Pacific Journal of Mathematics EXISTENCE OF PRINCIPAL VALUES OF SOME SINGULAR INTEGRALS ON CANTOR SETS, AND HAUSDORFF DIMENSION J ULIÀ C UFÍ , J UAN J ESÚS D ONAIRE , P ERTTI M ATTILA AND J OAN V ERDERA Volume 326 No. 2 October 2023 PACIFIC JOURNAL OF MATHEMATICS Vol. 326, No. 2, 2023 https://doi.org/10.2140/pjm.2023.326.285 EXISTENCE OF PRINCIPAL VALUES OF SOME SINGULAR INTEGRALS ON CANTOR SETS, AND HAUSDORFF DIMENSION J ULIÀ C UFÍ , J UAN J ESÚS D ONAIRE , P ERTTI M ATTILA AND J OAN V ERDERA Consider a standard Cantor set in the plane of Hausdorff dimension 1. If the linear density of the associated measure µ vanishes, then the set of points where the principal value of the Cauchy singular integral of µ exists has Hausdorff dimension 1. The result is extended to Cantor sets in Rd of Hausdorff dimension α and Riesz singular integrals of homogeneity −α, 0 < α < d: the set of points where the principal value of the Riesz singular integral of µ exists has Hausdorff dimension α. A martingale associated with the singular integral is introduced to support the proof. 1. Introduction Our main result deals with the Cauchy singular integral on Cantor sets in the plane and the proof extends with minor variations to the Riesz transforms in Rd . We first proceed to formulate the result for the Cauchy integral and then we take care of the Riesz transforms. The appropriate Cantor sets for the Cauchy integral are defined as follows. Let (λn )∞ n=1 a sequence of real numbers satisfying 4 ≤ λn ≤ λ < 2 . Let Q 0 := 1 1 [0, 1] × [0, 1] be the unit square. Take the 4 squares contained in Q 0 with sides of length λ1 parallel to the coordinate axis having a vertex in common with Q 0 (the 4 “corner squares” of side length λ1 ). Repeat in each of these 4 squares the same procedure with the dilation factor λ1 replaced by λ2 to get 16 squares of side length λ1 λ2 . Proceeding inductively we obtain at the n-th step 4n squares Q nj , 1 ≤ j ≤ 4n , of side length sn = λ1 · · · λn . Our Cantor set is 4 ∞ [ n \ K= Q nj . n=1 j=1 MSC2020: primary 42B20; secondary 30E20. Keywords: Cauchy singular integral, Riesz singular integral, Cantor set, Hausdorff dimension, martingale. © 2023 The Authors, under license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). Open Access made possible by subscribing institutions via Subscribe to Open. 286 JULIÀ CUF Í , JUAN JESÚS DONAIRE, PERTTI MATTILA AND JOAN VERDERA Let µ be the Borel probability measure on K with µ(Q nj ) = 4−n and denote by an the linear density at generation n, that is, 1 µ(Q j ) n an = n = ≤ 1. 4 sn sn Set Dn = {Q nj : j = 1, . . . , 4n } and D = ∞ S n=1 Dn . Theorem 1.1. If limn→∞ an = 0, then the set of points z ∈ K for which the principal value Z 1 (1-1) lim dµw ε→0 |w−z|>ε w − z exists has Hausdorff dimension greater than or equal to 1. This solves a problem posed in [Cufí et al. 2022, Open problem 5.5, p. 1621]. If an = 1 for all n, then K is the famous Garnett–Ivanov Cantor set, which has positive and finite one-dimensional Hausdorff measure but zero analytic capacity. In this case it was noticed in [Cufí et al. 2022] that the principal value does not exist at any point of K . If an → 0, then the Hausdorff dimension of K is greater than or equal to 1 and it has non-sigma finite one-dimensional Hausdorff measure. If in addition n an < ∞, then the principal value exists µ almost everywhere. P 2 So Theorem 1.1 is relevant only when an → 0 slowly. That the condition n an < ∞ implies the almost P 2 everywhere existence of principal values can be seen in two ways. First, we introduce a martingale (Sn )∞n=0 (see (2-1)) and show that the increments |Sn+1 (x)− Sn (x)| are bounded by C an , with the constant C independent of n and x. In Lemma 2.4 we prove that for any point x the principal value exists at x if and only if (Sn (x))∞ n=0 converges. If n an2 < ∞, then Sn is an L 2 martingale and consequently it converges P almost everywhere. Alternatively, the condition n an2 < ∞ implies that the Cauchy P