n k X X ωj jdim Vj = Q= j=1 j=1 the homogeneous dimension of G. The operator L= q X Xj2 j=1 is called sublaplacian; it is left invariant, homogeneous of degree two and, by Hörmander’s Theorem, hypoelliptic. Structure of left- and right-invariant vector fields. It is sometimes useful to consider also the right-invariant vector fields XjR (j = 1, . . . , n), which agree with ∂xj (and therefore with Xj ) at 0; also these XiR are homogeneous of degree one for i ≤ q. (For the following properties of invariant vector fields see [27, pp.606-621] or [8]). As to the structure of left (or right) invariant vector fields, it can be proved that the systems {Xi } and  the XiR have the following “triangular form” with respect to Cartesian derivatives: n X X i = ∂x i + XiR = ∂xi + k=i+1 n X qik (x) ∂xk (6) q ki (x) ∂xk (7) k=i+1 where qik , q ki are polynomials, homogeneous of degree ωk − ωi (the ωi ’s are the dilation exponents). When the triangular form of the Xi ’s is not important, we will keep writing (5), more compactly. The identities (6)-(7) imply that any Cartesian derivative ∂xk can be written as a linear combination of the Xi ’s (and, analogously, of the XjR ’s). In particular, any homogeneous differential operator can be rewritten as a linear combination of left invariant (or, similarly, right invariant) homogeneous vector fields, with polynomial coefficients. The above structure of the Xi ’s also implies that the L2 transpose Xi∗ of Xi is just −Xi (as in a standard integration by parts). From the above equations we also find that Xi = n X cki (x) XkR k=i where cki (x) are polynomials, homogeneous of degree ωk − ωi . In particular, since ωk − ωi < ωk , cki (x) does not depend on xh for h ≥ k and therefore commutes with XkR , that is Xi f = n X XkR cki (x) f k=i 4
(i = 1, . . . , n) for every test function f . The above identity can be sharpened as follows; taking its value at the origin we get: q X cki ∂xk ∂xi = k=i cki (x) since for k > q the are homogeneous polynomials of positive degree, and therefore vanish at the origin. Hence cki = δik for i, k = 1, 2, ..., q, that is: Xi f = XiR f + n X XkR cki (x) f k=q+1
for i = 1, . . . , q. (8) Moreover, the operators XkR , with k > q, can be expressed in terms of the X1R , . . . , XqR , i.e., for every k = q + 1, . . . , n there are constants ϑki1 i2 ...iω such that k X R R R k Xk = ϑi1 i2 ...iω Xi1 Xi2 ...XiRω . (9) k k 1≤ij ≤q Convolutions. The convolution of two functions in G is defined as Z Z g(y −1 ◦ x) f (y) dy, f (x ◦ y −1 ) g(y) dy = (f ∗ g)(x) = Rn Rn for every couple of functions for which the above integrals make sense. From this definition we see that if P is any left invariant differential operator, then P (f ∗ g) = f ∗ P g (provided the integrals converge). Note that, if G is not abelian, we cannot write f ∗ P g = P f ∗ g. Instead, if X and X R are, respectively, a left invariant and right invariant vector field which agree at the origin, the following hold (see [27], p.607)

(Xf ) ∗ g = f ∗ X R g ; X R (f ∗ g) = X R f ∗ g. Explicitly this means Z Rn X R g(y −1 ◦ x) f (y) dy = Z Rn g(y −1 ◦ x) Xf (y) dy (10) for any f, g for which the above integrals make sense. Heat kernels. If G = (Rn , ◦, D(λ)) is a Carnot group, we can naturally define its parabolic version setting, in Rn+1 :
(t, u) ◦P (s, v) = (t + s, u ◦ v) ; DP (λ) (t, u) = λ2 t, D (λ) u .
If we define the parabolic homogeneous group GP = Rn+1 , ◦P , DP (λ) , its homogeneous dimension is Q + 2. Let us now consider the heat operator in GP , H = ∂t − q X j=1 Xj2 = ∂t − L which is translation invariant, homogeneous of degree 2 and hypoelliptic. By a general result due to Folland, see [16], H possesses a homogeneous fundamental solution h (t, x) , usually called the heat kernel on G. The next theorem collects several well-known important facts about h, which are useful in the sequel; the statements (i)-(v) below can be found in [16, Section 1G], while the estimates (11)-(12) are contained in [28, Section IV.4]. 5 Theorem 1 There exists a function h(t, x) defined in Rn+1 with the following properties:
(i) h ∈ C ∞ Rn+1 \ {0} ;
(ii) h λ2 t, D (λ) x = λ−Q h (t, x) for any t > 0, x ∈ Rn , λ > 0; (iii) h (t, x) = 0 for any t < 0, x ∈ Rn ; Z h (t, x) dx = 1 for any t > 0; (iv) Rn
(v) h t, x−1 = h (t, x) for any t > 0, x ∈ Rn . Setting ht (x) = h(t, x), let us introduce the heat semigroup, defined as follows for any f ∈ L1 (Rn ): Z
