Comparison of Hausdorff measures with respect to the Euclidean and the Heisenberg metric

Publ. Mat. 47 (2003), 237–259 COMPARISON OF HAUSDORFF MEASURES WITH RESPECT TO THE EUCLIDEAN AND THE HEISENBERG METRIC Zolt´ an M. Balogh, Matthieu Rickly and Francesco Serra Cassano∗ Abstract We compare the Hausdorff measures and dimensions with respect to the Euclidean and Heisenberg metrics on the first Heisenberg group. The result is a dimension jump described by two inequalities. The sharpness of our estimates is shown by examples. Moreoveracomparison between Euclidean and H-rectifiability is given. 1. Introduction In this paper we consider the Heisenberg group H=H1=(R3,∗) as a homogeneous group endowed with the left invariant, homogeneous Heisenberg distance dHdefined as follows. The group multiplication ∗:H×H→His given by (x, y, t)∗(x,y,t ):=(x+x,y+y,t+t+2(xy−yx)).(1.1) His endowed with the homogeneous norm pH:= ((x2+y2)2+t2)1 4 (1.2) if p=(x, y, t)∈H, which induces the Heisenberg distance dH(p, p):=p−1∗pH.(1.3) It is well-known that the topological dimension of His 3, since H coincides with R3as a smooth manifold (see Lemma 2.1). On the other hand, the Hausdorff measures and dimensions of subsets of H≡R3with respect to either dHor the Euclidean metric dEcan be very different. For 2000 Mathematics Subject Classification. 28A78, 43A80. Key words. Hausdorff measures, Hausdorff dimension, Heisenberg metric. ∗F.S.C. is supported by GNAMPA of INDAM, project “Analysis in metric spaces and subelliptic equations”, by MURST, Italy, and University of Trento, Italy. Part of the work was done while F.S.C. was a visitor at the University of Bern. He wishes to thank the Institute of Mathematics for its hospitality. 238 Z. M. Balogh, M. Rickly, F. Serra Cassano instance, the Hausdorff dimension of (R3,d H)is4,while the dimension of a regular surface in (R3,d H)is3.However, if we consider a regular curve in (R3,d H), then it may have Hausdorff dimension both 1 and 2 (see [Gro] and also [Str]). The purpose of this paper is to describe in details this dimension jump phenomenon between the Heisenberg and the Euclidean Hausdorff dimensions on subsets of R3. Indeed we perform a comparison between the α-dimensional Hausdorff measures induced on R3by dHand dE, which we respectively denote with Hα Hand Hα E. More precisely our first result reads as follows: Theorem 1.1 (Dimension jump theorem).Let α≥0. Then (i) Hmin{2α,α+1} HH α E,(1.4) i.e. Hmin{2α,α+1} His absolutely continuous with respect to Hα E. (ii) Hmin{α,1+ α 2} EH α H,(1.5) i.e. Hmin{α,1+ α 2} Eis absolutely continuous with respect to Hα H. Our second result shows that the estimates of Theorem 1.1 are sharp: Theorem 1.2 (Sharpness of the dimension jump). (i) Given 0<α≤3, there is a compact subset Aαof Hsatisfying Hα E(Aα)<∞and Hmin{2α,α+1} H(Aα)>0. (ii) For 0<α<2and α=4, there is a compact subset Aαof H satisfying Hα H(Aα)<∞and Hmin{α,1+ α 2} E(Aα)>0. For 2≤α<4and 0<δ<1, there is a compact subset Aα,δ of H satisfying Hα H(Aα,δ)=0 and Hmin{α,1+ α 2}−δ E(Aα,δ)=H1+ α 2−δ E(Aα,δ)>0. Remark 1.1.We conjecture that given 2 ≤α<4, there is a compact subset Aαof Hsuch that Hmin{α,1+ α 2} E(Aα)=H1+ α 2 E(Aα)>0 and Hα H(Aα)<∞. However, we have not yet been able to