singular integral operator is bounded in L 2 (µ). In [Mattila and Verdera 2009] it was shown in a very general setting that L 2 boundedness together with zero density of the measure yields the almost everywhere existence of principal values. The main argument in the proof of Theorem 1.1 deals with case where n an2 = ∞. P It is a variation of a line of reasoning used in other situations (see [Donaire et al. 2014]). We use a stopping time argument to show that (Sn (x))∞ n=0 converges to 0 in a set of Hausdorff dimension 1 (indeed, given any complex number z 0 the martingale (Sn (x))∞ n=0 converges to z 0 in a set of Hausdorff dimension 1). We get the dimension 1 conclusion by applying a lemma of Hungerford [1988]. For the sake of the reader we present a proof of Hungerford’s lemma in our context in Appendix A. Our proof extends with only technical modifications to cover the case of other odd kernels, for instance, zm , m = 1, 2, . . . z m+1 EXISTENCE OF PRINCIPAL VALUES OF SINGULAR INTEGRALS ON CANTOR SETS 287 But one of the ingredients of our method fails for the odd kernel (z + z)/z 2 and we do not know whether Theorem 1.1 holds in this case. The difficulty is indicated at the fifth line after the statement of Lemma 3.1. In Rd our proof works for the Riesz transforms of any homogeneity −α, 0 < α < d. These are the vector valued singular integrals with kernel x R α (x) = 1+α , 0 < α < d. |x| The appropriate Cantor sets for the α-Riesz transform are those of Hausdorff dimension α. They are constructed by the procedure outlined before in the planar case with dilation factors that satisfy 2−d/α ≤ λn ≤ λ < 2−1 . At generation n one has 2dn cubes Q nj of side length sn = λ1 · · · λn . The Cantor set is defined by 2 ∞ [ dn \ K= Q nj n=1 j=1 and the canonical measure on K by µ(Q nj ) = 2−dn , 1 ≤ j ≤ 2dn . The αdensity is an = 2−dn sn−α = µ(Q nj )sn−α ≤ 1. For λn = 2−d/α , n = 1, 2 . . . , one gets the self similar Cantor set of dimension α. If an → 0 then our Cantor set has Hausdorff dimension ≥ α and non σ finite Hausdorff α-dimensional measure. Theorem 1.2. If limn→∞ an = 0, then the set of points x ∈ K for which the principal value Z (1-2) lim R α (y − x) dµy ε→0 |y−x|>ε exists has Hausdorff dimension greater than or equal to α. In Appendix B we give some indications on how to adapt the proof for the Cauchy kernel to the Riesz transforms in higher dimensions. We let diam(A) denote the diameter and dim A the Hausdorff dimension of a set A. We use the notation a ≲ b to mean that a ≤ C b for some constant C which may depend on λ and d, and a ∼ b for a ≲ b and b ≲ a. 2. Martingales Let C be the Cauchy kernel, C(x) = 1/x for x ∈ C, x ̸= 0. For each x ∈ K let Q n (x) be the square in Dn containing x. Define the truncated Cauchy integral at generation n as Z Tn (x) = C(x − y) dµy, x ∈ K , K \Q n (x) and a martingale (Sn (x))∞ n=0 by Z (2-1) Sn (x) = S Q n (x) = / Tn dµ, x ∈ K. Q n (x) 288 JULIÀ CUF Í , JUAN JESÚS DONAIRE, PERTTI MATTILA AND JOAN VERDERA Remark 2.1. That Sn is a martingale is easily checked. The reader will realise that the martingale condition also holds for kernels K (x, y) satisfying the antisymmetry condition K (x, y) = −K (y, x). We shall prove: Theorem 2.2. If limn→∞ an = 0, then the set of points x ∈ K for which (Sn (x))∞ n=0 converges has Hausdorff dimension greater than or equal to 1. We first show that the martingale (2-1) has uniformly bounded increments. Lemma 2.3. There exists a positive constant C = C(λ) such that (2-2) |Sn+1 (x) − Sn (x)| ≤ Can , n = 0, 1, . . . and x ∈ K . Thus if n an converges, (Sn (x))∞ P n=0 converges for all x ∈ K . As mentioned in the introduction, even the weaker condition n an2 < ∞ implies that (Sn (x))∞ P n=0 converges for µ almost all x ∈ K . Hence we shall assume that n an2 = ∞. Under P this assumption one proves in [Cufí et al. 2022] that the set where the principal values fail to exist has full µ measure. In Lemma 2.4 below we show that principal values exist if and only if the