Wt f (x) = h t, y −1 ◦ x f (y)dy = (f ∗ ht )(x). Rn 1 n Then, for any f ∈ L (R ) and t > 0, we have Wt f ∈ C ∞ (Rn ) and the function u (t, x) = Wt f (x) solves the equation Hu = 0 in (0, ∞) × Rn . Moreover, u (t, x) → f (x) strongly in L1 (Rn ) as t → 0. Finally, for every nonnegative integers j, k, for every x ∈ Rn , t > 0, the following Gaussian bounds hold: 2 c−1 t−Q/2 e−|x| k /c−1 t 2 ≤ h(t, x) ≤ ct−Q/2 e−|x| Xi1 · · · Xij (∂t ) h (t, x) ≤ c(j, k)t /ct (11) −(Q+j+2k)/2 −|x|2 /ct e (12) for i1 , i2 , ..., ij ∈ {1, 2, ..., q} . Here c ≥ 1 is a constant only depending on G, while c(j, k) depends on G, j, k. Remark 2 The Gaussian estimates in this context usually involve the so-called homogeneous norm and are more precise than those we use here, that are stated in terms of the Euclidean norm. We have stated the Gaussian bounds in the present form because we don’t need the estimates in their full strength and (11), (12) follow immediately from the usual estimates In this way we avoid the introduction of the homogeneous norm, which we don’t need. The bound on the derivatives Xi1 · · · Xij ∂tk h (t, x) still holds, in the same form, if the vector fields Xi1 , · · · , Xij are replaced with any family of j vector fields, homogeneous of degree one; for instance, we will apply this bound to derivatives with respect to the right invariant vector fields XiR . Moreover, a homogeneity argument shows that if a (x) is a homogeneous function of degree j ′ , then Xi1 ...Xij ∂tk [a (x) h (t, x)] ≤ c (j, k, a) t−(Q+j−j 2.2 ′ +2k)/2 −|x|2 /ct e . BV functions Let us define the Sobolev spaces W 1,p (G), 1 ≤ p < ∞, and the space BV (G) of functions of bounded variation in G and list their main properties. We remark that the definition of BV (G) goes back to [10] and refer to [1] and to [18] for more information on the Euclidean and the subriemannian case, respectively. We start from the Sobolev case. 6 Definition 3 For 1 ≤ p < ∞, we say that f ∈ Lp (Rn ) belongs to the Sobolev space W 1,p (G) if there are f1 , . . . , fq ∈ Lp (Rn ) such that Z Z g(x)fi (x)dx, i = 1, . . . , q, f (x)Xi g(x)dx = − Rn Rn for all g ∈ C01 (Rn ). In this case, we denote fi as Xi f and set ∇X f = (X1 f, X2 f, ..., Xq f ) . We also let k∇X f kL1 (Rn ) = Z Rn |∇X f | dx = Z Rn We now consider functions of bounded variation. r Xq i=1 2 |Xi f | dx. Definition 4 For f ∈ L1 (Rn ) we say that f ∈ BV (G) if there are finite Radon measures µi such that Z Z f (x)Xi g(x)dx = − g(x)dµi , i = 1, . . . , q, Rn Rn C01 (Rn ). for all g ∈ In this case, we denote µi by Xi f and notice that the total variation of the Rq -valued measure DG f = (X1 f, . . . , Xq f ) is given by Z  |DG f | (Rn ) = sup (13) f (x)divG g(x)dx : g ∈ C01 (Rn , Rq ) , kgk∞ ≤ 1 Rn where divG g(x) = q X Xi gi (x), if g(x) = (g1 (x), . . . , gq (x)) i=1 and kgk∞ = supx∈Rn |g(x)|. With the same proof contained e.g. in [1, Prop. 3.6], it is possible to show that if f belongs to BV (G) then its total variation |DG f | is a finite positive Radon measure and there is a |DG f |-measurable function σf : Rn → Rq such that |σf (x)| = 1 for |DG f |-a.e. x ∈ Rn and Z Z hg, σf id|DG f | (14) f (x)divG g(x)dx = Rn Rn C01 (G, Rq ) q for all g ∈ where, here and in the following, we denote by h·, ·i the usual inner product in R . We denote by DG f the vector measure −σf |DG f |, so that Xi f is the measure (−σf )i |DG f | and the following integration by parts holds true Z Z g (x) d (Xi f ) (x) (15) f (x)Xi g(x)dx = − Rn Rn for all g ∈ C01 (G). Note also that |Xi f | (Rn ) ≤ |DG f | (Rn ) . (16) To visualize the above definitions we note that, whenever f is a smooth function, by the Euclidean divergence theorem we have: σf = − ∇X f ; |∇X f | d |DG f | (x) = |∇X f (x)| dx; 7 (whenever |∇X f | 6= 0) and d (Xi f ) (x) = Xi f (x) dx 1,1 It is clear that W (G) functions are BV (G) functions whose measure gradient is absolutely continuous with respect to Lebesgue measure. Moreover, since it is the supremum of L1 continuous functionals, the total variation is L1 (Rn ) lower semicontinuous (see [18, Theorem 2.17]), i.e., fk → f in L1 (Rn ) implies that |DG f |(Rn ) ≤ lim inf |DG fk |(Rn ). k→∞ (17) Namely: Z Rn fk (x)divG g(x)dx ≤ |DG fk |(Rn ), ∀g ∈ C0∞ (Rn , Rq ) , kgk∞ ≤ 1. Passing to the liminf as k → ∞ we get Z f (x)divG g(x)dx ≤ lim inf |DG fk |(Rn ) k→∞ Rn and taking the supremum over all possible g’s we get (17). Definition 5 (Sets of finite G-perimeter) If χE is the characteristic function of a measurable set E ⊂ Rn and |DG χE | is finite, we say that E is a set of finite G-perimeter and we write PG (E) instead of |DG χE |, PG (E, F ) instead of |DG χE |(F ) for F Borel. Also, we call (generalized inward) G-normal the q-vector νE (x) = −σχE (x) . Recall that |νE (x)| = 1 for PG (E)-a.e. x ∈ Rn . In this case (14) takes the form Z Z hg, νE idPG (E). divG g(x)dx = − (18) Rn E Moreover, notice that if E has finite perimeter, then by the isoperimetric inequality in Carnot groups, see e.g. [10, Theorem 1.18 and Remark 1.19], either E or its complement E c has finite measure. Remark 6 To help the reader to visualize the above definitions, let us specialize them to the case of a bounded smooth domain E, see [18, Prop. 2..22]. Let nE be the Euclidean unit inner normal at ∂E, and consider the q-vector v whose i-th component is defined by vi = n X qij (x) (nE )j (x) j=1 (where the qij are the coefficients of the vector fields Xi , defined in (5)). Then Z Z hg, vidH n−1 (x) divG g(x)dx = − ∂E E from which we read that in this case v , dPG (E) = |v| d H n−1 νE = |v|
∂E (x) (19) at least at those points of the boundary where v 6= 0 (noncharacteristic points). Here H n−1 is the Euclidean (n − 1)-dimensional Hausdorff measure and denotes the restriction of the measure. 8 Identity (10) can be extended to the case g ∈ C 1 (G), f ∈ BV (G) as follows: Z Z −1 R g(y −1 ◦ x) d (Xi f ) (y) Xi g(y ◦ x) f (y) dy = (20) Rn Rn where the last integral is made with respect to the measure defined by Xi f . To see this, it is enough to take a sequence fk ∈ W 1,1 (G) such that fk → f in L1 (Rn ) and Xi fk is weakly∗ convergent to Xi f (the existence of such a sequence is proved in [17]), so that Z Z lim XiR g(y −1 ◦ x) fk (y) dy = lim g(y −1 ◦ x) Xi fk (y) dy k→∞ Rn k→∞ Rn Z g(y −1 ◦ x) d (Xi f ) (y). = Rn Let us come to some finer properties of BV functions that we need only in the proof of Theorem 14. We refer to [17, Theorem 2.3.5] for a proof of the following statement. Proposition 7 (Coarea formula) If f ∈ BV (G) then for a.e. τ ∈ R the set Eτ = {x ∈ Rn : f (x) > τ } has finite G-perimeter and |DG f | (Rn ) = +∞ Z −∞ |DG χEτ |(Rn )dτ. (21) R +∞ Conversely, if f ∈ L1 (Rn ) and −∞ |DG χEτ |(Rn )dτ < +∞ then f ∈ BV (G) and equality (21) holds. Moreover, if g : Rn → R is a Borel function, then Z Rn g(x)d |DG f | (x) = Z +∞ −∞ Z Rn g(x)d|DG χEτ |(x)dτ. (22) Next we have to introduce the reduced boundary. This definition relies on a particular notion of balls in G. We will denote by B∞ (x, r) the d∞ -balls of center x and radius r,
where d∞ (x, y) = d∞ 0, y −1 ◦ x and d∞ (0, x) is the homogeneous norm in G which is considered in [18, Theorem 5.1]. We do not recall its explicit analytical definition since we will never need it. The distance d∞ can be used to define the spherical Hausdorff measures k S∞ through the usual Carathéodory construction, see e.g. [15, Section 2.10.2]. Definition 8 Let E ⊂ Rn be measurable with finite G-perimeter. We say that x ∈ ∂G∗ E (reduced boundary of E) if the following conditions hold: (i) PG (E, B∞ (x, r)) > 0 for all r > 0; (ii) the following limit exists lim r→0 and equals νE (x). DG χE (B∞ (x, r)) |DG χE |(B∞ (x, r)) (iii) the equality |νE (x)| = 1 holds. In order to state the next result, let us introduce some notation. For E ⊂ Rn , x0 ∈ Rn , r > 0 we consider the translated and dilated set Er,x0 defined as Er,x0 = {x ∈ Rn : x0 ◦ D(r)x ∈ E} = D(r−1 )(x−1 0 ◦ E). 9 For any vector ν ∈ Rq we define the sets SG+ (ν) = {x ∈ Rn : hπx, νi ≥ 0} , TG (ν) = {x ∈ Rn : hπx, νi = 0} where π : Rn → Rq is the projection which reads the first q components. Hence the set TG (ν) is an Euclidean ((n − 1)-dimensional) “vertical plane” in Rn ; it is also a subgroup of G, since on the first q components the Lie group operation is just the Euclidean sum. We are now in a position to state the following result (see [18, Theorem 3.1]). Theorem 9 (Blow-up on the reduced boundary) Let G be of Step 2. If E ⊂ Rn is a set of finite G-perimeter and x0 ∈ ∂G∗ E then in L1loc (Rn ). lim χEr,x0 = χS + (νE (x0 )) r→0 G (23) (Note that νE (x0 ) is pointwise well defined for x0 ∈ ∂G∗ E, by point (ii) in Definition 8). Moreover, PG (E)-a.e. x ∈ Rn belongs to ∂G∗ E. The last statement in the above theorem allows us to rewrite formula (18) as an integral on the reduced boundary with respect to the Q − 1 spherical Hausdorff measure as follows: Z Z Q−1 divG g(x)dx = −θ∞ hg, νE idS∞ , (24) E ∂G∗ E (here θ∞ is a constant depending on G, see [18, Theorem 3.10]), which looks much closer to the classical divergence theorem. Moreover, the following analogue of (19) holds: Q−1 PG (E, ·) = θ∞ S∞ In order to apply the L1loc (∂G∗ E) . (25) convergence of (23) we will need also the following n 1 n 1 n Remark 10 If E, {Ek }∞ k=1 ⊂ R are measurable, χEk → χE in Lloc (R ) and f ∈ L (R ) ∩ 1 n ∞ n L (R ), then f χEk → f χE in L (R ). In fact, given ε > 0, there is a compact set K such that Z |f |dx < ε, whence Rn \K Z Rn |(χEk − χE ) f | dx ≤ ε + kf k∞ and the last integral tends to 0 as k → ∞. 