C A ] 8 N ov 2 01 9 SINGULAR INTEGRALS ON REGULAR CURVES IN THE HEISENBERG GROUP

We show that smooth  ́1-homogeneous horizontally odd kernels in the Heisenberg group induce Calderón-Zygmund operators on regular curves. This extends a theorem of G. David from 1984 to the Heisenberg group. CONTENTS

that the β-numbers on regular curves in γ Ă C satisfy the following square function estimate:ˆR 0ˆBpx 0 ,Rq β γ pBpx, rqq 2 dH 1 | γ pxq dr r R, Bpx 0 , Rq Ă C. (1. 3) The case of Lipschitz graphs was already contained in [32], where Jones deduced the L 2 -boundedness of C on Lipschitz graphs from the geometric condition (1.3). The square function estimate (1.3) is also valid for regular curves in R n , as shown by Okikiolu [42].
More recently, Tolsa [48] introduced the notion of α-numbers. These are, roughly speaking, measure-theoretic versions of Jones' β-numbers. Tolsa showed that odd mdimensional C 2 -smooth kernels in R n are CZ kernels for any m-regular measure µ on R n whose α-numbers satisfy a square function estimate analogous to (1.3). This improves on the result of David [17], since only C 2 -regularity of the kernel is required. Moreover, as in Jones' argument, the proof deduces the L 2 -boundedness of SIOs directly from bounds on a square function involving the α-numbers, without passing via Lipschitz graphs. However, the oddness of the kernels seems to be indispensable.
Investigating the connections between Lipschitz graphs, sets with BPLG, or admitting CDLG, square function estimates involving α's, β's, or other geometric quantities, and the L 2 -boundedness of SIOs, is known as the theory of uniform rectifiability. For more information, see [14,15,49].

Singular integrals on regular curves in H.
What are the natural kernels in H? In R n , the oddness assumption is prevalent, so why not study odd kernels in H? In R n , oddness is not only a matter of technical convenience: instead, it stems from the existence of "natural" odd kernels in R n , such as the Cauchy kernel 1 z , and its higher-dimensional counterpart, the Riesz kernel ∇|x| 2´n . SIOs associated to these kernels are of key importance in the theory of partial differential equations, see for example [21,50]. Following this train of thought in H, one is led to consider the "H-Riesz kernel" kppq " ∇ H }p}´2, and the question of whether k is a CZ kernel for 3-regular surfaces in H. This problem was first raised in [4]. Here, and in the introduction, }p} refers to the Korányi norm of p P H. For currently up-to-date results on the H-Riesz kernel, see [23].
The kernel kppq " ∇ H }p}´2 is not odd. Instead, as noted in [5], it is horizontally odd: 1 The property of a good kernel "k being a CZ kernel for a regular curve γ" means the same as above, namely that the maximal SIO induced by pk, H 1 | γ q defines an operator bounded on L p pH 1 | γ q, for 1 ă p ă 8. See Definition 2.15 for a more formal treatment.
1.3. Previous work. Above, we already discussed previous work concerning SIOs on 3-regular surfaces in H. SIOs on 1-regular subsets of H were first studied by Chousionis and Li in [7]. The kernels k : H z t0u Ñ C considered in [7] are not "good" in the sense of Definition 1.5. Instead, they are non-negative´1-homogeneous kernels of the form k α px, y, tq " p a |t|{}p}q α }p} , p " px, y, tq P H zt0u, α ě 1.
Chousionis and Li proved that k 8 is a CZ kernel for regular curves γ Ă H, and with Zimmerman they found a generalisation of this result to arbitrary Carnot groups [8].
Conversely, they also showed in [7] that if E Ă H is 1-regular, and k 2 is a CZ kernel for E, then E is contained on a regular curve. It may sound astounding that non-negative kernels could ever be CZ kernels. A partial explanation comes from noting that k α vanishes identically on the plane tpx, y, tq : t " 0u. Consequently, if ℓ Ă H is a horizontal line (see Definition 3.37), then the (maximal) SIO induced by pk α , H 1 | ℓ q is the zero operator.
In contrast, our good kernels vanish identically on the axis tpx, y, tq : x " 0 " yu, and the induced SIOs on horizontal lines can behave like the Hilbert transform. It is natural to long for a result which simultaneously generalises the work in [7], and the present paper. Here is one suggestion (caveat emptor!): Question 1. Let k : H z t0u Ñ C be a smooth´1-homogeneous function which is a CZ kernel for horizontal lines, with uniform constants. Is k then a CZ kernel for regular curves? 1.4. The proof of Theorem 1.6: an outline. In Section 1.1, we mentioned two approaches for studying SIOs on regular curves in R n : either reduce matters to the special case of Lipschitz graphs via "big piece" or "corona" methods, or take a more direct route via geometric square functions (α-numbers or β-numbers). In this paper, we take the former approach(es), as the latter appears to be difficult to execute for two separate reasons: ‚ The oddness of kernels in R n is critical in "quasiorthogonality" arguments, see [48], and horizontal oddness seems to be a poor substitute in this regard. ‚ Analogues of Jones' β-numbers have been extensively studied in H, see [24,35,37,38,36]. A surprising example of Juillet [35] shows that the L 2 -integral of the βnumbers appearing in (1.3) need not be bounded by H 1 pγq, for rectifiable curves γ Ă Bpx 0 , Rq. Instead, Li and Schul [37] proved a version of (1.3) where the exponent "2" is replaced by "4". This fact was used in [7] to show that k 8 is a CZ kernel for regular curves, but we were not able to employ it in the proof of Theorem 1.6 (except to deduce the "WGL", see Lemma 6.53). We then discuss the former approach. Heisenberg analogues of Lipschitz graphs are known as intrinsic Lipschitz graphs (iLGs), and they were introduced by Franchi, Serapioni, and Serra Cassano [25] in 2006. Their rectifiability properties, both qualitative and quantitative, have been investigated vigorously in recent years, see [6,11,22,26,39,40,43,44]. However, many of these papers have focused on 1-co-dimensional iLGs, whereas the objects relevant here are the 1-dimensional iLGs over horizontal subgroups of H, see Section 3.3. The first objective en route to Theorem 1.6 is to establish the result in the special case of 1-dimensional iLGs in H: Theorem 1.7. Good kernels are CZ kernels for iLGs over horizontal subgroups in H.