construct such sets. Comparison of Hausdorff Measures 239 The technique involved in the proof of Theorem 1.1 is based on an optimal covering of Heisenberg balls by smaller Euclidean balls and viceversa. This kind of mutual coverings have already been proposed in [Gro]. The proof of Theorem 1.2 relies on some more delicate arguments involving recent results on the size of Cantor-type and characteristic sets of regular surfaces in the metric spaces (R3,d H) and (R3,d E) (see [Ba1], [Ba2]). Another striking feature of the Heisenberg group is the lack of m-rectifiable subsets in the metric space (H,d H) according to the classical notion of rectifiability in a metric space due to Federer [Fe]. Indeed, it has been proved that (H,d H)ispurely m-unrectifiable for m=2,3,4 (see [AK, Theorem 7.2]). This has led to a more intrinsic definition of rectifiability in (H,d H), the 3-dimensional H-rectifiability, introduced in [FSSC1]. We will use our results from Theorem 1.2 to compare the classical notion of 2-Euclidean rectifiability in (R3,d E)tothe notion of 3-dimensional H-rectifiability (see Theorem 5.1). The paper is organized as follows: in Section 2 we recall some basic facts about the Heisenberg group and the Hausdorff measures defined on it. In Section 3 we prove our first theorem. The second theorem is proved in Section 4. In the last section we apply our results to compare 2-rectifiable subsets of (H,d E) and 3-dimensional H-rectifiable subsets of (H,d H). Acknowledgements. We thank Pertti Mattila for useful discussions on the subject. 2. Basic definitions and preliminary results In this section, we briefly restate the basic definitions and results needed in the rest of the article. We use the model of the first Heisenberg group H=H1with underlying space R3and group operation given by (1.1). Notice that p−1=−p and that 0 = (0,0,0) is the unit element of the group H. The dilation by ris the automorphism δr:H→Hgiven by δr(x, y, t):=(rx,ry,r2t).(2.1) For p∈Hthe left translation by pis the automorphism lp:H→H defined by lp(p):=p∗p. 240 Z. M. Balogh, M. Rickly, F. Serra Cassano The 3-dimensional Lebesgue measure L3in R3≡His a bi-invariant Haar measure for the group and satisfies L3(δr(A)) = r4L3(A) whenever A⊆Hand r>0 (see [FoSt, Proposition 1.2 (c)]). The Lie algebra of His spanned by the left invariant vector fields X=∂ ∂x +2y∂ ∂t,Y=∂ ∂y −2x∂ ∂t,T=∂ ∂t. The Lie algebra structure of His determined by the only non-trivial commutator relation [X,Y ]=−4T. The vector fields Xand Yspan a vector bundle, the so-called horizontal bundle HH, where HHp:= span{X(p),Y(p)}for all p∈H, which can be canonically identified with a vector subbundle of the tangent vector bundle TH≡TR3. The following relationship between the distances dEand dHin R3can be easily verified. Lemma 2.1. Let Abeabounded subset of (R3,d E)and let b≥1be a bound for Awith respect to dE. Then there is a positive constant c=c(b) such that for p, p∈Awe have 1 cdE(p, p)≤dH(p, p)≤c(dE(p, p))1 2.