martingale converges. Hence (Sn (x))∞ n=0 is not convergent for µ almost all x ∈ K . By a standard result in martingale theory (see, for example, [Shiryaev 1996, Corollary 6, p. 561]) we get (2-3) lim sup |Sn (x) − Sm (x)| = ∞, for all m = 0, 1, . . . and µ a.e. n→∞ Proof of Lemma 2.3. Set Q n = Q n (x), x ∈ K , n = 1, 2, . . . Then Z Z Z Z Sn+1 (x)− Sn (x) = / C(z − y) dµy dµz − / C(w− y) dµy dµw Q n+1 K \Q n+1 Qn K \Q n Z Z Z  = / / C(z − y) dµz − C(w− y) dµw dµy K \Q n Q n+1 Qn Z Z + / C(z − y) dµz dµy. Q n \Q n+1 Q n+1 The last double integral is ≲ an , where the implicit constant depends on λ here and for the rest of the proof. To estimate the first summand above we remark that for each z ′ ∈ Q n+1 and w ′ ∈ Q n we have Z Z / C(z − y) dµz − / C(w − y) dµw Q n+1 Qn Z Z C(z − y) − C(z − y) dµz − / C(w − y) − C(w′ − y) dµw ′   =/ Q n+1 Qn + C(z ′ − y) − C(w ′ − y). EXISTENCE OF PRINCIPAL VALUES OF SINGULAR INTEGRALS ON CANTOR SETS 289 Clearly C(z ′ − y) − C(w ′ − y) ≲ sn |x − y|−2 , y ∈ K \ Qn , x ∈ Qn . Hence Z C(z − y) − C(z ′ − y) dµz ≲ sn |x − y|−2  / Q n+1 and Z C(w − y) − C(w ′ − y) dµz ≲ sn |x − y|−2 .  / Qn Setting R j = Q j \ Q j+1 , the absolute value of the first summand of Sn+1 (x) − Sn (x) is Z n−1 X ≲ sn |x − y|−2 dµy ∼ sn j µ(R j ) s −2 K \Q n j=0 n−1 X = sn s −2 j 4 −j ≲ sn sn−2 4−n = an , j=0 because s −2 − j ≤ (s −2 λ2 )4− j = (4λ2 )s −2 4− j−1 , so s −2 4− j ≲ (4λ2 )n− j s −2 4−n . j 4 j+1 j+1 j n Hence |Sn+1 (x) − Sn (x)| ≲ an . □ By the following lemma Theorem 2.2 is equivalent to Theorem 1.1. Lemma 2.4. If limn→∞ an = 0, then for each x ∈ K the principal value (1-1) exists if and only if the sequence (Sn (x))∞ n=0 converges. Proof. Set Q n = Q n (x) for x ∈ K and n = 1, 2, . . . Then by the proof of Lemma 2.3 Z Z Z   1 1 1 Sn (x) − dµy = / ′ − dµy dµx ′ K \Q n x − y Q n K \Q n x − y x−y ≤ C an , where the constant depends on λ. Compare now a given truncation K \B(x,ε) x−y1 R dµy, 0 < ε < 1, with K \Q n x−y 1 dµy where n is chosen so that diam(Q n ) ≤ ε < R diam(Q n−1 ). Since Q n ⊂ B(x, ε) we have Z Z Z 1 1 1 dµy − dµy = dµy K \Q n x − y K \B(x,ε) x − y B(x,ε)\Q n x − y µB(x, ε) ≤C , sn 290 JULIÀ CUF Í , JUAN JESÚS DONAIRE, PERTTI MATTILA AND JOAN VERDERA with C = C(λ). To complete the proof just remark that, since ε < diam(Q n−1 ), B(x, ε) can intersect at most N squares in Dn , with N an absolute constant. Hence µB(x, ε) ≤ C µ(Q n ). □ We proceed now to discuss relative martingales. For x ∈ R ⊂ Q, Q ∈ Dm , R ∈ Dn , m < n, we define the relative martingale starting at Q as Z Z S Q,R (x) = S Q,R = / C(z − y) dµy dµz. R Q\R Then for some constant C, (2-4) |S R − S Q − S Q,R | ≤ C am . Indeed, we have Z Z Z Z SR − SQ = / C(z − y) dµy dµz − / C(w − y) dµy dµw R K \R Q K \Q Z Z Z  = / C(z − y) dµz − / C(w − y) dµw dµy K \Q R Q Z Z + / C(z − y) dµz dµy Q\R R Z Z Z  = / C(z − y) dµz − / C(w − y) dµw dµy + S Q,R . K \Q R Q The first summand above is bounded in absolute value by a constant times am by the same argument as in the proof of (2-2). As for (2-2) we have for R ⊂ R̃ ⊂ Q, Q ∈ Dm , R̃ ∈ Dn , R ∈ Dn+1 , (2-5) |S Q,R − S Q, R̃ | ≤ C an . 3. The stopping time argument The proof of Theorem 2.2 is based on a stopping time argument for which we need some preliminary facts. Given a nonzero complex number z consider the sector σ (z, θ ), 0 < θ < π , with vertex at z and aperture θ whose axis is the semiline emanating from z and passing through 0. That is, w ∈ σ (z, θ ) if and only if w − z −z θ     , ≥ cos |w − z| |z| 2 where ⟨ ·, · ⟩ denotes the scalar product in the plane. EXISTENCE OF PRINCIPAL VALUES OF SINGULAR INTEGRALS ON CANTOR SETS 291 The octants with vertex 0 are the eight sectors π π n o σ j = w ∈ C : w = |w|eiφ , ( j − 1) ≤ φ ≤ j , 1 ≤ j ≤ 8. 