This result is the main news of the paper. Once it has been established, we still need to complete David's approach in [16], and prove the following statements: Theorem 1.8. Regular curves in H have big pieces of intrinsic Lipschitz graphs (BPiLG) over horizontal subgroups.
"Theorem". Let pX, dq be a proper metric space, let G be a family of m-regular sets in pX, dq, and let K be an m-dimensional standard kernel on X which is a CZ kernel for all G P G, uniformly. Then K is a CZ kernel for any m-regular set B Ă X which has "big pieces" of sets in G.
For a more precise statement, see Theorem 6.3. The proof is a straightforward adaptation of [18,Proposition 3.2] to proper metric spaces, and we claim very little originality: the main point is to check that the Besicovitch covering theorem is not used in an essential way. Regarding Theorem 1.8, we follow an approach of David and Semmes [20], by showing, first, that regular curves have big horizontal projections (BHP), and satisfy the weak geometric lemma for Jones' β-numbers. Then, a combination of these properties yields BPiLG. These arguments are quite well-known, and have even been adapted to 1-co-dimensional iLGs in H n , see [6,22]. Only verifying the BHP property for regular curves produces a minor "new" problem. The details are contained in Section 6.2.
So, the heart of the matter is Theorem 1.7, whose proof indeed takes up most of the paper. Adapting some arguments from [12], the proof of Theorem 1.7 may be reduced to a problem concerning certain 1-dimensional SIOs on R. More precisely, one is led to consider the standard kernel K B px, yq " 1 x´y expˆ2πi " B 2 pxq´B 2 pyq´1 2 rB 1 pxq`B 1 pyqspx´yq px´yq 2 ˙, (1.9) where B " pB 1 , B 2 q : R Ñ R 2 is a tame map. This simply means that B 1 is Lipschitz, and 9 B 2 " B 1 . Tame maps are quite entertaining, and they are thoroughly investigated in Section 3.1. The kernel K B is not antisymmetric, but we nevertheless manage to prove in Theorem 4.10 that K B is a CZ kernel on R. Unfortunately, this is not quantitative enough: to apply the kernels K B in the context of Theorem 1.7, we need to know that the CZ constant of K B , denoted }K B } C.Z. , depends polynomially on the "tameness constant" of B. A similar problem for Lipschitz functions (and graphs) already appears in David's work [16,18], but the solution is easier there: it is based on the "big piece theorem" stated below Theorem 1.8, plus the simple -and ingenious -observation that "L-Lipschitz graphs have big pieces of 9 10 L-Lipschitz graphs", see [18, p. 66]. We were not able to prove an analogue of this property for tame maps, see Question 2.
Instead, we found a weaker substitute: tame maps admit "corona decompositions" by tame maps with a smaller constant. More precise statements can be found in Section 3.1.1. We mentioned in Section 1.1 that Semmes [45] used corona decompositions (by Lipschitz graphs) to reduce SIO problems on regular curves to SIO problems on Lipschitz graphs. Applying his mechanism, and the tame-corona decomposition mentioned above, we can finally infer the polynomial dependence of }K B } C.Z. on the "tameness" of B. We refer to Section 5 for details.
We have now summarised the proof of Theorem 1.6, and explained most of the structure of the paper. Let us add that in Section 2, we merely collect standard preliminaries on Calderón-Zygmund theory. In Section 3, we introduce tame maps, the Heisenberg group, and intrinsic Lipschitz graphs, and prove the corona decomposition for tame maps. In Section 4, we reduce the proof of Theorem 1.7 to the study of the kernel K B -or, as it really turns out, K A,B -and establish the "qualitative" fact that K A,B is a CZ kernel on R. The quantitative version is the main content of Section 5, and this section concludes the proof of Theorem 1.7. Finally, in Section 6, we prove the "BPiLG" Theorem 1.8 and use it to deduce Theorem 1.6 from Theorem 1.7.
Acknowledgements. This research was mostly conducted during the Simons Semester "Geometry and analysis in function and mapping theory on Euclidean and metric measure spaces" at IM PAN, Warsaw. We would like thank all the organisers, in particular Tomasz Adamowicz, and the staff at IM PAN, for their support and hospitality during our stay in Warsaw.
A standard kernel (SK), without reference to the dimension, will mean a 1-SK.