2.3 Z K |χEk − χE | dx Main results We state here the main results of this paper, namely Theorems 11 and 14. Theorem 11 Let f ∈ L1 (Rn ). The quantity lim sup k∇X Wt f kL1 (Rn ) t→0 is finite if and only if f ∈ BV (G). In this case the following holds: |DG f | (Rn ) ≤ lim inf k∇X Wt f kL1 (Rn ) ≤ lim sup k∇X Wt f kL1 (Rn ) t→0 t→0 ≤(1 + c) |DG f | (Rn ) , where c ≥ 0 is a constant depending only on G. 10 Theorem 11 shows that even in Carnot groups Definition 4 is equivalent to the analogue of De Giorgi’s original definition (3) given in the Euclidean case, even though we don’t know if the limit exists. Section 3 is devoted to the proof of Theorem 11. The next result requires the additional hypothesis that G is a Carnot group of Step 2. As we explain in Section 4, the reason is that our proof uses the rectifiability of the reduced boundary of a set of finite perimeter in G, which is known only in the Step 2 case, see [18]. Our argument works whenever the rectifiability theorem is true, and then would extend immediately to all cases where the rectifiability result were extended. We start by defining the function Z φG (ν) = h(1, x)dx, (26) TG (ν) for vectors ν ∈ Rq . Here the integral is taken over TG (ν), which is a hyperplane in Rn ; to simplify notation, we keep denoting by dx the (n − 1)-dimensional Lebegue measure over TG (ν). The function φG is continuous and, using the Gaussian bounds (11), there are constants c1 , c2 , depending on the group, such that 0 < c1 ≤ φG (ν) ≤ c2 . (27) Remark 12 (Rotation invariance of heat kernels) Notice that if the heat kernel is invariant under horizontal rotations (that is, under Euclidean rotations in the space Rq of the first variables x1 , x2 , ..., xq of Rn ), then φG is actually a constant (that is, independent of ν). This happens for instance in all groups of Heisenberg type (briefly called H-type groups), in view of the known formula assigning the heat kernel in that context (for a discussion of H-type groups, see for instance [7, Chap. 18]; for the computation of the heat kernel in this context, see [26]). By the way, we point out that in the Heisenberg groups with more than one vertical direction the heat kernel is invariant for horizontal rotations, but the sublaplacian is not. This can be seen through a direct computation based on (3.14) in [7]. On the other hand, from the general formula assigning the heat kernel in Carnot groups of step two, proved by [12] (see also [5]), one can expect the existence of groups of step two (more general than H-type groups) where the heat kernel is not invariant under horizontal rotations and the function φG is not constant. Let us first state our next result in the case of perimeters. Theorem 13 Let G be of Step 2. If E ⊂ Rn is a set of finite G-perimeter, the following equality holds Z Z 1 Q−1 √ φG (νE )dS∞ . (28) Wt χE (x)dx = lim t→0 2θ∞ t E c ∂G∗ E Conversely, if either E or E c has finite measure and Z 1 Wt χE (x) dx < +∞, lim inf √ t→0 t Ec then E has finite perimeter, and (28) holds. For general BV functions, we have the following. 11 Theorem 14 Let G be of Step 2. Then, if f ∈ BV (G) the following equality holds: Z Z 1 φG (σf )d |DG f | = lim √ |f (x) − f (y)|h(t, y −1 ◦ x)dxdy, t→0 n n n 4 t R R ×R (29) where σf is defined in (14). Conversely, if f ∈ L1 (Rn ) and the liminf of the quantity on the right hand side is finite, then f ∈ BV (G) and (29) holds. Remark 15 Note that, in view of (25), (27) and Remark 12, the right-hand side of (28) is always equivalent to, and in groups of Heisenberg-type is a multiple of, PG (E); the lefthand side of (29) is always equivalent to, and in groups of Heisenberg-type is a multiple of, |DG f | (Rn ). The proofs of the above results will be given in Section 4. 