4 4 These are the sectors with vertex the origin of amplitude 45◦ degrees and having an edge over a coordinate axis. It will be convenient to expand these octants so that they have the same axis and amplitude of 75◦ . In other words, we are adding 15◦ in each direction. Denote the expanded sectors by σ̃ j . The octants with vertex z are the sectors σ j (z) = z + σ j , 1 ≤ j ≤ 8, and the expanded octants σ̃ j (z) = z + σ̃ j . We have the following obvious lemma. Lemma 3.1. Given any sector σ of vertex z and amplitude 120◦ there exists an octant with vertex z, say σ j (z) for some index j between 1 and 8, such that σ̃ j (z) ⊂ σ . Consider the symmetries with respect to the coordinate axis and the main diagonal. That is, f 1 (x + i y) = −x + i y, f 2 (x + i y) = x − i y and f 3 (x + i y) = y + i x for x + i y ∈ C. For any j, k = 1, . . . , 8, by composing two such symmetries we obtain a linear mapping f j,k that maps the octant σ j onto the octant σk . Observe that C( f j (z)) = f j (C(z)) for j = 1, 2, and C( f 3 (z)) = − f 3 (C(z)). It is precisely this last identity that fails for the kernel (z + z)/z 2 . Let Q ∈ D and let c Q be its centre. Define f Q, j,k (x) = f j,k (x − c Q ) + c Q , x ∈ Q, j, k = 1, . . . , 8, so that f Q, j,k (x) − f Q, j,k (y) = f j,k (x − y), x, y ∈ Q, j, k = 1, . . . , 8. We claim that (3-1) S Q, f Q, j,k (R) = ε j,k f j,k (S Q,R ), R ⊂ Q, Q, R ∈ D, where ε j,k = ±1. We check (3-1) by the general formula for the image (push- forward) ν ♯, f of a measure ν under a Borel map f (see, for example, [Mattila 1995, Theorem 1.19]) Z Z g dν ♯, f = (g ◦ f ) dν. f (A) A The restriction of µ to Q is invariant under the maps f Q, j,k , i.e., (µ|Q)♯, f Q, j,k = µ|Q. Hence, since Q \ f Q, j,k (R) = f Q, j,k (Q \ R) and C f Q, j,k (z) − f Q, j,k (w) = ε j,k f j,k (C(z − w)),  we obtain Z Z Z Z  C(z − w) dµz dµw = ε j,k f j,k C(z − w) dµz dµw , Q\ f Q, j,k (R) f Q, j,k (R) Q\R R from which (3-1) follows. 292 JULIÀ CUF Í , JUAN JESÚS DONAIRE, PERTTI MATTILA AND JOAN VERDERA Assume that we have fixed an octant σ j and that for some square R ∈ D contained in Q we have S Q,R ∈ σk with k ̸= j. We claim that we can find a square R ′ ∈ D contained in Q, of the same size as R, such that |S Q,R ′ | = |S Q,R | and S Q,R ′ ∈ σk . If ε j,k = 1 then the value of the relative martingale at the square f Q, j,k (R) is f j,k (S Q,R ) ∈ σ j . Note that the size of f Q, j,k (R) is exactly the size of R and |S Q, f Q, j,k (R) | = |S Q,R |. To treat the case ε j,k = −1 let us introduce the mapping γ : Q → Q defined by γ (x) = −(x −c Q )+c Q . Then γ 2 is the identity mapping on Q and S Q,γ (R) = −S Q,R for each square R ∈ D contained in Q. Setting R ′ = (γ ◦ f Q, j,k )(R) we get f jk (S Q,R ) = −S Q, f Q, j,k (R) = S Q,(γ ◦ f Q, j,k )(R) = S Q,R ′ . We shall need the following elementary lemma. Lemma 3.2. If z ∈ C, w ∈ σ (z, 120◦ ) and 0 < |w − z| < |z|/2, then |w| ≤ |z| − |w − z|/4. Proof. Let R = |z|, r = |w − z| and let v be the third vertex, in addition to 0 and z, of the equilateral triangle containing w. Under the assumptions of the lemma |w| is maximized when w lies on the side connecting z and v. Assuming that w is on that side, project w on the side connecting 0 and z and apply Pythagoras to obtain √ |w|2 = (R − r/2)2 + ( 3r/2)2 = r 2 + R 2 − r R ≤ (R − r/4)2 = (|z| − |w − z|/4)2 because of the assumption r < R/2. □ Proof of Theorem 2.2. We assume, as we may, that n an2 = ∞. Then for µ almost P all x the sequence (Sn (x))∞ n=0 diverges and (2-3) holds. Let M be a big positive integer to be chosen later. We replace (an )∞ n=0 by the nonincreasing sequence bn = C maxm≥n am , where C is as in inequalities (2-2), (2-4) and (2-5), which now read (3-2) |Sn+1 (x) − Sn (x)| ≤ bn , n = 0, 1, . . . and x ∈ K , (3-3) |S R − S Q − S Q,R | ≤ bm , Q ∈ Dm , R ∈ Dn , R ⊂ Q, (3-4) |S Q,R − S Q, R̃ | ≤ bn , Q ∈ Dm , R ∈ Dn+1 , R̃ ∈ Dn , R ⊂ R̃ ⊂ Q. We plan to define a sequence of stopping time conditions. At each step a family of stopping time squares will arise, which is going to be the family Fn in Lemma A.1 (Hungerford’s lemma). The first stopping time is special and its goal is to have a family of squares with relatively large |S Q | for each square Q in the family. The first stopping time condition