An important class of SKs are those induced by good kernels k : H z t0u Ñ C, recall Definition 1.5. Setting Kpp, qq :" kpq´1¨pq, one obtains an SK satisfying Definition 2.1(1)-(2) with α " 1 2 , see Proposition 3.35. Further, the kernels Kpp, qq " kpq´1¨pq "evaluated on" intrinsic Lipschitz graphs yield another class of interesting SKs, this time in R. We record some details right away: Example 2.2. Let A : R Ñ R be an M -Lipschitz function, and let B " pB 1 , B 2 q : R Ñ R 2 be an N -tame function (here we just need to know that B 1 is N -Lipschitz, and 9 B 2 " B 1 ; see Section 3.1), where M, N ě 1. Let k : RˆR z △ Ñ C be an SK. Then, the kernel K k,A,B px, yq :" kpx, yqe A,B px, yq :" kpx, yq expˆ2πi " Apxq´Apyq x´y`B 2 pxq´B 2 pyq´1 2 rB 1 pxq`B 1 pyqspx´yq px´yq 2 i s an SK, with }K k,A,B } α,strong }k} α,strong maxtM, N u. To see this, fix x, x 1 , y P R with |x´x 1 | ď |x´y|{2, and write |K k,A,B px, yq´K k,A,B px 1 , yq| ď |kpx, yq´kpx 1 , yq|`|kpx 1 , yq||e A,B px, yq´e A,B px 1 , yq|, and use the SK estimates for k. The problem then reduces to estimating |e A,B px, yq´e A,B px 1 , yq|, which further reduces (using that t Þ Ñ e 2πit is 2π-Lipschitz) at finding upper bounds for We leave it to the reader to check that apx, x 1 , yq M |x 1´x |{|x´y|. To see that also |bpx, x 1 , yq| N |x 1´x |{|x´y|, we first infer from the tameness of B that B 2 P C 1 pRq, and 9 B 2 " B 1 , see Remark 3.2. Therefore, for x ‰ y, 3) The tameness of B also implies that B 1 is N -Lipschitz, so a little computation shows that the x and y derivatives of the right hand side are N {|x´y| almost everywhere. Now it follows from the fundamental theorem of calculus that bpx, x 1 , yq N |x´x 1 |{|x´y|, as claimed.
The kernel K k,A,B with kpx, yq " px´yq´1 will have special significance in the paper, and it will be denoted simply K A,B .
2.2. Generalised standard kernels and CZOs. In Section 5, we will encounter kernels which are not quite SKs in the sense above, but satisfy the following relaxed conditions: Definition 2.4. Let pX, dq be a proper metric space. A Borel function K : XˆX z △ Ñ C is a k-dimensional generalised standard kernel (k-GSK) if the "size" condition in Definition 2.1(1) holds with constant C ě 1, and moreover K satisfies the following two inequalities for all Radon measures µ on X, for all f P L 1 loc pµq, and for all closed balls B Ă X: Here M µ,k is the "radial" maximal function of order k: The best constant "C" here will be denoted by }K}.
On first sight, it may appear odd that the constant "C" needs to be independent of the choice of the Radon measure µ on X. However, Proposition 2.7 below shows that any k-SK K : XˆX z △ Ñ C is a k-GSK, with }K} α }K} α,strong . Proposition 2.7. Let pX, dq be a proper metric space, let k ą 0, and let K : XˆX z △ Ñ C a k-SK. Then (2.5)-(2.6) hold with a constant C α,k }K} α,strong .
The operator T µ,ǫ is called the ǫ-SIO induced by pK, µq. We also define the maximal SIO If the ǫ-SIOs are uniformly bounded on L 2 pµq, 16) we say that K is a Calderón-Zygmund kernel (CZ kernel) for µ, and we write Remark 2.17. In the introduction -notably the statements of the main theorems -we used the terminological convention that K is a CZ kernel for µ if }Tμ } L p pµqÑL p pµq ă 8 for all 1 ă p ă 8. There is no serious conflict: if µ is a measure on a proper metric space pX, dq satisfying the growth condition (2.14), and K : XˆX z △ Ñ C is a k-SK, then the condition (2.16) implies that }Tμ } L p pµqÑL p pµq ă 8 for all 1 ă p ă 8, see [41,Theorem 1.1]. In particular, all of this is true for kernels of the form pp, qq Þ Ñ kpq´1¨pq, where k : H z t0u Ñ C is a good kernel, and for H 1 measures restricted to regular curves in H.
The reason why we chose to define "CZ kernels" as in Definition 2.15 is that we, sometimes, want to apply the definition to GSKs: the maximal SIO characterisation above may well remain valid in this generality, but at least we have not seen it written down.
For a big part of this paper, we will only be concerned with CZOs, ǫ-SIOs, and maximal SIOs induced by GSKs on R, and the measure µ " L 1 . We will drop the sub-index "L 1 " in this situation, and write T, T ǫ , T˚in place of T L 1 , T ǫ,L 1 , TL 1 . Also, on R, we will only consider CZ kernels for L 1 , and write }K} C.Z. :" }K} C.Z.pL 1 q .
We will now gather some basic facts about the case X " R (although many of these statements have generalisations to metric spaces, see for example [41]).
Here M is the (non-centred) Hardy-Littlewood maximal function on R.
Theorem 2.21 (T 1 theorem). Let T be an operator induced by a bounded SK K : RˆR z △ Ñ C. Then, T is a CZO if and only if T 1, T t 1 P BMO, and T satisfies the weak boundedness property (WBP). In this case, For a proof, see [28,Theorem 8.3.3], or the original reference [19].
Definition 2.23 (Definitions of T 1, T t 1, and WBP). Under the assumptions of the T 1 theorem, the condition T 1 P BMO means that there exists a constant C ě 1 with the following property. If ϕ P C 8 pRq is a "smooth H 1 -atom" supported on a ball B 0 , i.e. satisfies The best constant "C", as above, is the definition of the quantity "}T 1} BMO " in (2.22). The condition T t 1 P BMO means, by definition, that (2.25) holds with xT pϕq, by on the left hand side. Finally, the WBP means that if ϕ, ψ are smooth non-negative functions supported on Bp0, 1q Ă R, with maxt}ϕ} C 5 , }ψ} C 5 u ď 1, then |xT pϕ x,r q, ψ x,r y| ď Cr´1, x P R, r ą 0. (2.26) Here f x,r pyq :" r´1¨f ppy´xq{rq. The best constant "C" in (2.26) is the definition of the quantity "}T } WBP " in (2.22).
We have established the following corollary of the T 1 theorem: Corollary 2.29. Let K : RˆR z △ Ñ C be an SK, and assume that the testing conditions (2.27) hold for some C ě 1, uniformly for ǫ ą 0. Then }K} C.Z. α C`}K} α,strong .