3 Proof of Theorem 11 In this section we give the proof of Theorem 11; first of all, we notice that Wt f → f in L1 as t → 0; moreover Wt f ∈ C ∞ (Rn ), hence Z |DG Wt f |(Rn ) = |∇X Wt f (x)|dx. Rn Therefore by the lower semicontinuity of the variation, that is (17), the inequality Z |∇X Wt f (x)|dx |DG f |(Rn ) ≤ lim inf t→0 (30) Rn holds, and then it remains to prove only the upper bound. We also notice that inequality (30) implies that if f ∈ L1 (Rn ) is such that the right hand side of (30) is finite, then f ∈ BV (G). So, Theorem 11 will follow if we prove that Z lim sup |∇X Wt f (x)|dx ≤ (1 + c) |DG f |(Rn ) (31) t→0 Rn for all f ∈ BV (G). We start with the following result, which will be useful in the proof of Theorem 11. It is a consequence of the algebraic properties of G and the Gaussian estimates on the heat kernel h (see Section 2.1). Lemma 16 For i, j ∈ {1, . . . , q}, let Gij be the kernel defined by the identity: n X k=q+1 XkR (cki (·)h(t, ·))(z) = q X j=1 = q X XjR n X X k=q+1 1≤il ≤q XjR Gij (t, z), ϑkji2 ...iω XiR2 ...XiRω k k
cki (·) h (t, ·) (z) (32) j=1 where the functions cki and the constants ϑki1 i2 ...iω are those introduced in (8) and (9), k respectively. They have the following properties:
(i) Gij λ2 t, D (λ) z = λ−Q Gij (t, z) for any λ > 0, t > 0, z ∈ Rn ; 12 (ii) there is a positive constant c, independent of t, such that Z Gij (t, z) dz ≤ c; Rn (iii) Z Rn Gij (t, z) dz = 0 for any t > 0; (iv) for every ε > 0, t0 > 0 there exists R > 0 such that Z |Gij (t, z)|dz < ε for any 0 < t ≤ t0 . Rn \BR Proof. (i) holds because cki (·) is homogeneous of degree ωk − ωi = ωk − 1 (for i = 1, 2, ..., q), h(t, ·) is homogeneous of degree −Q, and XiR2 ...XiRω is a homogeneous differential operator k of degree ωk − 1. For the same reason, the Gaussian estimates on h also imply, by Remark 2,
2 |Gij (t, z)| = XiR2 . . . XiRω cki (·) h (t, ·) (z) ≤ ct−Q/2 e−|z| /ct (33) k √
To prove (ii), with the change of variables z = D t w and using (i) and (33), we compute: Z Z  √   i t w tQ/2 dw Gij t, D Gj (t, z) dz = Rn Rn Z Z 2 ce−|w| /c dw (34) Gij (1, w) dw ≤ = Rn Rn which is a finite constant. (iii) holds because Gij is, by definition, a linear combination of derivatives of the kind h
i XiR2 XiR3 ...XiRω cki (·) h (t, ·) = XiR2 H (t, ·) , k
R T T R we know that R Xi R = −Xi (where () denotes transposition) and therefore we may deduce that Rn Xi f (x) dx = 0 for any f for which this integral makes sense. To prove (iv), it suffices to modify the computation (34) as follows: Z Z −|w|2 /c i dw < ε Gj (t, z) dz ≤ √ ce Rn \BR |w|>R/ t for R large enough and t ≤ t0 . The following result contains the main estimate on the commutator between the derivative Xi and the heat semigroup Xi (Wt f ) − Wt (Xi f ). Proposition 17 For any f ∈ BV (G) and i ∈ {1, 2, . . . , q}, t > 0, there exists an L1 function µit on Rn such that Xi (Wt f ) (x) = Wt (Xi f ) (x) + µit (x), where Wt (Xi f ) is the function defined by Z Wt (Xi f ) (x) := Rn h(t, y −1 ◦ x)dXi f (y). Moreover, for any t > 0, kµit kL1 (Rn ) ≤ cq|DG f |(Rn ), where c is the constant in Lemma 16, (ii). 13 (35) Proof. Let f ∈ BV (G) and fix i ∈ {1, 2, ..., q}. By (8) and (20) we have Z
Xi h t, y −1 ◦ x f (y) dy Xi Wt f (x) = (36) Rn = Z = Z
XiR h t, y −1 ◦ x f (y) dy + Rn Rn + = Z Rn n X X Rn k=q+1 ϑki1 i2 ...iω XiR1 XiR2 ...XiRω k
h t, y −1 ◦ x d (Xi f ) (y) n X + n X

XkR cki (·) h (t, ·) y −1 ◦ x f (y) dy
XiR h t, y −1 ◦ x f (y) dy Rn k=q+1 1≤i ≤q j Z Z X k=q+1 1≤ij ≤q =Wt (Xi f ) (x) + Z k

cki (·) h (t, ·) y −1 ◦ x f (y) dy ϑki1 i2 ...iω k q Z X
Gij t, y −1 ◦ x d (Xj f ) (y) , Rn j=1 Rn XiR2 ...XiRω k

cki (·) h (t, ·) y −1 ◦ x d (Xi1 f ) (y) where the kernels Gij are defined in Lemma 16. Set µit (x) = q Z X j=1 Rn
Gij t, y −1 ◦ x d (Xj f ) (y), (37) then by Lemma 16 and (16) we have µit L1 (Rn ) ≤c q Z X j=1 Rn |d (Xj f )| (y) ≤ cq |DG f | (Rn ) that is (35). By (36) and (37), we can write: Xi (Wt f ) (x) = Wt (Xi f ) (x) + µit (x). (38) Proof of Theorem 11. Let us conclude with the proof of (31). By Proposition 17 we have (with the supremum taken, accordingly to Definition 4, on all the functions g ∈ C01 (Rn , Rq ) such that kgk∞ ≤ 1): (Z ) Z q X gi (x)Xi (Wt f )(x)dx |∇X Wt f (x)|dx = sup g Rn = sup g (Z q X Rn i=1 gi (x) Rn i=1 Z Rn h(t, y −1  Z ◦ x)dXi f (y) dx + q X Rn i=1 14 ) gi (x)µit (x)dx since h(t, z −1 ) = h(t, z) ( q Z Z X = sup g = sup g i=1 R ( q Z X i=1 ( = sup − g n −1 h(t, x Rn  Z ◦ y)gi (x) dx dXi f (y) + Rn i=1 Wt gi (y)dXi f (y) + i=1 Z q X Rn i=1 Rn q Z X q X Xi (Wt gi )(y)f (y)dy + Z ) gi (x)µit (x)dx q X Rn i=1 Rn ) gi (x)µit (x)dx ) gi (x)µit (x)dx where in the last identity we have used (15). We now exploit the fact that the supremum on all compactly supported functions g in the above expression can be replaced by the supremum on functions φ rapidly decreasing at infinity (as Wt g is, by the Gaussian estimates) verifying