is (3-5) |S Q | > M b0 . Declare Q a stopping time square of first generation if Q is a square in D for which |S Q | > M b0 and |S Q ′ | ≤ M b0 , Q ⊊ Q ′ . We call F1 the set of stopping time squares EXISTENCE OF PRINCIPAL VALUES OF SINGULAR INTEGRALS ON CANTOR SETS 293 of first generation. One may think of this as a process as follows. One takes a point x ∈ K and looks at the squares in D containing x. One examines all those squares, starting at Q 0 and checks whether condition (3-5) is satisfied. If it is not, then one proceeds to the square containing x in the next generation. The process stops when one finds a square Q containing x for which (3-5) holds. Note that the set of x for which the process never stops has vanishing µ measure by (2-3). Hence Q∈F1 µ(Q) = 1. Since S Q 0 = 0, it follows from (3-2) that it is necessary P to descend at least M + 1 generations to find the first stopping time square. The second stopping time condition is slightly different. Let Q ∈ F1 . The second stopping time is performed on the relative martingale associated with Q and its condition is (3-6) |S Q,R | > M b M . A stopping time square R of second generation satisfies |S Q,R | > M b M and |S Q,R ′ | ≤ M b M , R ′ ∈ D, R ⊊ R ′ ⊂ Q. By (2-3) and (3-3) the stopping time squares of second generation cover almost all Q. Again, by (3-4) and the fact that S Q,Q = 0 one has to descend through at least M + 1 generations to find a stopping time square of second generation. Hence if R is a stopping time square of second generation and R ∈ Dn then n ≥ 2(M + 1). We do not put all stopping time squares of second generation in F2 (Q). We put a stopping time square of second generation R in F2 (Q) provided S R ∈ σ (S Q , 120◦ ). That there are many such stopping time squares can be shown as follows. Let R be a stopping time square of second generation. Let α denote the angle between the vectors S R − S Q and S Q,R . Then by (3-3), |S R − S Q | ≥ |S Q,R | − b M ≥ (M − 1)b M and |S R − S Q − S Q,R | bM 1 0 ≤ | sin α| ≤ ≤ = < sin 15◦ , |S R − S Q | (M − 1)b M M −1 provided M −1 > 1/ sin 15◦ , which we assume. Since |S R − S Q − S Q,R | < |S R − S Q | and S R,Q = S R − S Q + (S Q,R − S R − S Q ), we see that cos α > 0. Thus |α| < 15◦ . By Lemma 3.1 there is j with 1 ≤ j ≤ 8 such that σ̃ j (S Q ) ⊂ σ (S Q , 120◦ ). If we are lucky enough that we have S Q,R ∈ σ j and so S R − S Q ∈ σ˜j , which yields S R ∈ σ˜j (S Q ) ⊂ σ (S Q , 120◦ ). But it may occur that S Q,R ∈ σk , k ̸= j. Applying two symmetries f Q, j,k of Q, or a symmetry of the form γ ◦ f Q, j,k in the worst case, as we discussed before Lemma 3.2, we obtain a stopping time square R ′ of second generation and of the same size as R such that S Q,R ′ ∈ σ j and so S R ′ ∈ σ˜j (S(Q)) ⊂ σ (S Q , 120◦ ), as desired. 294 JULIÀ CUF Í , JUAN JESÚS DONAIRE, PERTTI MATTILA AND JOAN VERDERA Therefore, subdividing the stopping time squares of second generation in eight classes, according to the octant to which S Q,R belongs, we get X (3-7) µ(R) ≥ 18 µ(Q). R∈F2 (Q) Define F2 = Q∈F1 F2 (Q). S Let us obtain some properties of stopping time squares R in F2 (Q). Let R̃ be the father of R. Then |S Q, R̃ | ≤ M b M and so |S R̃ − S Q | ≤ |S Q, R̃ | + |S R̃ − S Q − S Q, R̃ | ≤ (M + 1)b M and |S R − S Q | ≤ |S R − S R̃ | + |S R̃ − S Q | ≤ b M + (M + 1)b M = (M + 2)b M . Now two possibilities appear. If |S Q | ≤ 2|S R − S Q | ≤ 2(M + 2)b M , then |S R | ≤ |S R − S Q | + |S Q | ≤ 3(M + 2)b M . If |S Q | > 2|S R − S Q |, since S R ∈ σ (S Q , 120◦ ) we can apply Lemma 3.2 to get |S R | ≤ |S Q | − |S R − S Q |/4 ≤ |S Q | − (M − 1)b M /4 ≤ |S Q | − b M provided M ≥ 5. Therefore at least one of the following two inequalities holds: either (3-8) |S R | ≤ 3(M + 2)b M , or (3-9) |S R | ≤ |S Q | − b M . We can proceed to define inductively Fn for n ≥ 3, in a way analogous to what we did to define F2 from F1 . Assume that we have defined Fn−1 = Q∈Fn−2 Fn−1 (Q). S Given Q ∈ Fn−1 we set the n generation stopping time in the relative martingale associated with Q as |S Q,R | > Mb(n−1)M If R is a stopping time square of n-th generation then besides the previous inequality one has |S Q,R ′ | ≤ Mb(n−1)M , R ′ ∈ D, R ⊊ R ′ ⊂ Q, whence (3-10) |S R ′ − S Q | ≤ |S Q,R ′ | + b(n−1)M ≤ (M + 1)b(n−1)M . EXISTENCE OF PRINCIPAL VALUES OF SINGULAR INTEGRALS ON CANTOR SETS 295 Note that if R is a stopping time square of generation n, we can take advantage of the symmetries of Q, as before, to find another one, say R ′ , of the same size with the additional property that S R ′ ∈ σ (S Q , 120◦ ). Define Fn (Q) as the stopping time squares R of generation n such that S R ∈ σ (S Q , 120◦ ) and Fn = Q∈Fn−1 Fn (Q). S We then have X (3-11) µ(R) ≥ 18 µ(Q). R∈Fn (Q) Given R ∈ Fn (Q), we have as before that at least one of the following two inequal- ities holds: either (3-12) |S R | ≤ 3(M + 2)b(n−1)M or (3-13) |S R | ≤ |S Q | − b(n−1)M . T∞ S Set F = n=1 Q∈Fn Q. To complete the proof we shall show that the hypo- theses of Hungerford’s Lemma A.1 are fulfilled and that (3-14) lim Sm (x) = 0, x ∈ F. m→∞ For (b) in Hungerford’s Lemma A.1 recall that each stopping time square has descended at least M + 1 generations from the generating square in the previous family. Then one has (b) with ε replaced by 1/4 M and taking M big enough one has 1/4 M < ε. Condition (c) with c = 1/8 is (3-11). To prove (3-14), take x ∈ F. For every n = 1, 2, . . . , there is a unique Q n ∈ Fn such that x ∈ Q n . Let m n be the unique positive integer satisfying Q n ∈ Dm n . Clearly the sequence m n is increasing and m n > M n. Since S Q n = Sm n (x) we have by (3-12) and (3-13) that either (3-15) |Sm n (x)| ≤ 3(M + 2)b(n−1)M or (3-16) |Sm n (x)| ≤ |Sm n−1 (x)| − b(n−1)M , n = 1, 2, . . . For m n−1 < m < m n we have by (3-10) (3-17) |Sm (x) − Sm n−1 (x)| ≤ (M + 1)b(n−1)M . To conclude that limm→∞ Sm (x) = 0 it is enough to show that limn→∞ Sm n (x) = 0. We say that n ∈ N1 , if (3-16) holds and n ∈ N2 , if (3-15) holds and (3-16) fails. Because n bn diverges and (bn )∞ P P n=1 is nonincreasing, n b(n−1)M also diverges. It follows that (3-16) cannot hold for infinitely many consecutive n, whence N2 is infinite. 296 JULIÀ CUF Í , JUAN JESÚS DONAIRE, PERTTI MATTILA AND JOAN VERDERA Let n ∈ N2 and let N > n be such that k ∈ N1 for all n < k < N . Then by (3-16) and (3-15) for n < k < N , |Sm k (x)| ≤ |Sm n (x)| ≤ 3(M + 2)b(n−1)M . It follows that limm→∞ Sm (x) = 0. □ Appendix A. A lemma on Hausdorff dimension Let µ be the canonical measure associated with a Cantor set in Rd , as defined in the Introduction before the statement of Theorem 1.2. Denote by Dn the set of all cubes Q nj , 1 ≤ j ≤ 2dn , appearing at the n-th generation of the construction and S D = n Dn . The following lemma is due to Hungerford [1988], who worked in a one- dimensional context. Lemma A.1. Let 0 < ε < c < 1 and let Fn be a disjoint family of cubes in D, for n = 0, 1, 2, . . . , satisfying the following. (a) F0 = {Q 0 }. (b) If Q ∈ Fn+1 , then there exists Q̃ ∈ Fn with Q ⊂ Q̃ and µ(Q) ≤ εµ( Q̃). (c) If Q ∈ Fn , then X µ(R) ≥ c µ(Q). R⊂Q, R∈Fn+1 T S Let E = n Q∈Fn . Then dim E ≥ α(1 − log c/ log ε). Proof. Set β = α(1 − log c/ log ε). We will construct a Borel probability measure ν with ν(E) = 1 such that for some constant C and for all balls B(x, r ) centred at x of radius r one has (A-1) ν(B(x, r )) ≤ Cr β for x ∈ E, 0 < r ≤ 1. Then Frostman’s lemma will give the result. Let us define the functions νn : Fn → R, n = 0, 1, 2 . . . , setting first ν0 (Q 0 ) = 1. Suppose that ν1 , . . . , νn−1 are defined and let for Q ∈ Fn , with Q̃ as in (b), νn−1 ( Q̃) νn (Q) = P µ(Q). R∈Fn ,R⊂ Q̃ µ(R) Then we define the Borel measures νn setting X νn (Q) νn (A) = µ(A ∩ Q) for A ⊂ Rd . µ(Q) Q∈Fn EXISTENCE OF PRINCIPAL VALUES OF SINGULAR INTEGRALS ON CANTOR SETS 297 Then for Q ∈ Fn , X νn+1 (Q) = νn+1 (R) R∈Fn+1 ,R⊂Q X νn (Q) = µ(R) µ(P) P R∈Fn+1 ,R⊂Q P∈Fn+1 ,P⊂Q = νn (Q). Iterating this we have (A-2) νm (Q) = νn (Q) for Q ∈ Fn , m > n. In particular, each νn is a