INTRINSIC LIPSCHITZ GRAPHS AND TAME MAPS
3.1. Tame maps. We say that a map pφ 1 , φ 2 q : We make a few hopefully clarifying remarks about the definition of tameness. First, condition (3.1) is implied (with twice the constant) by a "1-sided" version of itself:ˇˇˇφ Indeed, just apply the inequality above to both px, yq and py, xq to arrive at (3.1). Second, (3.1) implies that φ 1 is L-Lipschitz (by the triangle inequality). Third, assume that E contains an open interval I. Then (3.1) clearly implies that 9 φ 2 exists on I, and 9 φ 2 " φ 1 . Conversely, assume that φ " pφ 1 , φ 2 q : I Ñ R 2 , where I Ă R is an open interval, φ 1 is L-Lipschitz, and 9 φ 2 " φ 1 . Then (3.3) is satisfied, because, for x ă y, |rφ 2 pxq´φ 2 pyqs´φ 1 pxqpx´yq| ďˆy So, (3.1) and (3.3) are essentially short ways of writing that 9 φ 2 " φ 1 for a " L-Lipschitz function φ 1 without actually mentioning the derivative of φ 2 . We also note for future reference that the class of L-tame maps is preserved under the following operations: (1) Pre-composing with a translation in R.
(2) Adding a map of the form L a,b pxq :" pa, ax`bq, with a, b P R. In fact, the second point is just a special case of the fact that adding an L 1 -tame map to an L 2 -tame map produces an pL 1`L2 q-tame map: note that L a,b is 0-tame for any a, b P R.
The next lemma observes that tameness is preserved under parabolic rescaling: Lemma 3.5. Let B " pB 1 , B 2 q : E Ñ R 2 be L-tame, where E Ă R, and let r ą 0. Then, the map B r : r´1¨E Ñ R 2 , defined by B r pxq :" pB r 1 pxq, B r 2 pxqq :"`1 r B 1 prxq, 1 r 2 B 2 prxqȋ s also L-tame.
Proof. For x, y P R, x ‰ y, fixed, we note thaťˇˇˇB r 2 pxq´B r 2 pyq x´y´B r 1 pxqˇˇˇˇ" as desired.
We then record an extension result: Proposition 3.6. An L-tame map defined on E Ă R extends to an 18L-tame map defined on R.
Proof. Let φ " pφ 1 , φ 2 q : E Ñ R 2 be L-tame. By assumption, φ 1 is Lipschitz, and also φ 2 is locally Lipschitz by (3.1). So, extending φ 1 , φ 2 to continuous maps onĒ is no problem, and then (3.1) remains valid onĒ. So, we may assume that E is closed to begin with, and we write where I are the components of R z E. We will extend φ to each interval in I individually.
There are at most two unbounded intervals I P I. Both of them have an endpoint in E, and we define φ 1 on I to be the constant attained at the endpoint, say x. Then, we define φ 2 pyq :"ˆy x φ 1 psq ds, y P I.
Evidently φ 1 remains L-Lipschitz, and we will worry about condition (3.1) later. Next, fix I " rx, ys P I with x, y P E and x ă y. Assume for minor notational convenience that This can be achieved by applying the operations (1)-(2) described above. To understand the problem we are now facing, consider any extension of φ " pφ 1 , φ 2 q to I, denoted by φ I " pφ I 1 , φ I 2 q. Then, if φ I is supposed to be tame, we should have 9 φ I 2 " φ I 1 , and this forces So, φ I 1 needs to be chosen so that (3.8) holds -and on the other hand φ I 1 needs to be a " L-Lipschitz extension of φ 1 . In fact, we claim that φ I 1 can be taken 7L-Lipschitz. Let us first attempt the linear extensioñ This is an L-Lipschitz extension of φ 1 , but which may not agree with φ 2 pyq, i.e. the left hand side of (3.8). However, we are not too far off the mark. Recalling (3.7), and then using the tameness assumption (3.1), we havěˇˇˇφ Now, to fix the discrepancy between (3.9) and (3.8), we choose a 6L-Lipschitz function η I : r0, ys Ñ R satisfying η I p0q " 0 " η I pyq andˆy 0 η I psq ds " φ 2 pyq´φ 1 pyqy 2 . (3.11) For example, one can take η I " cη 0 , where |c| ď 1, and η 0 psq " # 6Ls, s P r0, y 2 s, 6Lpy´sq, s P r y 2 , ys, which coincides with the upper bound in (3.10). Finally, we set φ I 1 :"φ I 1`ηI , which is a 7L-Lipschitz extension of φ 1 (by the first point in (3.11)), and we define φ I 2 in the only possible way: This function extends φ 2 by a combination of (3.9) and the second point in (3.11). It remains to check that the tameness condition (3.1) is satisfied on R, with constant 18L; in fact, we check the 1-sided condition (3.3) with constant 9L. Pick distinct x, y P R. If x, y P E, there is nothing to prove. The same is true if x, y are contained on (the closure of) a common interval in I, because 9 φ 2 " φ 1 on these intervals, and recalling the estimate (3.4). So, assume that x P E and y P I P I with x ă y, say. Let x 1 P E X rx, yq be the left endpoint of I. Then, use the triangle inequality multiple times: |rφ 2 pxq´φ 2 pyqs´φ 1 pxqpx´yq| ď |rφ 2 pxq´φ 2 px 1 qs´φ 1 pxqpx´x 1 q| |rφ 2 px 1 q´φ 2 pyqs´φ 1 px 1 qpx 1´y q| This completes the proof.
3.1.1. Corona decomposition for tame maps. In this section, we prove the first main result of this paper, a corona decomposition for maps that are tame in the sense of (3.1). We start with the following rather obvious definition: Definition 3.13 (Tame-linear and tame-affine maps). A map φ " pφ 1 , φ 2 q : R Ñ R 2 is called tame-linear (or affine) if φ 1 : R Ñ R is linear (or affine) and 9 It would be nice to know the answer to the following question: Question 2. Does there exist a constant δ ą 0 with the following property? Let φ : r0, 1s Ñ R 2 be 1-tame. Then there exist a tame-linear map L : R Ñ R 2 and a p1´δq-tame map φ δ : r0, 1s Ñ R 2 such that |tx P r0, 1s : φpxq " rφ δ`L spxqu| ě δ.
