SINGULAR INTEGRALS ON SELF-SIMILAR SETS AND REMOVABILITY FOR LIPSCHITZ HARMONIC FUNCTIONS IN HEISENBERG GROUPS

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1 SINGULAR INTEGRALS ON SELF-SIMILAR SETS AND REMOVABILITY FOR LIPSCHITZ HARMONIC FUNCTIONS IN HEISENBERG GROUPS VASILIS CHOUSIONIS AND PERTTI MATTILA Abstract. In this paper we study singular integrals on small (that is, measure zero and lower than full dimensional) subsets of metric groups. The main examples of the groups we have in mind are Euclidean spaces and Heisenberg groups. In addition to obtaining results in a very general setting, the purpose of this work is twofold; we shall extend some results in Euclidean spaces to more general kernels than previously considered, and we shall obtain in Heisenberg groups some applications to harmonic (in the Heisenberg sense) functions of some results known earlier in Euclidean spaces. 1. Introduction The Cauchy singular integral operator on one-dimensional subsets of the complex plane has been studied extensively for a long time with many applications to analytic functions, in particular to analytic capacity and removable sets of bounded analytic functions. There have also been many investigations of the same kind for the Riesz singular integral operators with the kernel x/ x m 1 on m-dimensional subsets of R n. One of the general themes has been that boundedness properties of these singular integral operators imply some geometric regularity properties of the underlying sets, see, e.g., [DS], [M1], [M3], [Pa], [T] and [M5]. Standard self-similar Cantor sets have often served as examples where such results were first established. This tradition was started by Garnett in [G1] and Ivanov in [I1] who used them as examples of removable sets for bounded analytic functions with positive length. Later studies of such sets include [G], [J], [JM], [I], [M], [MTV1], [MTV] and [GV] in connection with the Cauchy integral in the complex plane, [MT] and [T4] in connection with the Riesz transforms in higher dimensions, and [D] and [C] in connection with other kernels. In this paper we shall first derive criteria for the unboundedness of very general singular integral operators on self-similar subsets of metric groups with dilations and then give explicit examples in Euclidean spaces and Heisenberg groups on which these criteria can be checked. Today quite complete results are known for the Cauchy integral and for the removable sets of bounded analytic functions. The new progress started from Melnikov s discovery in [Me] of the relation of the Cauchy kernel to the so-called Menger curvature. This relation was applied by Melnikov and Verdera in [MeV] to obtain a simple proof of the L - boundedness of the Cauchy singular integral operator on Lipschitz graphs, and in [MMV] in order to get geometric characterizations of those Ahlfors-David regular sets on which 000 Mathematics Subject Classification. Primary 4B0,8A75. Key words and phrases. Singular integrals, self-similar sets, removability, Heisenberg group. Both authors were supported by the Academy of Finland. 1

2 VASILIS CHOUSIONIS AND PERTTI MATTILA the Cauchy singular integral operator is L -bounded and of those which are removable for bounded analytic functions. This progress culminated in David s characterization in [D1] of removable sets of bounded analytic functions among sets of finite length as those which intersect every rectifiable curve in zero length, and in Tolsa s complete Menger curvature integral characaterization in [T1] of all removable sets of bounded analytic functions. Much less known is known in higher dimensions for the Riesz transforms and removable sets for Lipschitz harmonic functions, for some results, see e.g. [MPa], [M3], [Vi], [L], [Vo], [T3], [T4] and [ENV]. There are various reasons why the Lipschitz harmonic functions are a natural class to study, one of them is that by Tolsa s result in the plane the removable sets for bounded analytic and Lipschitz harmonic functions are exactly the same. Also the analog for the Lipschitz harmonic functions of the above mentioned David s result for sets of finite length was first proved in [DM]. In [CM] analogs of the results in [MPa] and [M3] were proven in Heisenberg groups for Riesz-type kernels. They imply in particular that the operators are unbounded on many self-similar fractals. An unsatisfactory feature is that these kernels are not related to any natural function classes in Heisenberg groups in the same way as the Riesz kernels are related to harmonic functions in R n. This is one of the main reasons why we wanted to study more general kernels in this paper. Our kernels are now such that they include the (horizontal) gradient of the fundamental solution of the sub-laplacian (or Kohn- Laplacian) operator which is exactly what is needed for the applications to the related harmonic functions. For many other recent developments on potential theory related to sub-laplacians, see [BLU] and the references given there. We shall now give a brief description of the main results of the paper. In Section we study a general metric group G which is equipped with natural dilations δ r : G G, r > 0. All Carnot groups are such. For a kernel K : G G \ {(x, y) : x = y} R and a finite Borel measure µ on G the maximal singular integral operator TK is defined by TK(f)(x) = sup K(x, y)f(y)dµy. ε>0 G\B(x,ε) Suppose that C = N S i(c) is a self-similar Cantor set generated by the similarities S i = τ qi δ ri, i = 1,..., N, where τ qi is the left translation by q i G and r i (0, 1). Let s > 0 be the Hausdorff dimension of C and suppose that the kernel K := K Ω is s-homogeneous: K Ω (x, y) = Ω(x 1 y), x, y G \ {(x, y) : x = y}, d(x, y) s where Ω : G R is a continuous and homogeneous function of degree zero, that is, Ω(δ r (x)) = Ω(x) for all x G, r > 0. In Theorem.3 we prove that if there exists a fixed point x for some iterated map S w := S i1 S in such that K Ω (x, y)dh s y 0 C\S w(c)

3 SINGULAR INTEGRALS, SELF-SIMILAR SETS AND REMOVABILITY IN H n 3 then TK Ω is unbounded in L (H s C), where H s C is the restriction of the s-dimensional Hausdorff measure H s to C. We shall give a simple example in the plane where this criterion can be applied. In Section 3 we shall study in the Heisenberg group H n removable sets for the H - harmonic functions, that is, solutions of the sub-laplacian equation H f = 0. As in the classical case in R n, see [Ca], the removable sets for the bounded H -harmonic functions can be characterized as polar sets or the null-sets of a capacity with the critical Hausdorff dimension Q where Q = n + is the Hausdorff dimension of H n, see Remark of [BLU]. We shall verify in Theorem 3.13 that the critical dimension for the Lipschitz H -harmonic functions is Q 1, in accordance with the classical case, by proving that for a compact subset C of H n, C is removable, if H Q 1 (C) = 0, and C is not removable, if the Hausdorff dimension dim C > Q 1. For this and the later applications to selfsimilar sets, we need a representation theorem for Lipschitz functions which are H - harmonic outside a compact set C with finite (Q 1)-dimensional Hausdorff measure. This is given in Theorem 3.1 and it tells us that such a function can be written in a neighborhood of C as a sum of a H -harmonic function and a potential whose kernel is the fundamental solution of H. Finally in Section 4 we present a family of self-similar Cantor sets with positive and finite (Q 1)-dimensional Hausdorff measure which are removable for Lipschitz H -harmonic functions. Throughout the paper we will denote by A constants which may change their values at different occurrences, while constants with subscripts will retain their values. We remark that as usual the notation x y means x Ay for some constant A depending only on structural constants, that is, the dimension n, the regularity constants of certain measures and the constant arising from the global equivalence of the metrics d and d c defined in Section 3. We would like to thank the referee for many useful comments.. Singular Integrals on self-similar sets of metric groups Throughout the rest of this section we assume, as in [M4], that (G, d) is a complete separable metric group with the following properties: (i) The left translations τ q : G G, τ q (x) = q x, x G, are isometries for all q G. (ii) There exist dilations δ r : G G, r > 0, which are continuous group homomorphisms for which, (a) δ 1 = identity, (b) d(δ r (x), δ r (y)) = rd(x, y) for x, y G, r > 0, (c) δ rs = δ r δ s. It follows that for all r > 0, δ r is a group isomorphism with δr 1 = δ 1. The closed and open balls with respect to d will be denoted by B(p, r) and U(p, r). By d(e) we will denote the diameter of E G with respect to the metric d. r