probability measure and some subsequence of (νn ) converges weakly to a probability measure ν such that ν(Q) = νn (Q) for Q ∈ Dn . Since  [  X X ν Q = ν(Q) = νn (Q) = 1, Q∈Fn Q∈Fn Q∈Fn we have ν(E) = 1. Therefore ν(E \ S Q∈Fn Q) = 0 for every n, so X (A-3) ν(Q) = ν(R), Q ∈ Fn . R⊂Q,R∈Fn+1 It remains to verify (A-1). First of all we have by condition (c) for Q ∈ Fn , n ≥ 2, ν(Q) νn (Q) νn−1 ( Q̃) ν( Q̃) = =P ≤ , µ(Q) µ(Q) R⊂ Q̃,R∈Fn µ(R) cµ( Q̃) and by induction, ν(Q) (A-4) ≤ c−n for Q ∈ Fn , n = 1, 2 . . . . µ(Q) Now let us prove that (A-5) ν(Q) ≤ Cd(Q)β for Q ∈ D. Take n such that εn+1 ≤ µ(Q) < ε n . We may assume that ν(Q) > 0. Then Q intersects a square R in the family Fn+1 . Since by (b) µ(R) ≤ εn+1 ≤ µ(Q), one has R ⊂ Q. We have, by (A-3) and (A-4), X X ν(Q) = ν(R) ≤ c−n−1 µ(R) ≤ c−n−1 µ(Q). R⊂Q,R∈Fn+1 R⊂Q,R∈Fn+1 298 JULIÀ CUF Í , JUAN JESÚS DONAIRE, PERTTI MATTILA AND JOAN VERDERA Since µ(Q) ≤ d(Q)α it is enough to show that c−n µ(Q) ≤ µ(Q)β/α which is c−n ≤ µ(Q)− log c/ log ε , that is, −n log c ≤ −(log c/ log ε) log µ(Q), or n ≤ log µ(Q)/ log ε, which is a consequence of µ(Q) < ε n . To finish, let x ∈ E and 0 < r ≤ 1. For some n, x belongs to a square Q ∈ Dn with d(Q)/4 ≤ r ≤ d(Q). Then B(x, r ) can meet at most 4d squares of Dn , and so by (A-5), ν(B(x, r )) ≤ 4d ν(Q) ≤ 4d C d(Q)β ≤ 4β+d C r β and (A-1) follows. □ Appendix B. The Riesz transforms in Rd We first slightly modify the argument in [Cufí et al. 2022] to show that ∞ 2 P n=1 an = ∞ yields divergence a.e. of the martingale. If the martingale converges in a set of positive measure, then also the principal values of the Riesz transform exist in a set E of positive measure, by the analog of Lemma 2.4. By a result of Tolsa [2014, Theorem 8.13] we find a set F ⊂ E of positive measure on which the singular Riesz transform operator is bounded on L 2 (µ|F ). In particular, the capacity of F associated with the Riesz kernel is positive and so also that of the Cantor set. The main result of [Mateu and Tolsa 2004] (see Theorem 1.2, p. 678 and its extension in the last formula in p. 696) states that the α-Riesz capacity of the Cantor set is comparable to P∞ 2 −1/2 n=1 an , so that positive capacity yields a convergent series. We remark that the previous argument uses very strong results, in particular the nonhomogeneous T (1)-Theorem of Nazarov, Treil and Volberg, to extract the subset F on which the singular Riesz transform is L 2 (µ|F ) bounded. In [Cufí et al. 2022] one resorts to Menger curvature, which is not available for kernels of homogeneity −α with 1 < α < d, and the proof is slightly simpler. It would be desirable to have a direct argument relating the series to the convergence of the martingale. The part of the stopping time argument of Section 3 that does not obviously extend to higher dimensions is related to the sector σ (z, 120◦ ). In particular, one should replace the 45◦ degrees sectors centred at the origin with one edge on a coordinate axis with other regions. We proceed as follows. Divide Rd into 2d regions (which in R3 are the usual octants) by requiring that each coordinate has a definite sign. For example, O = {x ∈ Rd : x1 ≥ 0, x2 ≥ 0, . . . xd ≥ 0} or O ′ = {x ∈ Rd : x1 ≤ 0, x2 ≥ 0, . . . xd ≥ 0} are such regions. Let us concentrate in the region O. Divide O in the d! subregions determined by a permutation σ of the d variables Oσ = {x ∈ Rd : 0 ≤ xσ (1) ≤ xσ (2) ≤ · · · ≤ xσ (d) }. EXISTENCE OF PRINCIPAL VALUES OF SINGULAR INTEGRALS ON CANTOR SETS 299 Note that the maximal angle between two vectors lying in a subregion Oσ is precisely arccos(d −1/2 ), which approaches 90◦ as d → ∞. Given a cone 0 with vertex at the origin and aperture θ , we would like to find a region Oσ contained in the cone 0. This can be done as follows. The axis of the cone is a ray emanating from the origin contained in Oσ for some σ . Taking θ = θ (d) < π close enough to π one can achieve Oσ ⊂ 0. Indeed, something stronger can be obtained: there exists a sufficiently small