(2.2) In particular the identity map Id: (R3,d E)→(R3,d H)is a homeomorphism. In the sequel, BE(p, r) and BH(p, r) will denote the closed ball of radius r>0 centered at pin (H,d E) and (H,d H) respectively. ·will denote the Euclidean norm on Rnfor n=2,3. Let us now recall the definition of Hausdorff and spherical measures on a metric space (X,d). Given a subset Aof X, the diameter of Ais diamd(A):=sup{d(a, a)|a, a∈A}. We write diam instead of diamdwhen there is no risk of confusion, and we let diamE:= diamdEand diamH:= diamdH. Observe that for p∈H and r>0wehave diamH(BH(x, r)) = 2r.(2.3) Comparison of Hausdorff Measures 241 For 0 ≤s<∞,0<δ≤∞and A⊆X,welet Hs d,δ(A):=inf∞  n=1 (diam(En))s|En⊆X, diam(En)≤δ, A ⊆ n∈N En and Ss d,δ(A):=inf ∞  n=1 diam(B(xn,r n))s|diam(B(xn,r n)) ≤δ, A⊆ n∈N B(xn,r n), where B(x, r) denotes the closed ball of radius rcentered at x. Since 0 < δ1≤δ2clearly implies Hs d,δ2(A)≤H s d,δ1(A) and Ss d,δ2(A)≤Ss d,δ1(A), we may define the s-dimensional Hausdorff measure Hs don (X,d) and the s-dimensional spherical measure Ss don (X,d)by Hs d(A):=lim δ↓0Hs d,δ(A) and Ss d(A):=lim δ↓0Ss d,δ(A) respectively. We will write Hs E,Ss Einstead of Hs dE,Ss dEand Hs H,Ss Hinstead of Hs dH,Ss dHand we will generally consider these measures on R3.By abuse of notation we shall also write Hs Efor the s-dimensional Hausdorff measure on (Rm,d E) for m=1,2. The Hausdorff dimension of a set A⊆(X,d)is H−dimd(A):=inf{s≥0|H s d(A)=0}. In the following proposition we collect some general properties of Hausdorff measures that will be needed below. Proposition 2.2. (i) Let (X,d)be a metric space. Then Hs dand Ss dare regular Borel (outer) measures and Hs d(A)≤Ss d(A)≤2sHs d(A) for all A⊆X,s≥0. 242 Z. M. Balogh, M. Rickly, F. Serra Cassano (ii) Let (Xi,d i)(i=1,2)betwo metric spaces and let f:(X1,d 1)→ (X2,d 2)be a L-Lipschitz continuous map, i.e. d2(f(x),f(y)) ≤L·d1(x, y)(∀x, y ∈X1). Then Hs d2(f(A)) ≤Ls·Hs d1(A) for all A⊆X1,s≥0. (iii) Let (X,d)be a metric space and let dt(x, y):=(d(x, y))tfor x, y ∈ Xand t∈(0,1). Then dtis a distance on Xand Hs d(A)=H s t dt(A) for all A⊆X,s≥0and t∈(0,1). (iv) Let (X,d)beametric space and let Y⊆X. Denote by dYthe metric on Yinduced by d. Then Hs d(A)=Hs dY(A) for all A⊆Y,s≥0. (v) Properties (ii) and (iii) hold if we replace the Hausdorff measure Hs dby the spherical measure Ss d. Proof: All these statements are classical and can be found in [Fa], [Fe] or [Ma]. Remark 2.1.Using (ii) of the above proposition, one easily sees that if d1 and d2are two bi-Lipschitz equivalent metrics on X, then Hs d1and Hs d2 are mutually absolutely continuous measures. In particular, if A⊆X then H−dimd1(A)=H−dimd2(A). Using the homogeneity and left invariance of dH, one has Hs H(lp(A)) = Hs H(A) and Hs H(δr(A)) = rsHs H(A) for any A⊆H,p∈H,s≥0 and r>0. Moreover it is easy to see that H4 H=cL3for a suitable positive constant c. In Subsection 4.1 we will make use of the following criterion giving a lower bound on the Hausdorff dimension of a set in a metric space. Lemma 2.3. Let (X,d)beaseparable metric space, let A⊆Xand let s>0. Suppose there is a measure µon Xwith the following properties: (i) µ(A)>0. (ii) µ(B(x, r)∩A)≤cµrsfor all x∈Xand 0<r≤R, where cµ, R>0are fixed constants. Then Hs d(A)>0. The proof is a simple exercise. Comparison of Hausdorff Measures 243 3. Proof of the dimension jump theorem In the following we prove the absolute continuity of H2α Hand Hα+1 H with respect to Hα E(Propositions 3.2 (i) and 3.6 (i) below) and