In other words: do 1-tame maps have big pieces of p1´δq-tame maps (up to subtracting a tame-linear map)? Since we were not able to answer this question, we show something slightly weaker, namely that 1-tame maps admit a "corona decomposition" with η-tame maps, for any η ą 0. To formulate the statement, we recall some terminology. Definition 3.14 (Dyadic intervals and trees). We write "D" for the standard dyadic intervals of R. For j P Z, we further write D j Ă D for the dyadic intervals Q of length |Q| " 2´j. A collection T Ă D is called a tree if (T1) T contains a "top interval" QpT q, that is, a unique maximal element.
Second, the intervals in G can be decomposed into a "forest" F of disjoint trees T , For every T P F there exists a 2-tame-linear map L T : R Ñ R 2 and an η-tame map ψ T : R Ñ R 2 such that ψ T`LT approximates φ well at the resolution of the intervals in T : In (3.19), d π refers to the parabolic metric on R 2 : d π ppx, sq, py, tqq :" maxt|x´y|, a |s´t|u, px, sq, py, tq P R 2 , and 2Q is the interval with the same midpoint but twice the length of Q. The proof of Theorem 3.15 uses, as a black box, the corona decomposition for R-valued Lipschitz functions on R. This statement looks very similar to the one of Theorem 3.15: Theorem 3.20. For every η P p0, 1q, there exists a constant C ě 1 such that the following holds. Let φ : R Ñ R be 1-Lipschitz. Then, there exists a decomposition D " B 9 YG with the properties (3.16), (3.17), (3.18), and such that the following holds. For every T P F there exists a 2-Lipschitz linear function L T : R Ñ R and an η-Lipschitz function ψ T : R Ñ R such that |φpsq´pψ T`LT qpsq| ď η|Q|, s P 2Q, Q P T . This statement follows, after a moment's thought, from the corona decomposition in [15, p.61, (3.33)]. We give the details in Appendix A. Before proving Theorem 3.15, we record version of Theorem 3.15 for N -tame maps with N ě 1. The main point here is that the Carleson packing constants do not depend on "N ", which only makes an appearance in the "quality of approximation" in (3.23). Corollary 3.22 (Corona for N -tame maps). For every η P p0, 1q, there exists a constant C ě 1 such that the following holds. Let φ : R Ñ R 2 be N -tame, N ě 1. Then, there exists a decomposition D " B 9 YG with the properties (3.16), (3.17), (3.18), and such that the following holds. For every T P F, there exists a 2N -tame-linear map L : R Ñ R 2 and an pηN q-tame map Proof. The mapφ :" N´1φ : R Ñ R 2 is 1-tame, so Theorem 3.15 applies to it verbatim. This yields the desired decomposition D " B 9 YG and, for each T P F, a 2-tame-linear map r L T : R Ñ R 2 , and an η-tame mapψ T : R Ñ R 2 , such that (3.19) holds forφ,ψ T , r L T . Now, we define the pηN q-tame map ψ T :" Nψ T , and the 2N -tame-linear map L T :" There is also a similar version of Theorem 3.20 for M -Lipschitz functions, M ě 1, but we omit stating this explicitly. We then turn to the proof of Theorem 3.15.
We apply the Lipschitz corona decomposition, Theorem 3.20, to φ 1 with the parameter δ :" mintη 2 {5, η{17u ą 0. The result is a decomposition D " B Y G of the type desired in the statement Theorem 3.15, accompanied with the trees T P F, and corresponding δ-Lipschitz functions φ T : R Ñ R and linear 2-Lipschitz maps L T : R Ñ R with the property that Fix a tree T P F, and consider the top interval QpT q " rx, ys. Based on the existence of the function φ T , we would now like to produce an η-tame function ψ T : rx, ys Ñ R 2 satisfying (3.19). The tame-linear part will be defined in the obvious way: To define ψ T , probably the first idea to try is to set ψ 1 :" φ T , and define The good news are that 9 ψ 2 " ψ 1 , and ψ 2 pxq " φ 2 pxq, so at least (3.19) is satisfied for s " x (recalling that (3.24) holds, and noting that φ 2 pxq " ψ 2 pxq`P T pxq). The bad news is that there is no a priori reason why |rψ 2`PT spsq´φ 2 psq| would be small for any s P px, ys. To fix this, we in fact need to modify φ T slightly before defining ψ 1 and ψ 2 exactly as above.
Let SpT q be the collection of minimal intervals in T (possibly an empty collection). Also, write for the set of points in QpT q in "infinite branches" of T . Observe that, by (3.24), we have Now, for S P SpT q fixed, we will slightly modify the restriction of φ T to 1 2 S, which is the interval with the same centre but half the length as S. The geometric feature of 1 2 S needed in the future is that if Q P T with |Q| ă |S|, then This is clear, because |Q| ă |S| forces Q X S " H by the minimality of S P SpT q.
While modifying φ T , we want to maintain the property that φ T is 17δ-Lipschitz, and that (3.24) holds with "δ" replaced by "5δ". However, in addition, we want to arrange that The idea is the same as the one already seen during the proof of Proposition 3.6: we want to find a 16δ-Lipschitz function η S : 1 2 S Ñ R with the properties that This is easily done, using the "triangle" function familiar from (3.12), and observing thaťˇˇˇˆS . Now, if we replace φ T by φ T`ηS on S, we find that the "new" φ T is 17δ-Lipschitz, and (3.27) holds. Moreover, since }η S } L 8 pSq ď 4δ|S|, there is some hope that (3.24) remains valid with "δ" replaced by "5δ". To prove this carefully, fix Q P T and s P 2Q. During the procedure above, we only modified φ T on sets of the form 1 2 S, with S P SpT q. So, if s R 1 2 S for any S P SpT q, then (3.24) is certainly valid, with original constant. So, assume that s P 1 2 S for some S P SpT q. Then s P 2Q X 1 2 S, so (3.26) forces |S| ď |Q|. Consequently, }η S } L 8 ď 4δ|S| ď 4δ|Q|.