4 4 VASILIS CHOUSIONIS AND PERTTI MATTILA We denote by H s, s 0, the s-dimensional Hausdorff measure obtained from the metric d, i.e. for E G and δ > 0, H s (E) = sup δ>0 Hδ s (E), where { Hδ(E) s = inf d(e i ) s : E } E i, d(e i ) < δ. i i In the same manner the s-dimensional spherical Hausdorff measure for E G is defined as S s (E) = sup δ>0 Sδ s (E), where { Sδ s (E) = inf ri s : E } B(p i, r i ), r i δ, p i G. i i Translation invariance and homogeneity under dilations of the Hausdorff measures follows as usual, therefore for A G, p G and s, r 0, H s (τ p (A)) = H s (A) and H s (δ r (A)) = r s H s (A) and the same relations hold for the spherical Hausdorff measures as well. Let µ be a finite Borel measure on G and let a Borel measurable K : G G \ {(x, y) : x = y} R be a kernel which is bounded away from the diagonal, i.e., K is bounded in {(x, y) : d(x, y) > δ} for all δ > 0. The truncated singular integral operators associated to µ and K are defined for f L 1 (µ) and ε > 0 as, T ε (f)(y) = K(x, y)f(y)dµy, G\B(x,ε) and the maximal singular integral is defined as usual, TK(f)(x) = sup T ε (f)(x). ε>0 We are particularly interested in the following class of kernels. Definition.1. For s > 0 the s-homogeneous kernels are of the form, K Ω (x, y) = Ω(x 1 y), x, y G \ {(x, y) : x = y}, d(x, y) s where Ω : G R is a continuous and homogeneous function of degree zero, that is, Ω(δ r (x)) = Ω(x) for all x G, r > 0. In the classical Euclidean setting homogeneous kernels have been studied widely, see e.g. [Gr]. The Hilbert transform in the line, the Cauchy transform in the complex plane and the Riesz transforms in higher dimensions are the best known singular integrals associated to homogeneous kernels. In [H] Huovinen studied general one-dimensional homogeneous kernels in the plane. In R n the lower dimensional coordinate s-riesz kernels, Rs(x, i y) = x i y i, s (0, n), i = 1,..., n, x y s+1 are often studied in conjunction with Ahlfors-David regular measures:

5 SINGULAR INTEGRALS, SELF-SIMILAR SETS AND REMOVABILITY IN H n 5 Definition.. A Borel measure µ on a metric space X is Ahlfors-David regular, or AD-regular, if for some positive numbers s and A, r s /A µ(b(x, r)) Ar s for all x spt µ, 0 < r < d(spt µ), where spt µ stands for the support of µ. The related central open question in R n asks if the L (µ)-boundedness of the s-riesz transforms, s N [1, n), forces the support of µ to be s-uniformly rectifiable or even simply s-rectifiable. In the case of s = 1 it was answered positively in [MMV], and it remains an open problem for s > 1. It originates from the fundamental work of David and Semmes, see e.g. [DS], and it can be heuristically understood in the following sense: Does the L (µ)-boundedness of Riesz transforms impose a certain geometric regularity on the support of µ? In order to achieve a better understanding for this problem, it is very natural to examine the behavior of Riesz transforms on fractals like self-similar sets. This is because although geometrically irregular they retain some structure. It should be expected that s-riesz transforms cannot be bounded on typical self similar sets. Indeed this is the case as follows by results proved in [M3] and [Vi]. In [CM] it was shown that an analogous result holds true even in the setting of Heisenberg groups. On the other hand it is not known if singular integrals associated to more general s-homogeneous kernels are unbounded on s-dimensional self-similar sets. In this direction Theorem.3 provides one criterion for unboundedness for homogeneous singular integrals valid in the general setting of this section. Let S = {S 1,..., S N }, N, be an iterated function system (IFS) of similarities of the form (.1) S i = τ qi δ ri where q i G, r i (0, 1) and i = 1,..., N. The self-similar set C is the invariant set with respect to S, that is, the unique non-empty compact set such that C = N S i (C). The invariant set C will be called a separated self-similar set whenever the sets S i (C) are pairwise disjoint for i = 1,..., N. It follows by a general result of Schief in [S] that separated Cantor sets satisfy 0 < H d (C) < for N ri d = 1, and the measure H d C is d-ad regular. We denote by I the set of all finite words w = (i 1,..., i n ) {1,..., N} n with n 0. Given any word w = (i 1,..., i n ) I its length is denoted by w = n and for m n, w m = (i 1,..., i m ). We also adopt the following conventions: S w := S i1 S in and C w = S w (C).

6 6 VASILIS CHOUSIONIS AND PERTTI MATTILA The fixed points of S are exactly those x C such that S w (x) = x for some w I. In this case {x} = S w k(c) k=1 where w k = k w and w k = (i 1,..., i n,......, i 1,..., i n ). Theorem.3. Let S = {S 1,..., S N } be an iterated function system in G generating a separated s-dimensional self-similar set C and let K Ω be an s-homogeneous kernel. If there exists a fixed point x for S, {x} = S w k(c), such that k=1 C\C w K Ω (x, y)dh s y 0, then the maximal operator T K Ω is unbounded in L (H s C), moreover T K Ω (1) L (H s C) =. Proof. Let µ = H s C which as explained earlier satisfies r s /A µ µ(b(x, r)) A µ r s for all x C, 0 < r < d(c). Without loss of generality we can assume that C\C w K Ω (x, y)dµy = η > 0. Notice that the homogeneity of Ω implies that for all v I, (.) Ω(S v (x) 1 S v (y)) = Ω(δ ri1...r i v (x 1 y)) = Ω(x 1 y). Therefore for all k N, after changing variables y = S w k(z) and recalling that S w k(x) = x, Ω(x 1 y) K Ω (x, y)dµy = dµy C w k \C w k+1 C w k \C w d(x, y) s k+1 Ω(x 1 S = wk(z)) (r C\C w d(x, S w k(z)) s i1... r i w ) ks dµz 1 Ω(S wk(s (x)) 1 S w = k w k(z)) C\C w d(s w k(s 1 (r (x)), S w k w k(z)) s i1... r i w ) ks dµz 1 Ω(S (x) 1 z) w = k C\C w d(s 1 dµz (x), z) w s k Ω(x 1 z) = dµz C\C w d(x, z) s = η.

7 SINGULAR INTEGRALS, SELF-SIMILAR SETS AND REMOVABILITY IN H n 7 Figure A. The set C is split into cylinders C,i having the same length as the grey shaded cylinder C, which is contained in C 1. Let M be an arbitrary big positive number and choose m N such that mη > M. Then m 1 K Ω (x, y)dµy = K Ω (x, y)dµy > M. C\C w m i=0 C w i\c w i+1 Therefore by the continuity of K Ω away from the diagonal there exist m, m N, m < m, such that (.3) K Ω (p, y)dµy > M for all p C. w m C\C w m To simplify notation let C 1 = C w m and C = C w m. Then C \ C = j C,i where the C,i s are cylinder sets belonging to the same generation with C, i.e., for all i = 1,..., j, C,i = C vi for some v i I with v i = w m, see Figure A for the case of a Cantor set in the plane. Let S,i, i = 1,..., j, be the iterated similarities such that C,i = S,i (C) and denote C,i 1 = S,i (C 1 ) and C,i = S,i (C ). Then exactly as before for i = 1,..., j, and p C,i, K Ω (p, y)dµy = K Ω (S 1,i (p), y)dµy > M C\C 1 C,i \C 1,i by (.3) since S 1,i (p) C. Continuing the same splitting process, we can write for n 3,