angle γ = γ (d) such that expanding Oσ in all directions by at most γ degrees one still remains in the cone 0. The planar argument now works with θ in place of 120◦ . One also needs to have enough linear isometries to transport one region Oσ into another Oσ ′ . Consider the following kinds of linear isometries. Fix a variable xi and take the mapping that leaves the other variables invariant and changes the sign to the xi variable. Given two variables xi and x j with i ̸= j consider the mapping that leaves the other variables invariant and interchanges xi and x j . Finally take the mapping x → −x. Let S be the set of such linear isometries. One can easily check that given two regions Oσ and Oσ ′ one can map one into the other by composing finitely many isometries in S . All these elements lead to a stopping time argument that proves Theorem 1.2. Acknowledgements The authors are grateful to X. Tolsa for various conversations on the subject and to the referee for many suggestions that have improved considerably the exposition. Pertti Mattila is grateful for the invitation to visit the Department of Mathematics at Universitat Autònoma de Barcelona. J. Verdera acknowledges support from the grants 2021-SGR-00071 (Generalitat de Catalunya), PID2020-112881GB-I00 and Severo Ochoa and Maria de Maeztu CEX2020-001084-M (Ministerio de Ciencia e Innovación). J. J. Donaire acknowledges support from the grant MTM2017-85666-P (Ministerio de Ciencia e Innovación) References [Cufí et al. 2022] J. Cufí, A. C. Ponce, and J. Verdera, “The precise representative for the gradient of the Riesz potential of a finite measure”, J. Lond. Math. Soc. (2) 106:2 (2022), 1603–1627. MR Zbl [Donaire et al. 2014] J. J. Donaire, J. G. Llorente, and A. Nicolau, “Differentiability of functions in the Zygmund class”, Proc. Lond. Math. Soc. (3) 108:1 (2014), 133–158. MR Zbl [Hungerford 1988] G. J. Hungerford, Boundaries of smooth sets and singular sets of Blaschke products in the little Bloch class, Ph.D. thesis, California Institute of Technology, 1988. [Mateu and Tolsa 2004] J. Mateu and X. Tolsa, “Riesz transforms and harmonic Lip1 -capacity in Cantor sets”, Proc. London Math. Soc. (3) 89:3 (2004), 676–696. MR Zbl [Mattila 1995] P. Mattila, Geometry of sets and measures in Euclidean spaces: fractals and rectifia- bility, Cambridge Studies in Advanced Mathematics 44, Cambridge University Press, 1995. MR Zbl 300 JULIÀ CUF Í , JUAN JESÚS DONAIRE, PERTTI MATTILA AND JOAN VERDERA [Mattila and Verdera 2009] P. Mattila and J. Verdera, “Convergence of singular integrals with general measures”, J. Eur. Math. Soc. 11:2 (2009), 257–271. MR Zbl [Shiryaev 1996] A. N. Shiryaev, Probability, 2nd ed., Graduate Texts in Mathematics 95, Springer, 1996. MR Zbl [Tolsa 2014] X. Tolsa, Analytic capacity, the Cauchy transform, and non-homogeneous Calderón– Zygmund theory, Progress in Mathematics 307, Springer, 2014. MR Zbl Received June 8, 2023. Revised September 26, 2023. 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PJM peer review and production are managed by EditF LOW® from Mathematical Sciences Publishers. PUBLISHED BY mathematical sciences publishers nonprofit scientific publishing http://msp.org/ © 2023 Mathematical Sciences Publishers PACIFIC JOURNAL OF MATHEMATICS Volume 326 No. 2 October 2023 Smooth local solutions to Schrödinger flows with damping term for 187 maps into symplectic manifolds B O C HEN and YOUDE WANG Modules over the planar Galilean conformal algebra arising from free 227 modules of rank one J IN C HENG , D ONGFANG G AO and Z ITING Z ENG F-algebroids and deformation quantization via pre-Lie algebroids 251 J OHN A LEXANDER C RUZ M ORALES , J IEFENG L IU and Y UNHE S HENG Existence of principal values of some singular integrals on Cantor sets, 285 and Hausdorff dimension J ULIÀ C UFÍ , J UAN J ESÚS D ONAIRE , P ERTTI M ATTILA and J OAN V ERDERA Certain Fourier operators and their associated Poisson summation 301 formulae on GL1 D IHUA J IANG and Z HILIN L UO