Since the "original" φ T only differs from the "new" φ T on 1 2 S by the function η S , we see that Now, assume that similar modifications to φ T have been performed inside all intervals S P SpT q, and in particular (3.27) holds for all S P SpT q. We infer the following corollary: if s P QpT q, and either s P E or s P BS with S P SpT q, thenˆs Recall that x is the left endpoint of QpT q. Now, with the fine-tuned definition of φ T , we proceed as planned, setting ψ 1 :" φ T and defining ψ 2 as in (3.25). Since the map ψ " pψ 1 , ψ 2 q : QpT q Ñ R is now 17δ-tame, and 17δ ď η by definition, it remains to check that (3.19) holds for all x P Q P T . This amounts to checking that First, consider s P E. Then, since 9 φ 2 " φ 1 , we have So, the difference in (3.29) is zero, as it should be. Next, fix some Q P T , and consider s P 2Q. Then, there exists a point satisfying |s´s 1 | ď |Q|. Then φ 2 ps 1 q " ψ 2 ps 1 q`P T ps 1 q, repeating the computation on line (3.30). Consequently, noting in the last inequality that rs 1 , ss Ă 2Q, so (3.24) (with "5δ" in place of "δ") holds for all points in rs 1 , ss. We conclude from this estimate and (3.24) that recalling that ? 5δ ď η. The proof is complete.
Tame maps will now go away for a moment, but they will return in Section 3.3, where we relate them to intrinsic Lipschitz functions on the Heisenberg group.

Definition 3.34 (Horizontal gradient).
Let Ω Ă H be an open set. The horizontal gradient of a C 1 function u : Ω Ñ R is defined by We record that good kernels (Definition 1.5) give rise to SKs.
Proof. The first claim follows immediately from the´1-homogeneity of k, as |kppq| "ˇˇk`δ }p}`δ}p}´1 ppq˘˘ˇˇ" }p}´1ˇˇk`δ }p}´1 p˘ˇˇď }p}´1 sup and hence, since k˝δ λ " λ´1k, The exponent α " 1 2 arises when verifying the Hölder continuity of q Þ Ñ Kpq´1¨pq. Definition 3.36 (Homogeneous subgroups). A subgroup of H is homogeneous if it is closed under dilations. Homogeneous subgroups of H are either contained in the xy-plane, in which case they are called horizontal, or they contain the t-axis, in which case they are said to be vertical.

Definition 3.38 (Projections and components).
Let W Ă H be a vertical subgroup of topological dimension 2. We associate to W the unique horizontal subgroup L Ă W, and the complementary horizontal subgroup V. The choice of V is somewhat arbitrary, but we declare here V to be the Euclidean orthogonal complement of L in the xy-plane. We write T for the t-axis. Then, every point p P H has a unique "coordinate" decomposition where w " l¨t " t¨l P W with l P L and t P T, and v P V. This decomposition gives rise to the vertical projections π W : H Ñ W and π T : H Ñ T, given by p Þ Ñ w and p Þ Ñ t, and the horizontal projections π V : H Ñ V and π L : H Ñ L, given by p Þ Ñ v and p Þ Ñ l, respectively. The horizontal projections are 1-Lipschitz group homomorphisms, while π W and π T are neither Lipschitz maps nor group homomorphisms. Nevertheless, π T and π W satisfy for some absolute constant C ě 1. If φ : X Ñ W is a map, where X is any set, we define the first and second components of φ to be the functions φ 1 " π L˝φ : X Ñ L and φ 2 " π T˝φ : X Ñ T.
Remark 3.40. If W " LˆT is a vertical subgroup with complementary subgroup V, we will write in coordinates W " ty¨t : y P L and t P Vu -tpy, tq : y, t P Ru " R 2 . Similarly, V will be identified with R. Under these identifications, the components φ 1 : V Ñ L and φ 2 : V Ñ T of any map φ : V Ñ W can be seen as functions R Ñ R, and in particular the derivative notation " 9 φ j " should be understood in this sense.
3.3. Intrinsic Lipschitz graphs. We define intrinsic Lipschitz functions and graphs over horizontal subgroups in H. On the one hand, this is just a special case of a definition of Franchi, Serapioni, and Serra Cassano [25]. On the other hand, intrinsic Lipschitz functions over horizontal subgroups have nicer properties than those over vertical subgroups, essentially because π V is a group homomorphism.
Proposition 3.43. A set Γ Ă H is an intrinsic Lipschitz graph over a horizontal subgroup V if and only if the horizontal projection π V restricted to Γ is injective with metric Lipschitz inverse Φ Γ : π V pΓq Ñ Γ.
Proof. Let Γ Ă H be an intrinsic L-Lipschitz graph over V. If p, q P Γ then which implies by the triangle inequality that }q´1¨p} ď p1`Lq}π V pqq´1¨π V ppq}. Consequently, the projection π V restricted to Γ is bilipschitz, so the map Φ Γ : π V pΓq Ñ Γ, given by the relation π V pΦ Γ pvqq " v, is well-defined and p1`Lq-Lipschitz Conversely, assume that Γ Ă H is a set such that the horizontal projection π V restricted to Γ is injective with L-Lipschitz inverse Φ. Then, if p " Φpvq, q " Φpv 1 q P Γ, we have which shows that Γ is an intrinsic CL-Lipschitz graph over V.