8 8 VASILIS CHOUSIONIS AND PERTTI MATTILA C \ where for all 3 k n: (.4) ( C j C,i j n 1 C n 1,i ) = j n C n,i, (i) The C k,i s for i = 1,..., j k, are cylinder sets in the same generation with any C k 1,i, i = 1,..., j k 1. (ii) C 1 k,i = S k,i(c 1 ), i = 1,..., j k where by S k,i we denote the iterated map such that S k,i (C) = C k,i. (iii) For all p C k,i = S k,i(c ), C k,i \C 1 k,i K Ω (p, y)dµy > M. Next we define the cylindrical maximal operator (.5) TC(f)(p) = sup K Ω (p, y)f(y)dµy v,w I C w\c v p C v C w for p C and f L 1 (µ). It follows by (.4) that for every n N, n, (.6) T C(1)(p) > M for p C n jk k= Ck,i. Let λ = µ(c ) < 1. Since µ(c \C ) = (1 λ)µ(c) it follows easily that for n N, n, µ(c) µ ( C \ ( C n k= j k Ck,i )) If n is chosen large enough such that (1 λ) n < 1, = (1 λ) n µ(c). µ({p C : T C(1)(p) > M}) µ(c n k= j k C k,i) > 1 µ(c). This implies that T C(1) L (µ) M and (TC(1)) dµ > M µ(c). Since M can be selected arbitrarily big, TC (1) L (µ) = and the operator TC is unbounded in L (µ). Notice that there exists a constant α C > 0, depending only on the set C, such that for every v I (.7) dist(c v, C \ C v ) α C d(c v ).

9 SINGULAR INTEGRALS, SELF-SIMILAR SETS AND REMOVABILITY IN H n 9 To see this first notice that since the sets C i = S i (C), i = 1,..., N, are disjoint there exists some α C > 0 such that dist(c i, C \ C i ) α C d(c i ). Hence for all v = (i 1,..., i v ) I, (.8) dist(c v, C v v 1 \ C v ) = dist(s v v 1 (S i v (C)), S v v 1 (C \ S i v (C))) = r i1... r i v 1 dist(c \ S i v (C)) α C r i1... r i v 1 d(s i v (C)) = α C r i1... r i v d(c) = d(c v ). Furthermore C \ C v = v j=1 C v j 1 \ C v j and this union is disjoint. Therefore using (.8) dist(c v, C \ C v ) = min dist(c v, C v j 1 \ C v j ) j=1,..., v min j=1,..., v dist(c v j, C v j 1 \ C v j ) α C min d(c v j ) j=1,..., v = α C d(c v ). Lemma.4. There is a constant A C depending only on the Cantor set C and the kernel K Ω such that for all w, v I and p H n for which C v C w and dist(p, C v ) α C d(c v ), K Ω (p, y)dµy K Ω (p, y)dµy + A C. C w\c v B(p, d(c w))\b(p, d(c v)) Proof. We can always assume that α C 1 and hence for p C v := {p : dist(p, C v ) α C d(c v )} (.9) C w \ C v = (C w \ B(p, d(c v ))) ((B(p, d(c v )) \ C v ) C w ) and (.10) B(p, d(c w )) \ B(p, d(c v )) = (B(p, d(c w )) \ (C w B(p, d(c v )) (C w \ B(p, d(c v ))). Using (.10) we replace the term C w \ B(p, d(c v )) in (.9) and we estimate K Ω (p, y)dµy K Ω (p, y)dµy C w\c v B(p, d(c w))\b(p, d(c v)) + K Ω (p, y)dµy B(p, d(c w))\(c w B(p, d(c v)) + K Ω (p, y)dµy. (B(p, d(c v))\c v) C w Let, I 1 = K Ω (p, y)dµy and I = K Ω (p, y)dµy. B(p, d(c w))\(c w B(p, d(c v)) (B(p, d(c v))\c v) C w Notice that C v C w implies that C v C w. Hence for p C v by (.7) d(p, C \ C w ) d(c w, C \ C w ) d(p, C w ) α C d(c w).

10 10 VASILIS CHOUSIONIS AND PERTTI MATTILA Therefore we can now estimate Ω L (µ) I 1 B(p, d(c w))\c w d(p, y) dµy Ω L (µ) µ(b(p, d(c w)) s s α s 4s A µ Ω L (µ). C d(c w ) s α s C Notice also that if p C v and y B(p, d(c v )) \ C v spt µ then d(p, y) α C d(c v ). Hence in the same way Ω L (µ) I B(p, d(c v))\c v d(p, y) dµy 4s A µ Ω L (µ). s α s C Lemma.4 implies that for all p C, T C(1)(p) T K Ω (1)(p) + A C, therefore T K Ω (1) L (µ) = and T K Ω is unbounded in L (µ). Remark.5. Even when the ambient space is Euclidean, Theorem.3 provides new information about the behavior of general homogeneous singular integrals on self-similar sets. For example it follows easily that the operator associated to the kernel z 3 / z 4, z C \ {0}, is unbounded on many simple 1-dimensional self-similar sets, perhaps the most recognizable among them being the Sierpiński gasket. Furthermore for any kernel K Ω (x) = Ω(x/ x ) x s, x R n \ {0}, s (0, n), where Ω is continuous one can easily find Sierpiński-type s-dimensional self-similar sets C s for which one can check using Theorem.3 that the corresponding operator T KΩ is unbounded. 3. H -removability and singular integrals For an introduction to Heisenberg groups, see for example [CDPT] or [BLU]. Below we state the basic facts needed in this paper. The Heisenberg group H n, identified with R n+1, is a non-abelian group where the group operation is given by, ( ) n p q = p 1 + q 1,, p n + q n, p n+1 + q n+1 (p i q i+n p i+n q i ). We will also denote points p H n by p = (p, p n+1 ), p R n, p n+1 R. Recall that for any q H n and r > 0, the left translations τ q : H n H n are given by τ q (p) = q p. Furthermore we define the dilations δ r : H n H n by These dilations are group homomorphisms. A natural metric d on H n is defined by where δ r (p) = (rp 1,..., rp n, r p n+1 ). d(p, q) = p 1 q p = ( (p 1,..., p n ) 4 R n + p n+1) 1 4.

11 SINGULAR INTEGRALS, SELF-SIMILAR SETS AND REMOVABILITY IN H n 11 The metric is left invariant, that is d(q p 1, q p ) = d(p 1, p ), and the dilations satisfy d(δ r (p 1 ), δ r (p )) = rd(p 1, p ). The (n + 1)-dimensional Lebesgue measure L n+1 on H n is left and right invariant and it is a Haar measure of the Heisenberg group. We stress that although the topological dimension of H n is n + 1 the Hausdorff dimension of (H n, d) is Q := n +, see e.g. [BLU], , which is also called the homogeneous dimension of H n. The Lie algebra of left invariant vector fields in H n is generated by X i := i + x i+n n+1, Y i := i+n x i n+1, T := n+1, for i = 1,..., n. The vector fields X 1,..., X n, Y 1,..., Y n define the horizontal subbundle HH n of the tangent vector bundle of R n+1. For every point p H n the horizontal fiber is denoted by HH n p and can be endowed with the scalar product, p and the corresponding norm p that make the vector fields X 1,..., X n, Y 1,..., Y n orthonormal. Often when dealing with two sections ϕ and ψ whose argument is not stated explicitly we will use the notation ϕ, ψ instead of ϕ, ψ p. Therefore for p, q H n, p, q = p, q R n and p = p n. Furthermore for a given p H n we define the projections n n π p (q) = q i X i (p) + q i+n Y i (p) for q H n. Definition 3.1. An absolutely continuous curve γ : [0, T ] H n will be called sub-unit, with respect to the vector fields X 1,..., X n, Y 1,..., Y n, if there exist real measurable functions a 1 (t),..., a n (t), t [0, T ], such that n j=1 a j(t) 1 and n n γ(t) = a j (t)x j (γ(t)) + a j+n (t)y j (γ(t)), for a.e. t [0, T ]. j=1 Definition 3.. For p, q H n their Carnot-Carathéodory distance is j=1 d c (p, q) = inf{t > 0 : there is a subunit curve γ : [0, T ] H n such that γ(0) = p and γ(t ) = q}. Remark 3.3. It follows by Chow s theorem that the above set of curves joining p and q is not empty and hence d c is a metric on H n. Furthermore the infimum in the definition can be replaced by a minimum. See [BLU] for more details. As well as with d the metric d c is left invariant and homogeneous with respect to dilations, see, for example, Propositions 5..4 and 5..6 of [BLU]. The closed and open balls with respect to d c will be denoted by B c (p, r) and U c (p, r). The following result is well known and can be found for example in [BLU] and [CDPT]. Proposition 3.4. The Carnot-Carathéodory distance d c is globally equivalent to the metric d. If f is a real function defined on an open set of H n its H-gradient is given by H f = (X 1 f,..., X n f, Y 1 f,..., Y n f).