the constraint kφk∞ ≤ 1; therefore the last expression is, by (35) ≤ |DG f | + q X i=1 kµit kL1 (Rn ) ≤ (1 + cq 2 ) |DG f | (Rn ). We have therefore proved that for any t > 0, Z |∇X Wt f (x)|dx ≤ (1 + cq 2 ) |DG f | (Rn ). Rn Passing to the limsup as t → 0 we are done. 4 Proof of Theorem 14 Theorem 14, thanks to coarea formula, follows from Theorem 13, so we first prove the latter. As a preliminary result, we present a characterization of BV (G) functions analogous to that in [9, Proposition 2.3]. Lemma 18 If f ∈ L1 (Rn ), z ∈ Rn and Z 1 lim inf |f (x ◦ D(t)z) − f (x)|dx < ∞, t→0 t Rn Pq then the distributional derivative of f along Z = j=1 zj Xj is a finite measure. (39) Proof. The proof is essentially the same contained in [9], but we repeat it for the reader’s convenience. We start by recalling the following identity (see [7, p. 17]): if we take φ ∈ Cc1 (Rn ) and Z is any vector field which at the origin equals z, then d [φ (x ◦ tz)]/t=0 . dt P Let us apply this identity to the vector field Z = qi=1 zi Xi which at the origin takes the value z ∗ = (z1 , z2 , ..., zq , 0, 0, ..., 0) . Zφ (x) = For any z ∈ Rn , d d [φ (x ◦ D(t)z)]/t=0 = [φ (x ◦ tz ∗ )]/t=0 dt dt 15 hence q X d zi (Xi φ) (x) . [φ (x ◦ D(t)z)]/t=0 = dt i=1 Since Z φ(x ◦ D(t)z −1 ) − φ(x) f (x ◦ D(t)z) − f (x) dx = φ(x)dx t t n n R R Z 1 ≤ kφk∞ |f (x ◦ D(t)z) − f (x)|dx, |t| Rn Z f (x) from (39) we deduce that the functional Z Z φ(x ◦ D(t)z −1 ) − φ(x) f (x)Zφ(x)dx dx = − Tf φ := lim f (x) t→0 Rn t Rn is well defined and satisfies the condition |Tf φ| ≤ Ckφk∞ . Then, there exists a measure µZ such that Z φ(x)dµZ (x), ∀φ ∈ Cc1 (Rn ). Tf φ = Rn n Cc1 (Rn ) By the density of in Cc (R ), we get the conclusion. Proof of Theorem 13. Let us introduce the functions Z Wt χE (x)dx g(t) = Ec Z
Wt2 χE (x)dx. F (t) = g t2 = Ec The statement we want to prove is: g (t) lim √ = 2θ∞ t→0 t First, notice that g(t) → R Ec Z ∂G∗ E Q−1 φG (νE )dS∞ . χE (x)dx = 0 as t ↓ 0. Indeed, Wt χE − χE = − (Wt χE c − χE c ) , (40) since χE + χE c = 1 = Wt 1 = Wt χE + Wt χE c and, since by the finiteness of the perimeter of E either E or E c has finite measure, √ kWt χE − χE kL1 (Rn ) → 0 as t ↓ 0. By De L’Hospital rule we can evaluate limt→0 g (t) / t as √
lim 2 tg ′ (t) = lim 2tg ′ t2 = lim F ′ (t) . t→0 t→0 t→0 Computing the derivative of F using the divergence theorem (24) and the properties of h(t, ·), we get: Z Z F ′ (t) = 2t LWt2 χE dx = 2t divG ∇X Wt2 χE dx Ec Ec Z Q−1 = 2θ∞ t h∇X Wt2 χE , νE idS∞ ∂G∗ E = 2θ∞ t Z ∂G∗ E h∇X Z E  Q−1 h(t2 , y −1 ◦ x)dy , νE (x)idS∞ (x). 16 
Now, taking into account the fact that h t, z −1 = h (t, z), we have     Z Z Z
1 −1 2 −1 2 −1 −Q x ◦ y dy h(t , y ◦ x)dy = h(t , x ◦ y)dy = t h 1, D t E E    E    Z
1 1 t−Q h 1, D = (y) dy x−1 ◦ D t t E     Z
1 x−1 ◦ z dz = h 1, D t D( 1t )E     Z 1 = (x) dz h 1, z −1 ◦ D t D( 1t )E hence * Z +     1 1 −1 Q−1 F (t) = 2θ∞ t ∇X h 1, z ◦ D (x) dz, νE (x) dS∞ (x) 1 ∗ t t ∂G E D( t )E *Z +   Z 1 −1 Q−1 = 2θ∞ ∇X h(1, z ◦ D (x) dz, νE (x) dS∞ (x) t D( 1t )E ∂G∗ E *Z + Z −1 Q−1 = 2θ∞ ∇X h(1, z )dz, νE (x) dS∞ (x) Z ′ ∂G∗ E D(1/t)(x−1 E) q Z X = 2θ∞ i=1 νE (x)i ∂G∗ E Z Xi h(1, z −1 D(1/t)(x−1 E) )dz ! Q−1 dS∞ (x). By Theorem 9 (blow-up of the reduced boundary), we have the L1loc convergence E1/t,x−1 = D(1/t)(x−1 E) → SG+ (νE (x)). (41) Since by the Gaussian estimates Xi h(1, z −1 ) ∈ L1 ∩ C ∞ (Rn ), the integral on E1/t,x−1 converges to the integral on SG+ (νE (x)) (see Remark 10), is a continuous function of x and we can apply the dominated convergence theorem, getting ! Z Z q X ′ −1 Q−1 νE (x)i Xi h(1, z )dz dS∞ (x) lim F (t) = 2θ∞ t→0 = 2θ∞ ∂G∗ E i=1 Z q X ∂G∗ E + SG (νE (x)) νE (x)i i=1 Z − SG (νE (x)) ! Q−1 Xi h(1, z)dz dS∞ (x) with the obvious meaning of the symbol SG− (νE (x)). Next, we perform a standard integration by parts in the inner integral, exploiting the fact that the (Euclidean) outer normal to the halfspace SG− (νE (x)) is nE (x) = (νE (x)1 , ..., νE (x)q , 0, ..., 0) while Xi h(1, z) = ∂zi h (1, z) + n X j=q+1 hence q X i=1 νE (x)i Z − SG (νE (x)) Xi h(1, z)dz ! = q X i=1 17