Remark 3.45. We record that every intrinsic L-Lipschitz graph Γ Ă H can be parametrised by an intrinsic L-Lipschitz function defined on E :" π V pΓq Ă V. Simply, let Φ Γ : E Ñ Γ be the map defined in Proposition 3.43, and let φ Γ pvq :" π W pΦ Γ pvqq. (3.46) Thus, Γ is parametrised by φ, and φ is intrinsic L-Lipschitz by definition. Proof. Indeed, recall from (3.46) that φpvq " π W pΦpvqq, where Φ : E Ñ Γ is the graph map of Γpφq. Consequently φ 1 " π L˝Φ . Then, using the fact that π L is a group homomorphism, we infer that We conclude this section with an area formula for intrinsic Lipschitz graphs over horizontal subgroups.
Proposition 3.48. Let φ " pφ 1 , φ 2 q : I Ă V Ñ W be an intrinsic Lipschitz map defined on an interval I Ă V, and let Φ be its graph map. Then, ΦpIq is a 1-regular subset of pH, dq and Proof. By Proposition 3.43, the map Φ : I Ñ pH, dq is a Lipschitz curve. Since Φ is injective, the length with respect to the metric d of a subcurve Φpra, bsq, ra, bs Ă I, agrees with H 1 pΦpra, bsqq, see for instance [2, Theorem 2.6.2.]. Moreover, length |¨| pπpΦpra, bsqqq ď length d pΦpra, bsqq ď length cc pΦpra, bsqq, (3.50) where the left-hand side denotes the Euclidean length of the image of Φpra, bsq under the projection π : H Ñ R 2 , px, y, tq Þ Ñ px, yq, and d cc is the standard sub-Riemannian distance on H, see [1]. Since π˝Φ is (Euclidean) Lipschitz, the left-hand side of (3.50) equalŝ b a |pπ˝Φq 1 pvq| dv, and the same is true for the right-hand side, cf. e.g. [29]. Using we have thus established (3.49) for A " ra, bs. The case of Borel sets A Ă I follows by approximation.
3.3.1. Connection between tame maps and intrinsic Lipschitz graphs. In this section, let W " tp0, y, tq : y, t P Ru, L " tp0, y, 0q : y P Ru, and V " tpx, 0, 0q : x P Ru. As we discussed in Remark 3.40, we will identify W -R 2 and V -R -L. With these identifications, we have the following relationship between intrinsic Lipschitz functions and tame maps.
Proof. A formula for the vertical projection π W is π W px, y, tq " py, t´x y 2 q, px, y, tq P H, while π V px, y, tq " x. The graph map of φ is given by Spelling out the last condition, one finds that  In conclusion, if E is an interval, the best constants in the inequalities (3.53) and (3.54) are actually within a multiple of "2" from each other.
Thanks to the connection between tame maps and intrinsic Lipschitz functions, Proposition 3.6 (extension of tame maps) implies an extension result for intrinsic Lipschitz graphs over horizontal subgroups.

THE EXPONENTIAL KERNEL APPEARS
4.1. Good kernels and intrinsic Lipschitz graphs. We fix a good kernel k : H zt0u Ñ C, and gradually start proving that it is a CZ kernel for (H 1 restricted to) any intrinsic Lipschitz graph over a horizontal subgroup in H. We fix a horizontal subgroup V with complementary vertical subgroup W, and an intrinsic L-Lipschitz function φ " pφ 1 , φ 2 q : V Ñ W, for L ě 1. We assume with no loss of generality that V -tpx, 0, 0q : x P Ru -R and W -tp0, y, tq : y, t P Ru -R 2 . The main point of this section is to show how Theorem 1.7 can be reduced to a statement involving only Lipschitz functions and tame maps defined on R, see Theorem 4.8 below.
Let Φ be the graph map of φ, and let Γ " ΦpVq Ă H be the intrinsic graph of φ. Write µ :" H 1 | γ , and let K : HˆH z △ Ñ C be the SK Kpp, qq :" kpq´1¨pq. We start by inferring from the area formula, Proposition 3.48, that for all w P R and g P Ť 1ăpă8 L p pRq. Since we are reduced to considering the ǫ-SIO T ε :" T ε,L 1 induced by the kernel pv, wq Þ Ñ KpΦpwq, Φpvqq, namely The truncations appearing in (4.1) and (4.2) are different, but the proof of Proposition 3.43 shows that |v´w| ď dpΦpvq, Φpwqq ď p1`Lq|v´w|, v, w P R. A standard maximal function argument then implies that there is a constant C ě 1, depending only on K and L, such that So, to prove that K is a CZ kernel for µ, it suffices to show that pv, wq Þ Ñ KpΦpwq, Φpvqq is a CZ kernel (for L 1 ). Recalling (3.52), and using the´1-homogeneity and horizontal oddness of k, we obtain the following explicit expression for the kernel of interest: Here φ 1 is L-Lipschitz by (3.53), and pφ 1 ,´φ 2 q : R Ñ R 2 is a 2L 2 -tame function by Proposition 3.51, so the terms are bounded by 1 in absolute value. So, the values of kpp2Lq´1, θ 1 , θ 2 q for pθ 1 , θ 2 q P R 2 outside r´1, 1s 2 never appear in the final expression on line (4.4), and having already arrived on this line, we may assume that pθ 1 , θ 2 q Þ Ñ kpp2Lq´1, θ 1 , θ 2 q is 2π-periodic in both variables in θ 1 , θ 2 (and evidently smooth as a function on R 2 ). We learned this trick from [18, p. 54]. Under this assumption, we may expand pθ 1 , θ 2 q Þ Ñ kpp2Lq´1, θ 1 , θ 2 q as a Fourier series kpp2Lq´1, θ 1 , Here Since pθ 1 , θ 2 q Þ Ñ kpp2Lq´1, θ 1 , θ 2 q is smooth, the constants c n decay rapidly as |n| Ñ 8. Now, going back to the original kernel KpΦpwq, Φpvqq, we note by combining (4.4) and (4.6) that Due to the rapid decay of the coefficients c n as |n| Ñ 8, it remains to show that }K n } C.Z. polyp|n|q. This can be deduced from the subsequent proposition, whose proof follows by combining techniques developed by Christ [10], David [17], Hofmann [30], and Semmes [46]: There exists a constant C ě 1 such that the following holds. Let M, N ě 1. Let A : R Ñ R be M -Lipschitz, and let B : R Ñ R 2 be N -tame. Then the kernel is a CZ kernel for L 1 with Theorem 4.8 will be proven in Section 5, in more general form, see Theorem 5.4.