the absolute continuity of Hα Eand H1+ α 2 Ewith respect to Hα H(Propositions 3.2 (ii) and 3.6 (ii) below). Indeed, notice that from Proposition 2.2 (i) we can indifferently estimate the Hausdorff or spherical measures. We start with a technical lemma: Lemma 3.1. A subset Aof His bounded with respect to dEif and only if it is bounded with respect to dH. More precisely, if bE≥1is a bound for Awith respect to dE, then 21 4bEis a bound for Awith respect to dH, and if bH≥1is a bound for Awith respect to dH, then 21 2b2 His a bound for Awith respect to dE. The proof is left to the reader. Proposition 3.2. The Hausdorff measures satisfy the following absolute continuity properties (α≥0arbitrary): (i) Hα HH α 2 E. (ii) Hα EH α H. Proof: (i) Suppose that H α 2 E(A)=0and denote by √dE:R3×R3→ [0,+∞) the distance √dE(p, p):=(dE(p, p))1 2if p,p∈R3. Then by Proposition 2.2 (iii) with t=1 2we also get 0 = H α 2 E(A)=Hα √dE(A). Let An:= A∩BE(0,n). Applying Lemma 2.1 with A=An, there is a positive constant cndepending only on n, such that for all p, p∈Anwe have 1 cn dE(p, p)≤dH(p, p)≤cndE(p, p).(3.1) Let d(n) Hand d(n) Edenote the restrictions of the distances dHand √dEto An.Bythe right inequality in (3.1) and Proposition 2.2 (ii) with X1=X2=An,f= Id, d1=d(n) E,d2=d(n) Hand L=cn Hα d(n) H (An)≤cα nHα d(n) E (An). On the other hand from Proposition 2.2 (iv) Hα H(An)=Hα d(n) H (An)≤cα nHα d(n) E (An)=Hα √dE(An)=0. Then, by passing to the limit when n→∞,weget Hα H(A)=0. 244 Z. M. Balogh, M. Rickly, F. Serra Cassano The statement (ii) can be proved by similar arguments using the left inequality in (3.1). In the following we shall prove Sα+1 HSα Eand S1+ α 2 ESα H, cf. Proposition 3.6 below. This is considerably more complicated than Proposition 3.2. The proof is based on more involved covering lemmas relating the Euclidean and Heisenberg balls. The statements of these covering lemmas can be found in [Gro, 0.6.C]. Before going into details we need one more technical lemma in order to control the distortion of the shape of Euclidean balls under group translation. Lemma 3.3. Let Abeabounded subset of (R3,d E)with bound b≥0. For r>0,p=(x, y, t)∈R3and p=(x,y,t )∈Awe have lp(BE(p, r)) ⊆BE(lp(p),(2b+1)r). Moreover if lp(p)+(x,y,t )∈lp(BE(p, r)) then (x,y)≤r. Proof: Let (x+x,y+y,t+t)∈BE(p, r). lp(x+x,y+y,t+t) =(x+x+x,y+y+y,t +t+t + 2((x+x)y−x(y+y))) =lp(p)+(x,y,t  +2(xy−xy)). Now (x,y)=(x,y)≤rand (x,y,t  +2(xy−xy))2 =x2+y2+|t +2(xy−xy)|2 ≤x2+y2+(|t|+2p·(x,y,t ))2 ≤(x,y,t )2+4b(x,y,t )·|t|+4b2(x,y,t )2 ≤r2+4br2+4b2r2=(2b+1) 2r2. The claim follows. The following statement exhibits a close-to-optimal covering of a Euclidean ball by Heisenberg balls. Comparison of Hausdorff Measures 245 Proposition 3.4. Let Abe abounded subset of (R3,d E)with bound b≥ 0. Then there exists N=N(b)∈Nsuch that for any Euclidean ball BE(p, r)with p∈Aand 0<r<1we can find Heisenberg balls BH(p1,r),...,B H(pk,r)satisfying: (i) BE(p, r)⊆k i=1 BH(pi,r). (ii) k≤N r. Remark 3.1.Notice that (ii) yields a sharp control on the number k of Heisenberg balls needed to cover the Euclidean ball in terms of its radius r. Proof: It is clearly enough to show that, given a bounded subset Aof (R3,d E) with bound b≥0, there is N=N(b)∈Nsuch that for p∈A and 0 <r<1wecan find BH(p1,r),...,B H(pk,r) satisfying: BE(p, r/2) ⊆ k  i=1 BH(pi,r) and k≤N r. The idea is to use a group translation to move the Euclidean ball to the origin and to perform the covering there with balls centered on the vertical axis Ot.Forp=(0,0,t)∈Otand r>0, let us denote by BH,∞(p, r) the set of points p=(x,y,t )∈Hsatisfying (x,y)≤r and |t−t|≤r2. Hence BH,∞(p, r)isaflat box of height 2r2centered at p∈Hand its orthogonal projection along the vertical axis is a disk of radius r.Nowlet p∈A,0<r<1 and let k:= [4b+2 r]+1. By Lemma 3.3 we can cover lp−1(BE(p, r 2)) with kboxes BH,∞(pi,r 2), where i∈{1,...,k}and (pi)i∈{1,...,k}are suitable points on the vertical axis. The reader can easily see that BH,∞(pi,r 2)⊆BH(pi,r) for 1 ≤i≤k, thus BE(p, r/2) ⊆lpk  i=1 BH(pi,r)= k  i=1 lp(BH(pi,r)) = k  i=1 (BH(p∗pi,r)). Finally, kr < 4b+3 rr=4b+3and the claim is proved with N:= [4b+ 3]+1. The following statement is in some sense the counterpart of Proposition 3.4. It presents a close-to-optimal covering of a Heisenberg ball by Euclidean balls. 252 Z. M. Balogh, M. Rickly, F. Serra Cassano Remark 4.1.Assume that the surface Sis given as the graph of a C1 function f:U→Roveradomain U⊆R2, i.e. S={(x, y, f(x, y)) | (x, y)∈U}. Then p=(x, y, t)∈Sis a characteristic point if and only if ∂f ∂x(x, y)=2yand ∂f ∂y(x, y)=−2x. In what follows we consider characteristic sets C(S)onsuch graphs. We collect the results from [Ba2] needed in the sequel in the following theorem: Theorem 4.2. Given 1<α<2, there exists a compact subset Qαof Q:= [−1 2,1 2]×[−1 2,1 2]and a C1,1smooth map f:Q→R, such that: (i) 0 <Hα E(Qα)<∞. (ii) {(x, y, f(x, y)) |(x, y)∈Qα}⊆C(S), where S={(x, y, f(x, y)) | (x, y)∈Q}. (iii) For any s≥0, there is a constant c=c(s)such that Hs H(A)≤ cHs E(A)whenever A⊆C(S). Remark 4.2.(i) and (ii) are shown in the proof of Theorem 1.4 in [Ba2]. (iii) is essentially Case 2 of Theorem 1.1 in [Ba2] where Ht H,∞(respectively Ht E,∞) has been replaced by Ht H(respectively Ht E). The proof given there only needs very minor changes in our case. The computations can be found in [Ri]. 2. Case 1 <α<2. Our set Aαis obtained from the characteristic set of a surface in the following way: Let Qαand f:Q→Rbe as in Theorem 4.2. Consider the mapping F:Q→R3given by F(x, y)= (x, y, f(x, y)) and set Aα:= F(Qα). Observe that F:(Qα,d E)→ (Aα,d E)isabi-Lipschitz mapping and therefore 1 kHα E(Qα)≤H α E(Aα)≤kHα E(Qα) for some k>0. This shows in particular that Hα E(Aα)>0. We apply Theorem 4.2 to estimate Hα H(Aα). Since Aα⊆C(S), it follows that Hα H(Aα)≤cHα E(Aα)≤ckHα E(Qα). 3. Case α=2.Given 0 <δ<2 arbitrary, consider the set Q2−δand the map Ffrom the Case 2 above. Then for Aα,δ := F(Q2−δ), we have H2−δ E(Aα,δ)>0 and H2 H(Aα,δ)=0byTheorem 4.2. Comparison of Hausdorff Measures 253 4. Case 2 <α<4. Given 0 <δ<1 arbitrary, consider the map F and the set F(Q2−δ) from 2. above. Our set Aα,δ will be a product of F(Q2−δ) with the Cantor set Cα−2 2lying in the vertical axis: Aα,δ := {(x, y, f(x, y)+t)|(x, y)∈Q2−δ,t∈Cα−2 2}. The estimate H1+ α 2−δ E(Aα,δ)>0isagain a consequence of the Euclidean product structure of Aα,δ.Tosee this, consider the map g:Q2−δ×Cα−2 2→Aα,δ;g(x, y, t):=F(x, y)+(0,0,t). gis a bijection, and the inverse is g−1:Aα,δ →Q2−δ×Cα−2 2;g−1(x, y, t):=(x, y, t)−(0,0,f(x, y)). It is easy to see that g:(Q2−δ×Cα−2 2,d E)→(Aα,δ,d E)isbi-Lipschitz. Hence, it is enough to show that H1+ α 2−δ E(Q2−δ×Cα−2 2)>0. But H2−δ E(Q2−δ)>0 and H α−2 2 E(Cα−2 2)>0, by Proposition 4.1 (ii) and Theorem 4.2, so H1+ α 2−δ E(Q2−δ×Cα−2 2)=H(2−δ)+( α−2 2) E(Q2−δ×Cα−2 2)>0 by [Ma, Theorem 8.10 (1)]. Next, we prove that Hα H,∞(Aα,δ)=0. This is a more difficult task since we have to consider coverings with Heisenberg balls. In our argument it is crucial that the shape of Heisenberg balls remains unchanged under vertical translations. Now remember that 0 <λ<1isrelated to Cα−2 2by the equation λα−2 2=1 2and let c:= (1 + λ−1 2)22α−2. Given 0 <<cλ 2, using the compactness of F(Q2−δ) and the fact that H2 H(F(Q2−δ)) = 0, we can cover F(Q2−δ) with Heisenberg balls (BH(pn,r n))n∈{1,...,N}in such a way that N n=1(2rn)2< /c. For each n∈{1,...,N}, let kn∈Nsuch that λkn+1 2≤2rn<λ kn 2 (2rn<(/c)1 2<λfor n=1,...,N by construction). For n=1,...,N and ln=1,...,2kn, let Bn,ln:= {(x, y, t +t)|(x, y, t)∈BH(pn,r n),t ∈Ikn,ln}, where Ikn,lnare the intervals appearing in the construction of the Cantor set Cα−2 2. Notice that Aα,δ ⊆n,lnBn,ln.Inwhat follows we estimate 254 Z. M. Balogh, M. Rickly, F. Serra Cassano the Heisenberg diameter of Bn,ln: Let (x, y, t),(x,y,t )∈BH(pn,r n) and let t1,t 2∈Ikn,ln. Then dH((x, y, t +t1),(x,y,t +t2)) =(x−x, y−y,(t−t)+(t2−t1)−2xy+2xy)H =(x−x, y−y,(t−t)−2xy+2xy)∗(0,0,t 2−t1)H ≤dH((x, y, t),(x,y,t )) + |t2−t1|1 2 ≤2rn+λkn 2. This computation gives diamH(Bn,ln)≤2rn+λkn 2.Now:  n,ln (diamH(Bn,ln))α = n,ln (diamH(Bn,ln))2(diamH(Bn,ln))α−2 ≤ n,ln (2rn+λkn 2)2(2rn+λkn 2)α−2≤ n,ln ((1 + λ−1 2)2rn)2(2λkn 2)α−2 = n,ln (1 + λ−1 2)2(2rn)22α−21 2kn = n (1 + λ−1 2)2(2rn)22α−2 =(1+λ−1 2)22α−2 n (2rn)2=c n (2rn)2<c c=. Hence, for an arbitrary 0 <<cλ 2, there exists a covering (Bn,ln)n,ln (n=1,...,N,ln=1,...,2kn)ofAα,δ such that n,ln (diamH(Bn,ln))α< . 5. Case α=4.This, again, is trivial. Take A4:= BE(0,1). Then 0<H3 E(A4)<∞and 0 <H4 H(A4)<∞. 5. An application to H-rectifiability In this section we will compare the classical notion of 2-Euclidean rectifiability and the notion of H-rectifiability introduced in [FSSC1]. Let us recall the classical notion of m-rectifiability in a general metric space due to Federer ([Fe, 3.2.14]): Given a metric space (X, d) and a Comparison of Hausdorff Measures 255 positive integer m,wesay that a Borel set E⊆Xis m-rectifiable if there exists a countable collection of Lipschitz maps fi:Ai⊆(Rm,d E)→ (X,d) such that Hm dE\∞  i=1 fi(Ai)=0. An m-dimensional Euclidean rectifiable set is an m-rectifiable subset of (Rn,d E). Remark 5.1.It is well known that m-dimensional Euclidean rectifiability is equivalent to the requirement that Hm dE-almost all of the set can be covered by a sequence of C1m-graphs of Rn(see [AFP, Chapter 2, Section 2.9]). It turned out however that this notion of rectifiability is not appropriate in the setting of the Heisenberg group endowed with the Heisenberg metric. Indeed, Ambrosio and Kirchheim (see [AK, Theorem 7.2]) proved that the Heisenberg group (H,d H)ispurely m-unrectifiable for m=2,3,4, i.e. that for any Lipschitz map f:A⊆(Rm,d E)→(H,d H) one has Hm H(f(A)) = 0. This lack of