12 1 VASILIS CHOUSIONIS AND PERTTI MATTILA The H-divergence of a function φ = (φ 1,..., φ n ) : H n R n is defined as n div H φ = (X i φ i + Y i φ i+n ). The sub-laplacian in H n is given by or equivalently H = n (X i + Y i ) H = div H H. Definition 3.5. Let D H n be an open set. A real valued function f C (D) is called H -harmonic, or simply harmonic, on D if H f = 0 on D. Actually, the assumption f C (D) is superfluous, since even the distributional solutions of H f = 0 are C, see [BLU]. Definition 3.6. Let D be an open subset of H n. We say that f : D R is Pansu differentiable at p D if there exists a homomorphism L : H n R such that f(τ p (δ r ν)) f(p) lim = L(ν) r 0 + r uniformly with respect to ν belonging to some compact subset of H n. Furthermore, L is unique and we write L := d H f(p). The proof of the following proposition, as well as a comprehensive discussion about calculus in H n, can be found in [FSSC1]. Proposition 3.7. If f is Pansu differentiable at p, then d H f(p)(ν) = H f(p), π p (ν) p. We shall consider removable sets for Lipschitz solutions of the sub-laplacian: Definition 3.8. A compact set C H n will be called removable, or H -removable for Lipschitz H -harmonic functions, if for every domain D with C D and every Lipschitz function f : D R, H f = 0 in D \ C implies H f = 0 in D. As usual we denote for any D H n and any function f : D R, Lip(f) := sup x,y D f(x) f(y), d(x, y) and we will also use the following notation for the upper bound for the Lipschitz constants in Carnot-Carathéodory balls: Lip B (f) := sup{lip(f Uc(p,r)) : p D, r > 0, U c (p, r) D}. The following proposition is known. It follows, for example, from the Poincaré inequality, see Theorem 5.16 in [CDPT] and the arguments for its proof on pages It is also essentially contained in a more general setting in Theorems 1.3 and 1.4 of [GN]. However, we prefer to give a simple direct proof.

13 SINGULAR INTEGRALS, SELF-SIMILAR SETS AND REMOVABILITY IN H n 13 Proposition 3.9. Let D H n be a domain and let f C 1 (D). Then Lip B (f) < if and only if H f <. More precisely, there is a constant c(n) depending only on n such that (3.1) H f Lip B (f) c(n) H f. Proof. By Pansu s Rademacher type theorem, see [Pan], f is a.e. Pansu differentiable in D. Let q be a point where f is Pansu differentiable, then for all ν H n, d H f(q)(ν) = lim f(q δ r (ν)) f(q) r 0 + r Lip B(f) ν. By Proposition 3.7, d H f(q)(ν) = H f(q), π q (ν) q = H f(q), ν R n, and choosing ν = ( H f(q), 0) we get H f(q) Lip B (f), and so H f Lip B (f). On the other hand we check that if H f <, then Lip B (f) <. For any q D there exists a radius r q such that U c (q, r q ) D. Then by the definition of the Carnot-Carathéodory metric for any p U c (q, r q ) there exists a subunit curve γ : [0, T ] U c (q, r q ), as in Definition 3.1, such that γ(0) = q, γ(t ) = p and T = d c (q, p). Then, T f(q) f(p) = d 0 dt (f(γ(t)))dt T = f(γ(t)), γ(t) dt 0 T n a j (t) f(γ(t)), X j (γ(t)) + a j+n (t) f(γ(t)), Y j (γ(t)) dt = 0 j=1 ( T n 0 T 0 T 0 ( j=1 ) 1 ( ) 1 n a j (t) f(γ(t)), X j (γ(t)) + f(γ(t)), Y j (γ(t)) dt j=1 n (X j f(γ(t))) + (Y j f(γ(t))) ) 1 dt j=1 H f(γ(t)) dt H f d c (q, p), where in the fourth line we used that f(γ(t)), X j (γ(t)) = X j (f)(γ(t)) and f(γ(t)), Y j (γ(t)) = Y j (f)(γ(t)). The inequality Lip B (f) c(n) H f follows from this and Proposition 3.4. Fundamental solutions for sub-laplacians in homogeneous Carnot groups are defined in accordance with the classical Euclidean setting. In particular in the case of the sub- Laplacian in H n : Definition 3.10 (Fundamental solutions). A function Γ : R n+1 \{0} R is a fundametal solution for H if: (i) Γ C (R n+1 \ {0}), (ii) Γ L 1 loc (Rn+1 ) and lim p Γ(p) 0,

14 14 VASILIS CHOUSIONIS AND PERTTI MATTILA (iii) for all ϕ C 0 (R n+1 ), R n+1 Γ(p) H ϕ(p)dp = ϕ(0). It also follows easily, see Theorem and Proposition of [BLU], that for every p H n, (3.) Γ H ϕ(p) = ϕ(p) for all ϕ C 0 (R n+1 ). Convolutions are defined as usual by f g(p) = f(q 1 p)g(q)dq for f, g L 1 and p H n. One very general result due to Folland, see [Fo], guarantees that there exists a fundamental solution for all sub-laplacians in homogeneous Carnot groups with homogeneous dimension Q >. In particular the fundamental solution Γ of H is given by Γ(p) = C Q p Q for p H n \ {0} where Q = n + is the homogeneous dimension of H n. The exact value of C Q can be found in [BLU]. Let K = H Γ, then K = (K 1,..., K n ) : H n R n where (3.3) K i (p) = c Q p i p + p i+n p n+1 p Q+ and K i+n (p) = c Q p i+n p p i p n+1 p Q+, for i = 1,..., n, p H n \{0} and c Q = ( Q)C Q. We will also use the following notation, (3.4) Ω i (p) = c Q p i p + p i+n p n+1 p 3 and Ω i+n (p) = c Q p i+n p p i p n+1 p 3, for i = 1,..., n and p H n \ {0}. Hence, (3.5) K i (p) = Ω i(p) and K(p) = Ω(p) p Q 1 p Q 1, for i = 1,..., n, Ω = (Ω 1,..., Ω n ) and p H n \ {0}. It follows that the functions Ω i are homogeneous and hence, recalling Definition.1, the kernels K i are (Q 1)-homogeneous. The following proposition asserts that K is a standard kernel. Proposition For all i = 1,..., n, (i) K i (p) p 1 Q for p H n \ {0}, (ii) H K i (p) p Q for p H n \ {0}, { (iii) K i (p 1 q 1 ) K i (p 1 d(q1, q ) q ) max d(p, q 1 ), d(q } 1, q ) Q d(p, q ) Q for q 1, q p H n. Proof. The size estimate (i) follows immediately by the definition of the kernel K. It also follows easily that for p H n \ {0}, j K i (p) 1 for j, i = 1,..., n, p Q