∂zj qji (z) h (1, z) (νE (x)i )2 Z h(1, z)dz = φG (νE ) TG (νE (x)) for any x ∈ ∂G∗ E, and we conclude that Z Z 1 Q−1 √ lim Wt χE (x)dx = φG (νE )dS∞ . t→0 2θ∞ t E c ∂G∗ E (42) In order to prove the converse, assume that E has finite measure and notice that for any z Z Z √ √ 1 χE (x ◦ D( t)z)(1 − χE (x))dx = χE (x ◦ D( t)z) − χE (x) dx, 2 Rn Rn and that the Lebesgue measure is right invariant as well. Therefore Z Z Z   √ Wt χE (x) dx = t−Q/2 h 1, D(1/ t)(y −1 ◦ x) (1 − χE (x)) dydx n Ec Z Z R E √ h(1, z)χE (x ◦ D( t)z −1 ) (1 − χE (x)) dzdx = Rn Rn Z Z √ 1 |χE (x ◦ D( t)w) − χE (x)| dxdw h(1, w−1 ) = 2 Rn n Z R Z √ 1 |χE (x ◦ D( t)w) − χE (x)| dxdw. h(1, w) = 2 Rn Rn Then, the finiteness of the liminf of the integral in the left hand side of (42) implies that there is a sequence tk ↓ 0 such that Z Z √ √ 1 h(1, w) |χE (x ◦ D( tk )w) − χE (x)| dxdw ≤ C tk 2 Rn n R for all k. On the other hand, by the lower Gaussian estimates on h (1, w) we can also write Z Z √ 1 h(1, w) |χE (x ◦ D( tk )w) − χE (x)| dxdw 2 Rn Rn Z Z √ 1 ≥ h(1, w) |χE (x ◦ D( tk )w) − χE (x)| dxdw 2 |w|≤2 Rn Z Z √ |χE (x ◦ D( tk )w) − χE (x)| dxdw ≥c |w|≤2 hence Z |w|≤2 Z Rn Rn √ √ |χE (x ◦ D( tk )w) − χE (x)| dxdw ≤ C tk . So, if we define the function Φtk (w) = Z Rn √ |χE (x ◦ D( tk )w) − χE (x)| √ dx, tk by Fatou’s Lemma we deduce that Φ0 (w) = lim inf Φtk (w) k→∞ is integrable on B = {|w| ≤ 2}. So, for any ε > 0, there are a set Bε of measure less than ε and a constant Cε > 0 such that for every w ∈ B \ Bε , Φ0 (w) ≤ Cε . In particular, we may 18 choose n linearly independent directions w1 , . . . , wn with Φ0 (wi ) ≤ Cε , and there is k0 ∈ N such that Z √ √ |χE (x ◦ D( tk )wi ) − χE (x)| dx ≤ (Cε + 1) tk , ∀ i = 1, . . . , n, k > k0 . Rn Then by Lemma 18 we can conclude that E has finite perimeter. We note that the above proof of Theorem 13 simplifies that of the analogous result in the Euclidean setting given in [24]. With an argument similar to that used in the proof of Proposition 8 in [25], it is also possible to prove the following Proposition 19 If E has finite perimeter, then Z 1 √ |Wt χE − χE | dx ≤ cG PG (E) , t > 0 4 t Rn with cG = Proof. First of all we note that Z Rn Z Rn (43) |∇X h (1, z)| dz. (Wt χE (x) − χE (x)) dx = 0. This follows by (iv) in Theorem 1 if E has finite measure; otherwise E c must have finite measure, and the same identity holds in view of (40). Recalling that 0 ≤ Wt χE (x) ≤ 1, we deduce that Z Z Z (Wt χE (x) − χE (x))+ dx (44) (Wt χE (x) − χE (x)) dx = Wt χE (x) dx = Rn Ec Ec Z Z 1 |Wt χE (x) − χE (x)| dx (Wt χE (x) − χE (x))− dx = = 2 Rn Rn (notice that the above equality holds under the hypothesis that either E or E c has finite measure). We can also write Z Wt χE (x) dx = Ec Z Ec (Wt χE (x) − χE (x)) dx = But then by the divergence theorem Z Z Z dPG (E) (x) LWt χE (x) dx = Ec Rn ∂G∗ E ≤ Z ∂G∗ E dPG (E) (x) Z Rn 19 Z 0 t ds Z LWt χE (x) dx. Ec h∇X h(s, y −1 ◦ x), νE (x)iχE (y) dy ∇X h(s, y −1 ◦ x) dy (45) By (44) and (45) the conclusion then follows since 1 √ 4 t Z t Z 1 LWs χE (x)dx ds Wt χE (x)dx = √ 2 t 0 Ec Ec ! Z Z t Z 1 −1 ≤ √ |∇X h(s, y ◦ x)|dy ds dPG (E) (x) 2 t 0 Rn ∂G∗ E ! Z Z t Z 1 1 = √ |∇X h(1, z)|dz ds dPG (E) (x) √ s Rn 2 t 0 ∂G∗ E 1 |Wt χE − χE |dx = √ 2 t Rn Z Z ≤ cG PG (E). Proof of Theorem 14. The derivation of Theorem 14 from Theorem 13 is based on the coarea formula, and is similar to that of Theorem 4.1 of [24]. Assume f ∈ BV (G) and set Eτ = {f > τ }. By Proposition 7, for a.e. τ ∈ R the set Eτ has finite perimeter, hence by Theorem 13 we can write Z Z 1 Q−1 √ (x) φG (νEτ (x))dS∞ Wt χEτ (x)dx = lim t→0 2θ∞ t E c ∗E ∂ τ G τ for a.e. τ . Moreover, using the same arguments as [1, Proposition 3.69] and the continuity of φG it is easily checked that the function φG (σf (x)) is (obviously bounded and) Borel. Comparing f with f ∨ τ χEτ and using [1, Proposition 3.73] we see that for a.e. τ ∈ R the equality Q−1 -a.e. x ∈ ∂G∗ Eτ , whence σf (x) = νEτ (x) holds for S∞ Q−1 -a.e.x ∈ ∂G∗ Eτ . φG (σf (x)) = φG (νEτ (x)) for a.e.τ ∈ R, S∞ We point out that if x belongs to the boundary of two different level sets Eτ , Eσ (which happens whenever f has a jump at x), then the above equality holds for both the levels because νEτ (x) = νEσ (x) for σ, τ ∈ [f − (x), f + (x)]. We also notice that, for almost every x, y ∈ Rn , Z R |χEτ (x) − χEτ (y)|dτ = |f (x) − f (y)|. (46) With the aid of coarea formula (22) with g(x) = φG (σf (x)), by (44), (46) and using the dominated convergence theorem, we obtain that Z Z Z Q−1 φG (σf (x))d |DG f | = θ∞ (x)dτ φG (νEτ (x))dS∞ Rn R ∂G∗ Eτ Z 1 lim √ Wt χEτ (x)dxdτ R t→0 2 t Eτc Z Z 1 √ = lim |Wt χEτ − χEτ |dxdτ t→0 R 4 t Rn Z Z 1 |χEτ (y) − χEτ (x)|h(t, y −1 ◦ x)dxdydτ = lim √ t→0 4 t R Rn ×Rn Z 1 |f (x) − f (y)|h(t, y −1 ◦ x)dxdy. (47) = lim √ t→0 4 t Rn ×Rn = Z 20 Conversely, assume that f ∈ L1 (Rn ) and that Z 1 lim inf √ |f (x) − f (y)|h(t, y −1 ◦ x)dxdy t→0 t Rn ×Rn is finite. Then, Z Z Z |f (x) − f (y)| h(t, y −1 ◦ x)dxdy = Rn ×Rn R Rn ×Rn |χEτ (x) − χEτ (y)| h(t, y −1 ◦ x)dxdydτ . R Since E c Wt χE ≥ 0, by the above equality and (44), which we are allowed to exploit because for f ∈ L1 (Rn ) either Eτ or its complement has finite measure, we get ! Z Z Z Z 1 1 √ lim inf √ Wt χEτ dx dτ ≤ lim inf Wt χEτ dx dτ 0≤ t→0 t→0 t Eτc t Eτc R R Z Z 1 ≤ lim inf √ |χEτ (x) − χEτ (y)| h(t, y −1 ◦ x)dx dy dτ < +∞ . t→0 2 t Rn ×Rn R In particular, by Theorem 13, for a.e. τ ∈ R the set Eτ has finite perimeter and the limit R lim √1t E c Wt χEτ dx exists. As a consequence, f ∈ BVloc (G) and we may use the above t→0 τ identity (47) to get Z 1 φG (σf )d |DG f | min φG Rn Z 1 1 lim inf √ |f (x) − f (y)| h(t, y −1 ◦ x)dx dy < +∞ ≤ min φG t→0 4 t Rn ×Rn |DG f |(Rn ) ≤ that is, f ∈ BV (G). References [1] Ambrosio, L., N. Fusco and D. Pallara. Functions of bounded variation and free discontinuity problems Oxford University Press, 2000. [2] Ambrosio, L., M. Miranda Jr. and D. Pallara. “Special Functions of Bounded Variation in Doubling Metric Measure Spaces.” in: Pallara, D. (ed.), Calculus of Variations: Topics from the Mathematical Heritage of Ennio De Giorgi. 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Brandolini. “Lp estimates for uniformly hypoelliptic operators with discontinuous coefficients on homogeneous groups.” Rend. Sem. Mat. Univ. Politec. Torino 58 (2000): 389-433. [9] Capogna, L. “Regularity of Quasi-Linear Equations in the Heisenberg Group.” Comm. Pure Appl. Math. 50 (1997): 867-889. [10] Capogna, L., D. Danielli and N. Garofalo. “The geometric Sobolev embedding for vector fields and the isoperimetric inequality.” Comm. Anal. Geom. 2 (1994): 203-215. [11] Carbonaro, A. and G. Mauceri. “A note on bounded variation and heat semigroup on Riemannian manifolds.” Bull. Australian Math. Soc. 76 (2007): 155-160. [12] Cygan, J. “Heat kernels for class 2 nilpotent groups.” Studia Math. 6, no.4 (1979): 227-238. [13] De Giorgi, E. “Su una teoria generale della misura (r − 1)-dimensionale in uno spazio ad r dimensioni.” Ann. Mat. Pura Appl. (4) 36 (1954): 191-213, and also Ambrosio, L., G. Dal Maso, M. Forti, M. Miranda and S. Spagnolo eds. Ennio De Giorgi: Selected Papers. Springer, 2006, 79-99. English translation, Ibid., 58-78. [14] De Giorgi, E. “Nuovi teoremi relativi alle misure (r − 1)-dimensionali in uno spazio a r dimensioni.” Ricerche Mat., 4 (1955): 95-113, and also Ambrosio, L., G. Dal Maso, M. Forti, M. Miranda and S. Spagnolo eds. Ennio De Giorgi: Selected Papers. Springer, 2006, 128-144. English translation, Ibid., 111-127. [15] Federer, H. Geometric Measure Theory. Springer, 1969. [16] Folland, G. B. and E. M. Stein. Hardy spaces on homogeneous groups. Princeton University Press, 1982. [17] Franchi, B., R. Serapioni and F. Serra Cassano. “Meyers-Serrin type theorems and relaxation of variational integrals depending on vector fields.” Houston J. Math. 22, no. 4 (1996): 859–890. [18] Franchi, B., R. Serapioni and F. Serra Cassano. “On the strcucture of finite perimeter sets in Carnot step 2 groups.” J. Geom. Anal., 13 (2003): 431-466. [19] Garofalo, N. and Nhieu, D.M. “Isoperimetric and Sobolev inequalities for Carnot– Carathéodory spaces and the existence of minimal surface.” Comm. Pure Appl. Math. 2, n.10 (1996): 1081-1144. [20] Kinnunen, J., R. Korte, N. Shanmugalingam and H. Tuominen. “The De Giorgi measure and an obstacle problem related to minimal surfaces in metric spaces.” J. Math. Pures Appl. 98 (2010): 599-622. [21] Ledoux, M. “Semigroup proofs of the isoperimetric inequality in Euclidean and Gauss space.” Bull. Sci. Math. 118 (1994): 485-510. 22 [22] Miranda Jr., M. “Functions of bounded variation on ‘good’ metric spaces.” J. Math. Pures Appl., 82 (2003): 975-1004. [23] Miranda Jr., M., D. Pallara, F. Paronetto and M. Preunkert. “Heat semigroup and functions of bounded variation on Riemannian manifolds.” J. Reine Ang. Math., 613 (2007): 99-119. [24] Miranda Jr., M., D. Pallara, F. Paronetto and M. Preunkert. “Short-time heat flow and functions of bounded variation in Rn .” Ann. Fac. Sci. Toulouse Math., 16 (2007): 125-145. [25] Preunkert, M. “A semigroup version of the isoperimetric inequality.” Semigroup Forum, 68 (2004): 233-245. [26] Randall, J. “The heat kernel for generalized Heisenberg groups.” J. Geom. Anal. 6 (1996): 287-316. [27] Stein, E. M. Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, 1993. [28] Varopoulos, N. Th., L. Saloff-Coste and T. Coulhon. Analysis and geometry on groups. Cambridge University Press, 1992. 23