rectifiable sets in the classical sense suggests that more intrinsic definitions of rectifiability could be useful instead. To this aim, in [FSSC1], an intrinsic definition of rectifiability in the Heisenberg group was introduced in the codimension one case. This was successfully used to study the structure of the sets of intrinsic finite perimeter (see [FSSC1], [FSSC2], [FSSC3]). The idea in [FSSC1]was to replace the images of Lipschitz mappings in Federer’s definition by surfaces given as level sets of H-differentiable functions. This led to the following definition: Definition 5.1. We shall say that S⊆His 3-dimensional H-rectifiable if there exists a sequence of H-regular hypersurfaces (Si)i∈Nsuch that H3 HS\ i∈N Si=0. In the above definition the notion of H-regular hypersurface appears. Here S⊆R3≡His called an H-regular hypersurface if it is locally the level set of an H-regular function f:H→R.H-regularity of fmeans that the horizontal gradient ∇Hf:= (Xf,Yf): H→R2 256 Z. M. Balogh, M. Rickly, F. Serra Cassano of fexists, is continuous and non-vanishing on S.Inparticular, if Sis a Euclidean C1regular surface without characteristic points, then Sis H-regular. It was proved in [FSSC1] that the essential boundary of a set Eof locally finite H-perimeter in His 3-dimensional H-rectifiable. The following comparison result between 2-dimensional Euclidean rectifiability and 3-dimensional H-rectifiability is in order: Theorem 5.1. (i) Each 2-dimensional Euclidean rectifiable set S⊆R3≡His 3-dimensional H-rectifiable. (ii) There are 3-dimensional H-rectifiable sets S⊆R3≡Hthat are not 2-dimensional Euclidean rectifiable. Proof: (i) By Remark 5.1, given a 2-dimensional Euclidean rectifiable subset S⊆R3,wecan assume without loss of generality that S=N∪ i∈N Si where H2 E(N)=0andSi⊆R3is a Euclidean C1regular hypersurface. Let ˜ Si:= Si\C(Si), where C(Si) denotes the set of characteristic points of Si. Then ˜ Siis an H-regular hypersurface, and S=˜ N∪ i∈N ˜ Si with ˜ N=N∪ i∈N C(Si). By Theorem 1.1 with α=2wehave H3 H(N)=0. According to a result from [Ba2], if Sis a Euclidean C1regular hypersurface, then H3 H(C(S)) = 0. This gives H3 H(C(Si)) = 0 for all i∈N. Hence H3 H(˜ N)=0and consequently Sis 3-dimensional H-rectifiable. (ii) Given 0 <δ<0.5, we know by Theorem 1.2 (ii) that there is a compact set N=A3,δ ⊆Hsuch that H3 H(N)=0 but H2.5−δ E(N)>0. 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Semmes,Onthe nonexistence of bi-Lipschitz parameterizations and geometric problems about A∞-weights, Rev. Mat. Iberoamericana 12(2) (1996), 337–410. [Str] R. S. Strichartz, Self-similarity on nilpotent Lie groups, in: “Geometric analysis” (Philadelphia, PA, 1991), Contemp. Math. 140, Amer. Math. Soc., Providence, RI, 1992, pp. 123–157. Comparison of Hausdorff Measures 259 Zolt´an M. Balogh: Institute of Mathematics University of Bern Sidlerstrasse 5 3012 Bern Switzerland E-mail address:[email protected] Matthieu Rickly: Institute of Mathematics University of Bern Sidlerstrasse 5 3012 Bern Switzerland E-mail address:[email protected] Francesco Serra Cassano: Dipartimento di Matematica Universit`adiTrento Via Sommarive 14 38050 Povo (Trento) Italy E-mail address:[email protected] Rebut el 4 de juny de 2002.