15 SINGULAR INTEGRALS, SELF-SIMILAR SETS AND REMOVABILITY IN H n 15 and n+1 K i (p) 1 for i = 1,..., n. p Q+1 Hence X i K j (p) 1 p and Y ik Q j (p) 1 for i = 1,..., n, j = 1,..., n, p Q and (ii) follows. For the proof of (iii) let q 1, q p H n. Without loss of generality assume that d c (q 1, p) d c (q, p). We are going to consider two cases. First Case: d c (q 1, q ) 1d c(q 1, p) Since d c is globally equivalent to d we can use (i) to obtain K i (p 1 q 1 ) K i (p 1 q ) 1 d c (q 1, p) + 1 Q 1 d c (q, p) Q 1 d c (q 1, p) Q 1 4d c(q 1, q ) d c (q 1, p) Q max { d(q1, q ) d(p, q 1 ) Q, d(q 1, q ) d(p, q ) Q }. Second Case: d c (q 1, q ) < 1 d c(q 1, p) By the definition of the Carnot-Carathéodory metric there is a sub-unit curve γ : [0, d c (q 1, q )] H n such that γ(0) = p 1 q 1 and γ(d c (q 1, q )) = p 1 q. Furthermore, (3.6) γ([0, d c (q 1, q )]) B c (p 1 q 1, d c (q 1, q )). Hence for every t [0, d c (q 1, q )], since by (3.6) d c (γ(t), p 1 q 1 ) d c (q 1, q ), (3.7) γ(t) d c (0, γ(t)) d c (0, p 1 q 1 ) d c (γ(t), p 1 q 1 ) d c (p, q 1 ) d c (q 1, q ) 1 d c(q 1, p).

16 16 VASILIS CHOUSIONIS AND PERTTI MATTILA Therefore if T = d c (q 1, q ) we can estimate as in Proposition 3.9 for i = 1,..., n: K i (p 1 q 1 ) K i (p 1 q ) = K i (γ(0)) K i (γ(t )) T = d 0 dt (K i(γ(t))dt ( T n ) 1 (X j (K i )(γ(t))) + (Y j (K i )(γ(t))) dt = 0 T 0 T 0 j=1 H K i (γ(t)) dt dt γ(t) Q d c(q 1, q ) d c (q 1, p) Q max where we used (ii) and (3.7) respectively. { d(q1, q ) d(p, q 1 ) Q, d(q 1, q ) d(p, q ) Q }. In the following we prove a representation theorem for Lipschitz harmonic functions of H n outside a compact set of finite H Q 1 measure. Theorem 3.1. Let C be a compact subset of H n with H Q 1 (C) < and let D C be a domain in H n. Suppose f : D R is a Lipschitz function such that H f = 0 in D \ C. Then there exist a bounded domain G, C G D, a Borel function h : C R and a H -harmonic function H : G R such that f(p) = Γ(q 1 p)h(q)dh Q 1 q + H(p) for p G \ C and h L (H Q 1 C) + H H 1. C Proof. It suffices to prove the theorem with H Q 1 replaced by S Q 1. Without loss of generality we can assume that D is bounded. Let D 1 be some domain such that C D 1 D and dist(d 1, H n \ D) > 0. For every m = 1,,... there exists a finite number j m of balls U m,j := U(p m,j, r m,j ) such that U m,j C, (3.8) C j m j=1 Furthermore let G m = jm j=1 U m,j and U m,j D 1, r m,j 1 m and j m j=1 r Q 1 m,j S Q 1 (C) + 1 m. 0 < ε m < min{dist(c, H n \ G m ), dist(d 1, H n \ D)}. By the Whitney-McShane extension Lemma there exists a Lipschitz function F : H n R such that F D = f and F is bounded.

17 SINGULAR INTEGRALS, SELF-SIMILAR SETS AND REMOVABILITY IN H n 17 Let J C0 (R n+1 ), J 0, such that spt J B(0, 1) and J = 1. For any ε > 0 let J ε (x) = ε Q J(δ 1/ε (x)). We consider the following sequence of mollifiers, f m (x) : = F J εm (x) = F (y)j εm (x y 1 )dy (3.9) = F (y 1 x)j εm (y)dy, for x H n. Since F is bounded and uniformly continuous and furthermore for all m N, (i) f m C, (ii) H f m H F <, (iii) f m is harmonic in the set f m F 0 D εm := {p D \ C : dist(p, (D \ C)) > ε m }. It follows from (iii) that every mollifier f m is harmonic in the set D 1 \ G m. We continue by choosing another domain D such that G m D D 1 for all m = 1,,..., and an auxiliary function ϕ C0 (R n+1 ) such that { 1 in D ϕ = 0 in H n \ D 1. For m = 1,,... set g m := ϕf m and notice that g m C 0 (R n+1 ) and H g m A 1 where A 1 does not depend on m. It follows by (3.) that for all m N, (3.10) g m (p) = Γ H g m (p) for all p H n. Notice that (i) g m = 0 in H n \ D 1, (ii) g m = f m in D \ G m and hence H g m = H f m = 0 in D \ G m. Therefore for all m N and p D \ G m, (3.11) f m (p) = Γ(q 1 p) H g m (q)dq + Γ(q 1 p) H g m (q)dq G m D 1 \D by (3.10). For m N set H m : D R, (3.1) H m (p) = Γ(q 1 p) H g m (q)dq D 1 \D and I m : D \ G m R, m = 1,,..., (3.13) I m (p) = Γ(q 1 p) H g m (q)dq. G m Since the functions H g m are uniformly bounded in D 1 \ D, for all m N

18 18 VASILIS CHOUSIONIS AND PERTTI MATTILA (i) H m is harmonic in D, (ii) H H m 1, since H Γ is locally integrable. The functions H m are C by Hörmander s theorem, see for example Theorem 1 in Preface of [BLU]. Thus we can apply Proposition 3.9 and conclude from (ii) that Lip B (H m ) 1. The functions I m can be expressed as, (3.14) I m (p) = div H,q (Γ(q 1 p) H g m (q))dq + H Γ(p 1 q), H g m (q) dq, G m G m where div H,q stands for the H-divergence with respect to the variable q and we also used the left invariance of H and the symmetry of Γ to get that H,q (Γ(q 1 p)) = H,q (Γ(p 1 q)) = H Γ(p 1 q). By the Divergence Theorem of Franchi, Serapioni and Serra Cassano, see [FSSC1] (in particular Corollary 7.7 ), div H,q (Γ(q 1 p) H g m (q))dq G (3.15) m = A Γ(q G 1 p) H g m (q), ν m (q) b(q)ds Q 1 q m where ν m is an S Q 1 -measurable section of HH n such that ν m (q) = 1 for S Q 1 -a.e q G m and b is a non-negative Borel function such that b L (S Q 1 ) A 3. 1 By (3.8), L n+1 (G m ) 0, therefore for p D \ C, (3.16) lim m H Γ(p 1 q), H g m (q) dq 0, G m since H g m is uniformly bounded in D and H Γ is locally integrable. Notice that the signed measures, (3.17) σ m = A H g m ( ), ν m ( ) bs Q 1 G m, have uniformly bounded total variations σ m. This follows by (3.8), as (3.18) σ m A H g m b L (S Q 1 )S Q 1 ( G m ) A 1 A A 3 S Q 1 ( U m,j ) = A 4 j j α(q 1)r Q 1 m,j A 5 (S Q 1 (C) + 1 m ), 1 The divergence theorem in [FSSC1] is stated in terms of the spherical Hausdorff measure S Q 1 with respect to the norm p := max{ p, p n+1 }. Since the corresponding norm d is globally equivallent to d we get that S Q 1 << S Q 1 << S Q 1 and the function b is the Radon-Nikodym derivative ds Q 1 ds Q 1.

19 SINGULAR INTEGRALS, SELF-SIMILAR SETS AND REMOVABILITY IN H n 19 for A 4 := A 1 A A 3, A 5 = α(q 1)A 4 and α(q 1) := S Q 1 ( B(0, 1)). Therefore, by a general compactness theorem, see e.g. [AFP], we can extract a weakly converging subsequence (σ mk ) k N such that σ mk σ. Furthermore spt σ := spt σ C. To see this let p / C. Let δ = dist(p, C) and choose i 0 big enough such that 1/m i0 < δ/4. Then by (3.8), p / G mi for all i i 0. Since spt σ mi G mi and B(p, δ ) G m i =, which implies that p / spt σ. Notice also that by (3.18) (3.19) σ lim inf k σ (U(p, δ/)) lim inf i σ m i (U(p, δ/)) = 0, σ m k A 5 S Q 1 (C). Finally combining (3.14)-(3.17) we get that for p D \ C, lim I m k (p) = Γ(q 1 p)dσq k and by (3.11)-(3.13) f(p) = C C Γ(q 1 p)dσq + lim k H mk (p). Since the sequence of harmonic functions (H mk ) is equicontinuous on compact subsets of D, the Arzela-Ascoli theorem implies that there exists a subsequence (H mkl ) which converges uniformly on compact subsets of D. From the Mean Value Theorem for sub- Laplacians and its converse, see [BLU], Theorems and 5.6.3, we deduce that (H mkl ) converges to a function H which is harmonic in D. Therefore for p D \ C, f(p) = Γ(q 1 p)dσq + H(p). C Furthermore the function H is C in D with Lip B (H) 1, therefore by Proposition 3.9 H H 1. Set µ = S Q 1 C. In order to complete the proof it suffices to show that (3.0) σ µ and h := dσ dµ L (µ). The proof of (3.0) is almost identical with the proof appearing in [MPa] but we provide the details for completeness. It is enough to prove that for every open ball U and its closure U (3.1) σ (U) A 5 µ(u). Then from (3.1) we deduce that for any closed ball B and open balls U i B, U i B, (3.) σ (B) lim i σ (U i ) lim i A 5 µ(u i ) = A 5 µ(b), which implies (3.0).

20 0 VASILIS CHOUSIONIS AND PERTTI MATTILA Suppose that there exist an open ball U and a positive number ε such that (3.3) σ(u) > A 5 (µ(u) + ε). In case C U, (3.19) implies that σ (U) A 5 µ(u) therefore we can assume that C \ U. There exists a compact set F such that (3.4) F C \ U and µ(f ) > µ(c \ U) ε 4. Let δ ε := dist(f, U) and choose k N large enough such that 1/m k < min{δ ε /4, ε/}. Then by (3.8) (3.5) max j j mk r mk,j 1 m k < δ ε 4 and j mk j=1 r Q 1 m k,j µ(c) + 1 m k. Let J 1 k = {j : U mk,j U }, J k = {j : U mk,j F }. It follows that F j J k U mk,j, therefore j Jk enough (3.6) r Q 1 m k,j r Q 1 m k,j µ(f ) ε 4. S Q 1 1/m k (F ). Choosing k large j J k It also holds that Therefore for k large enough, by (3.5), j J 1 k U mk,j 1 U mk,j = for j 1 J 1 k, j J k. r Q 1 m k,j + j J k r Q 1 m k,j j mk j=1 r Q 1 m k,j µ(c) + ε, and by (3.6) and (3.4) (3.7) j J 1 k r Q 1 m k,j µ(c) µ(f ) + 3ε 4 < µ(c) µ(c \ U) + ε = µ(u) + ε. For all k N large enough by the definition of σ mk, (3.17), and (3.7) we see as in (3.18) that

21 SINGULAR INTEGRALS, SELF-SIMILAR SETS AND REMOVABILITY IN H n 1 σ mk (U) σ mk ( U mk,j ) j J 1 k A 4 S Q 1 ( U mk,j ) j J 1 k = A 4 j J 1 k A 5 (µ(u) + ε). α(q 1)r Q 1 m k,j Since σ mk σ, we deduce that σ (U) lim inf k σ m k (U) A 5 (µ(u) + ε) which contradits (3.3) and thus the proof is complete. The following theorem, with Q replaced by n, is also valid for Lipschitz harmonic functions in R n. Theorem Let C be a compact subset of H n. (i) If H Q 1 (C) = 0, C is removable. (ii) If dim C > Q 1, C is not removable. Proof. The first statement follows from Theorem 3.1. To see this let D C be a subdomain of H n. Applying the previous Theorem we deduce that if f : D R is Lipschitz in D and H -harmonic in D \ C there exists a H -harmonic function H in a domain G, C G D such that f(p) = H(p) for p G \ C. This implies that f = H in G. Hence f is harmonic in G (and so also in D). Therefore C is removable. In order to prove (ii) let Q 1 < s < dim C. By Frostman s lemma in compact metric spaces, see [M1], there exists a Borel measure µ with spt µ C such that µ(b(p, r)) r s for p H n, r > 0. We define f : H n R + as f(p) = Γ(q 1 p)dµq. It follows that f is a nonconstant function which is C in H n \ C and H f = 0 on H n \ C.

22 VASILIS CHOUSIONIS AND PERTTI MATTILA Furthermore f is Lipschitz: For p 1, p H n exactly as in the proof of Proposition 3.11, we obtain f(p 1 ) f(p ) = Γ(q 1 p 1 )dµq Γ(q 1 p )dµq ( ) 1 1 d(p 1, p ) dµq + dµq d(p 1, q) Q 1 d(p, q) Q 1 d(p 1, p ). To prove the last inequality let p H n, and consider two cases. If dist(p, C) > d(c), 1 dµ µ(c) 1. d(p, q) Q 1 d(c) Q 1 If dist(p, C) d(c), then C B(p, d(c)). Let A = d(c), then 1 d(p, q) dµ dµq Q 1 d(p, q) Q 1 j=0 j=0 B(p, j A)\B(p, (j+1) A) µ(b(p, j A)) ( (j+1) A) Q 1 Q 1 A s (Q 1) j=0 ( s (Q 1) ) j 1. Since f 0 by a Liouville-type theorem for sub-laplacians, see Theorem of [BLU], we deduce that H f 0 on C and hence it is not removable. In the following we fix some notation. Notation Recalling (3.3), (3.4) and (3.5) for a signed Borel measure σ set T σ (p) := K(q 1 p)dσq, whenever it exists, and Tσ(p) ε := H n \B(p,ε) K(q 1 p)dσq Tσ (p) := sup Tσ(p). ε ε>0 Remark Vertical hyperplanes of the form {(x, t) H n : x W, t R}, where W is a linear hyperplane in R n, are homogeneous subgroups of H n, that is, they are closed subgroups invariant under the dilations δ r. Their Hausdorff dimension is Q 1. If V is any such vertical hyperplane and σ denotes the (Q )-dimensional Lebesgue measure on V it follows by [St], Theorem 4 p.63 and essentially Corollary p.36, that T σ is bounded in L (σ). This implies, for example by the methods used in [MPa], that the subsets of vertical hyperplanes of positive measure are not removable for Lipschitz harmonic functions. The proof of the following lemma is rather similar to that of Lemma 5.4 in [MPa].

23 SINGULAR INTEGRALS, SELF-SIMILAR SETS AND REMOVABILITY IN H n 3 Lemma Let σ be a signed Borel measure in H n and A σ a positive constant such that σ (B(p, r)) A σ r Q 1 for p H n, r > 0. Then T σ (p) T σ + A T for p H n, where A T is a constant depending only on σ. Proof. We can assume that L = T σ <. The constants A i that will appear in the following depend only on n and σ. For ε > 0 and p H n, 1 1 d σ qdz L n+1 (B(p, ε/)) B(p,ε/) B(p,ε) q 1 z Q 1 ε Q 1 d σ qdz B(p,ε/) B(p,ε) q 1 z Q 1 ε Q dz d σ q q 1 z Q 1 where we used Fubini and that B(p,ε) B(q,ε) ε 1 Q σ (B(p, ε)) A σ B(q,ε) dz ε, q 1 z Q 1 which is easily checked by summing over the annuli B(q, 1 i ε) \ B(q, i ε), i = 0, 1,.... Now because of the inequality established above we can choose z B(p, ε/) with T σ (z) L such that B(p,ε) Therefore, Tσ(p) ε T σ (z) = K(q 1 z) d σ q H n \B(p,ε) H n \B(p,ε) H n \B(p,ε) B(p,ε) 1 q 1 z Q 1 d σ q A 6. K(q 1 p)d σ q K(q 1 z)d σ q K(q 1 p) K(q 1 z) d σ q + K(q 1 p) K(q 1 z) d σ q + A 6. Furthermore, by Proposition 3.11 (iii), as z B(p, ε/), K(q 1 p) K(q 1 z) d σ q H n \B(p,ε) H n \B(p,ε) H n \B(p,ε) { d(p, z) max d(p, q), d(p, z) Q d(p, z) d(p, q) Q d σ q + d(z, q) Q H n \B(z,ε/) } d σ q B(p,ε) d(p, z) d(z, q) Q d σ q K(q 1 z) d σ q

24 4 VASILIS CHOUSIONIS AND PERTTI MATTILA Since H n \B(p,ε) and in the same way, we deduce that d(p, z) d(p, q) Q d σ q ε H n \B(p,ε) H n \B(z,ε/) ε j=0 j=0 A σ ε = A σ Q, B(p, j+1 ε)\b(p, j ε) σ (B(p, j+1 ε)) ( j ε) Q j=0 ( j+1 ε) Q 1 ( j ε) Q d(p, z) d(z, q) Q d σ q A σ Q+1, K(q 1 p) K(q 1 z) d σ q A 7. 1 d(p, q) Q d σ q Therefore Tσ(p) ε Tσ(p) ε T σ (z) + T σ (z) A 6 + A 7 + L. The lemma is proven. 4. H -removable Cantor sets in H n In this section we shall construct a self-similar Cantor set C in H n which is removable although 0 < H Q 1 (C) <. The construction is similar to the one used in [CM] and it is based on ideas of Strichartz in [Str]. Notice that in Theorem 4. there is one piece S 0 (C r,n ) of C r,n well separated from the others. This is in order to make the condition of Theorem.3 easily checkable. It is almost sure that also the more symmetric example used in [CM] would satisfy that condition, but the calculation would become much more complicated. Definition 4.1. Let Q = [0, 1] n R n, r > 0, N N be such that r < 1 < 1. Let N z j R n, j = 1,..., N n, be distinct points such that z j,i { l : l = 0, 1,, N 1} for N all j = 1,, N n and i = 1,.., n. The similarities S r,n = {S 0,.., S 1 N n+}, depending on the parameters r and N, are defined as follows, S 0 = δ r, S j = τ (z j N n, 1 + i N ) δ r, for i = 0,, N 1 and j = in n + 1,, (i + 1)N n. where j m := j mod m. Theorem 4.. Let C r,n be the self-similar set defined by, C r,n = 1 N n+ j=0 S j (C r,n ).

25 SINGULAR INTEGRALS, SELF-SIMILAR SETS AND REMOVABILITY IN H n 5 Then there exists a set R C r,n such that for all j = 0,..., 1 N n+, (i) S j (R) R and (ii) the sets S j (R) are disjoint. This implies that the sets S j (C r,n ) are disjoint for j = 0,..., 1 N n+ and 0 < H a (C r,n ) < with a = log( 1 N n+ + 1) log( 1 r ). Furthermore the measure H a C r,n is a-ad regular. Proof. The proof is almost identical with that of Theorem 4. of [CM] but we present it since later we shall need some of its components. Using an idea of Strichartz we show that there exists a continuous function ϕ : Q R such that the set R = {q H n : q Q and ϕ(q ) q n+1 ϕ(q ) + 1} satisfies (i) and (ii). This will follow if we find some continuous ϕ : Q R which satisfies for all j = 1,..., N n, (4.1) τ (zj,0)δ r (R) = {q H n : q Q j and ϕ(q ) q n+1 ϕ(q ) + r }, where Q j = τ (zj,0)(δ r (Q)). If (4.1) holds then it is readily seen that R satisfies (i). In order to see that R satisfies (ii) as well first notice that (4.1) implies that for j = in n + 1,, (i + 1)N n and i = 0,, N 1, (4.) S j (R) = τ (z j N n, 1 + i )δ r(r) N = {q H n : q Q j N n and ϕ(q ) i N q n+1 ϕ(q ) i N + r }. Now let j k {0,..., 1N n + } and let p S j (R) and q S k (R). We need to show that p q. If j N n k N n then p Q j N n, q Q k N n, therefore p q, and so p q. If j N n = k N n and j, k 0 (the case jk = 0 is similar and simpler), then there exist i l {0,..., N 1} such that j {in n + 1,, (i + 1)N n } and k {ln n + 1,, (l + 1)N n }. Assume without loss of generality that i > l. If p = q we have by (4.), since r < 1 < 1, N q n+1 ϕ(q ) Hence p q and S j (R) S k (R) =. Since l N + r < ϕ(q ) l + 1 ϕ(p ) + 1 N + i N p n+1. τ (zj,0)δ r (R) = {p H n : p Q j and r ϕ( p z j ) r n (z j,i p i+n z j,i+n p i ) p n+1 r ϕ( p z j ) r n (z j,i p i+n z j,i+n p i ) + r },

26 6 VASILIS CHOUSIONIS AND PERTTI MATTILA proving (4.1) amounts to showing that (4.3) ϕ(w) = r ϕ( w z n j ) (z j,i w i+n z j,i+n w i ) for w Q j, j = 1,..., N n. r As usual for any metric space X, denote C(X) = {f : X R and f is continuous}. Let B = N n j=1 Q j and L : C(B) C(Q) be a linear extension operator such that and L(f)(x) = f(x) for x B L(f) = f. Since the Q j s are disjoint the operator L can be defined simply by taking ε > 0 small enough and letting f(x) when x B, ε dist(x, B) L(f)(x) = f( x) when 0 < dist(x, B) < ε, ε 0 when dist(x, B) ε, where x B and dist(x, B) = d(x, x). Furthermore define the functions h : B R, f : B R, n h(w) = (z j,i w i+n z j,i+n w i ) for w Q j, f(w) = r f( w z j ) for f C(Q), w Q j, r and the operator T : C(B) C(Q) as, Then and for f, g C(B), T (f)(w) = r f( w z j ) r T (f) = L( f + h). n (z j,i w i+n z j,i+n w i ) for w Q j, T f T g = L( f g) = f g r f g. Hence T is a contraction and it has a unique fixed point ϕ which satisfies (4.3). The remaining assertions follow from [S]. Remark 4.3. Notice that, by (4.) in order for all p C r,n \S 0 (C r,n ) to satisfy p n+1 > 0 it suffices to have, (4.4) ϕ(w) > 1 N n for all w Q j. j=1

27 SINGULAR INTEGRALS, SELF-SIMILAR SETS AND REMOVABILITY IN H n 7 For w Q j = n [z j,i, z j,i + r], j = 1,.., N n, z j,i w i+n z j,i+n w i = z j,i w i+n w i w i+n + w i w i+n z j,i+n w i (z j,i w i )w i+n + w i (w i+n z j,i+n ) r, for all i = 1,..., n. Hence by (4.3) it follows that, n ϕ(w) r ϕ + z j,i w i+n z j,i+n w i r ϕ + 4nr. Therefore and (4.4) is satisfied if r < 1 16n. ϕ 4nr 1 r 8nr, Remark 4.4. Choose r 0 < 1 such that N 16n 0 = 1 r 0 C r,n0, r < r 0. Then for r (0, r 0 ), Furthermore {dim C r,n0 : r (0, r 0 )} = (0, log( 1N n+ log( 1 N n ) log(n 0 ) Therefore there exists some r Q 1 < 1 N 0 We will denote C rq 1,N 0 by C Q 1. > log( N 0 ) + log(n n+1 0 ) log(n 0 ) such that 0 < H n+1 (C rq 1,N 0 ) <. N and consider the self similar sets 0 + 1) log(n 0 ) ). > n + 1. Theorem 4.5. The Cantor set C Q 1 satisfies 0 < H Q 1 (C Q 1 ) < and is removable. Proof. Suppose that C Q 1 is not removable. Then there exists a domain D C Q 1 and a Lipschitz function f : D R which is H -harmonic in D \ C Q 1 but not in D. By Theorem 3.1 there exists a domain G, C Q 1 G D, a Borel function h : C R and a H -harmonic function H : G R such that f(p) = Γ(q 1 p)h(q)dh Q 1 q + H(p) for p G \ C Q 1 C Q 1 and h L (H Q 1 C Q 1 ) + H H 1. Let σ = hh Q 1 C Q 1. In this case by the left invariance of H as in (3.14) and recalling Notation 3.14 which implies that T σ (p) = H f(p) H H(p) for all p G \ C Q 1 (4.5) T σ (p) 1 for all p G \ C Q 1. Let δ = dist(c Q 1, H n \ G) > 0. Then for p H n \ G, 1 (4.6) T σ (p) q 1 p d σ q σ (C Q 1) 1. Q 1 δ Q 1

28 8 VASILIS CHOUSIONIS AND PERTTI MATTILA By (4.5) and (4.6) we deduce that T σ L. Hence, recalling Theorem 4., since the measure H Q 1 C Q 1 is (Q 1)-AD regular we can apply Lemma 3.16 and conclude that Tσ is bounded. Furthermore since f is not harmonic in C Q 1, h 0 in a set of positive H Q 1 measure. Therefore there exists a point p C Q 1 of approximate continuity (with respect to H Q 1 C Q 1 ) of h such that h(p) 0. Recalling that C rq 1,N 0 := C Q 1 and Definition 4.1 let w k {0,..., 1N n+ } k be such that p S wk (C Q 1 ). Then by the approximate continuity of h, r (1 Q)k (S 1 w k ) (σ S wk (C Q 1 )) h(p)h Q 1 C Q 1 as k, and the boundedness of T σ implies that T H Q 1 C Q 1 is bounded. To see this let z H n \ (C Q 1 k=1 S 1 w k (C Q 1 )). If dist(z, C Q 1 ) > α C Q 1 d(c Q 1 ), then (4.7) T H Q 1 C Q 1 (z) 1. Therefore we can assume that dist(z, C Q 1 ) α C Q 1 d(c Q 1 ). Recalling Remark 4.4 this implies that for any w I, (4.8) dist(s w (z), S w (C Q 1 )) = r w Q 1 dist(z, C Q 1) r w α CQ 1 Q 1 d(c Q 1 ) = α C Q 1 d(s w (C Q 1 )). Notice that the homogeneity of K implies that K(S 1 w k (q) 1 z) = r (Q 1)k K(q 1 S wk (z)) as in the proof of Theorem.3. Therefore by (.), h(p)t H Q 1 C Q 1 (z) = lim r (1 Q)k k K(q 1 z)d(s 1 w k ) (σ S wk (C Q 1 ))q = lim r (1 Q)k K(Sw k S 1 k (q) 1 z)dσq wk (C Q 1 ) = lim K(q k S 1 S wk (z))dσq wk (C Q 1 ) ( = lim k K(q 1 S wk (z))dσq C Q 1 Since z / k=1 S 1 w k (C Q 1 ), K(q 1 S wk (z))dσq C Q 1 T σ. C Q 1 \S wk (C Q 1 ) Furthermore by Lemma.4 and (4.8) we get that, K(q 1 S wk (z))dσq T σ + A CQ 1. C Q 1 \S wk (C Q 1 ) K(q 1 S wk (z))dσq ). Therefore, h(p)t H Q 1 C Q 1 (z) 3 T σ + A CQ 1,

29 and since SINGULAR INTEGRALS, SELF-SIMILAR SETS AND REMOVABILITY IN H n 9 L n+1 (C Q 1 k=1 S 1 w k (C Q 1 ) we get that T H Q 1 C Q 1 L. Hence by Lemma 3.16 T H Q 1 C Q 1 is bounded. On the other hand notice that K 1+n (q 1 )dh Q 1 q (4.9) C Q 1 \S 0 (C Q 1 ) = = C Q 1 \S 0 (C Q 1 ) C Q 1 \S 0 (C Q 1 ) ) = 0 ( q 1+n ) q ( q 1 )( q n+1 ) c Q dh Q 1 q q Q+ c Q q 1+n q + q 1 q n+1 q Q+ dh Q 1 q. Recalling Definition 4.1 for q C Q 1 \S 0 (C Q 1 ), q 1+n, q 1 [0, 1]\[0, r Q 1 ] and by Remark 4.3 we also have that q n+1 > 0. Hence q 1+n q + q 1 q n+1 > 0 for q C Q 1 \ S 0 (C Q 1 ) and by (4.9) K 1+n (q 1 )dh Q 1 q 0. C Q 1 \S 0 (C Q 1 ) Therefore, by Theorem.3 (recall the definition of fixed points of a family similarities given before it), since 0 is a fixed point for S rq 1,N 0, more precisely S 0 (0) = 0, TK n+1 (H Q 1 C Q 1 ) and hence T H Q 1 C Q 1 is unbounded. We have reached a contradiction and the theorem is proven. 5. Concluding comments and questions Here we shall discuss some questions that are left unanswered, or even not considered at all, so far. What (Q 1)-dimensional subsets of H n are not removable? The proof of Theorem 4.5 uses the special structure of C Q 1 only at the end to check the condition of Theorem.3. It is quite likely that the cases of self-similar sets where this condition fails are quite exceptional, but checking it could be technically very complicated. In our case we set up the example so that the integrand doesn t change sign, but even for the sets considered in [CM] one would need to compare carefully the positive and negative contributions. Note also that there are actually infinitely many sufficient conditions for the unboundedness in Theorem.3 corresponding to the dense set of fixed points. The related question is on what (Q 1)-dimensional subsets of H n the singular integral operator related to the kernel K = H Γ can be L -bounded. Or on what m-dimensional subsets of H n the singular integral operators related to appropriate m-homogeneous kernels can be L -bounded. As mentioned in the introduction essentially complete results are only known for the Cauchy kernel in the complex plane (or also for the Riesz kernel x x in R n ). For m-dimensional Ahlfors-David-regular sets E and m-homogeneous Riesz kernels in R n we know that the L -boundedness implies that m must be an integer, [Vi], and E must be well approximated by m-planes almost everywhere at some arbitrarily small scales, [MPa], [M3]. Similar results were proved for Riesz-type kernels in H n in [CM]. A property of these kernels R that was crucial for the proofs is that R(x) = R(y)