Proof of Theorem 1.7 assuming Theorem 4.8. From (4.3) and (4.7), we infer that To see that the right hand side is finite, it suffices by the discussion above to show that there exists a constant C ě 1 such that, for every n " pn 1 , n 2 q P Z 2 , the kernel

Calderón commutators appear.
Let A : R Ñ R be Lipschitz, let B : R Ñ R 2 be tame, and consider the SK We mention that Theorem 4.8 does not immediately, or even easily, follow from Theorem 4.10, because we are interested in the polynomial dependence on M and N . The sharper result will be derived "by induction" in Section 5, and the main result of this section will be the "base case" of that induction.
We will show the CZ property of K A,B by decomposing the kernel into a sum of simpler ones, resembling Calderón commutators, then proving separately that they are CZ kernels, and finally summing up the results. In fact, using that e 2πix " ř ně0 p2πixq n {n!, we first write Then, the terms S n are further decomposed as follows: Motivated by this decomposition, we define the standard kernels C m,n px, yq :" 1 x´y " Apxq´Apyq x´y ı m " are standard kernels with }K A } α,strong p1`M q}K} α,strong and }K B } α,strong p1`N q}K} α,strong .
For the second inequality, use expansion (2.3), which reduces matters to the Lipschitz constant of B 1 (i.e. N ). It follows, by iteration, that if A is 1-Lipschitz and B is 1-tame, the kernel C m,n satisfies }C m,n } strong ď C m`n`1 for some absolute constant C ě 1.
The proof of the following theorem will occupy most of this section. Theorem 4.14. Let A : R Ñ R be 1-Lipschitz, let B " pB 1 , B 2 q : R Ñ R 2 be 1-tame, and let m, n ě 0. Then }C m,n } C.Z. ď C m`n`1 , where C ě 1 is an absolute constant.
It follows immediately from Theorem 4.14 that S n is a also a CZ-kernel with and finally that K A,B is a CZ kernel with So, Theorem 4.10 follows from Theorem 4.14. We start with a few preparations to prove the latter. This is a special case of Jones' traveling salesman theorem [33], but the case for Lipschitz graphs in R 2 is much simpler, see the book of Garnett-Marshall, [27, Chapter X, Lemma 2.4]. The quadratic dependence on LippAq follows from the LippAq " 1 case by scaling (noting that β cA pBpx, sqq " cβ A pBpx, sqq). The following lemma shows that the β-number in (4.15) also controls deviations from affine maps defined via averaging the gradient: Lemma 4.17. Let ψ P C 8 pRq be a standard bump function: ψ " 1, ψ ě 0 and spt ψ Ă Bp0, 1q, and ψp´zq " ψpzq. For s ą 0, let ψ s pxq :" s´1¨ψpx{sq. For a Lipschitz function A : R Ñ R, x P R, and s ą 0, define the linear map y Þ Ñ L x,s pyq :" P s pA 1 qpxqy, . where P s pA 1 qpxq :" pA 1˚ψ s qpxq. 2 Then, |Apxq´Apyq´L x,s px´yq| s ψ β A pBpx, sqq, y P Bpx, sq.

Boundedness of the Calderón commutators.
In this section, we prove Theorem 4.14. To a large extent, we can use arguments in [10] and [30], but the details look a little different, so we record them fairly completely. Fix a 1-Lipschitz function A : R Ñ R, a 1-tame map B " pB 1 , B 2 q : R Ñ R 2 , and m, n ě 0. We abbreviate C m,n px, yq :" Kpx, yq :" 1 x´y

"
Apxq´Apyq x´y (4.21) so K is antisymmetric only when n is even. On the other hand, the kernels of standard Calderón commutators (i.e. the kernels K above with n " 0) are always antisymmetric.
Proof. We plan to verify the testing conditions (2.27), so let K ǫ px, yq :" ψ ǫ px´yqKpx, yq, as above (2.27), where ψ is even, and (4.22) For simplicity of notation, the smooth ǫ-SIO is denoted T : The presence of the "ǫ" will have a rather negligible impact on the argument. Let B 0 " Bpx 0 , Rq Ă R be a ball, and let η P C 8 pRq with 1 2B 0 ď η ď 1 3B 0 . After performing the changes of variables x Þ Ñ Rx 1 and y Þ Ñ Ry 1 , and using Lemma 3.5, we may reduce to the case R " 1. Then, pre-composing A, B with a translation, we may also take x 0 " 0. We claim that whenever b P C 8 pRq with 1 Bp0,2q ď b ď 1 Bp0,3q , then |T pbq| ď Cpm`1q andˆB p0,1q |T t pbq| ď Cpm`1q. (4.23) It is not a typo that the right hand sides do not depend on n; the reason is clear after Section 4.5. The kernel of the adjoint T t is K t ǫ px, yq " K ǫ py, xq " p´1q n`1 K ǫ px, yq by (4.21), so it suffices to prove the first estimate in (4.23). At this point, we already observe that, in proving (4.23), we may assume that the function B 1 appearing in the kernel of T satisfies B 1 p0q " 0 and spt B 1 Ă Bp0, 10q.
(4.24) In fact, the value of the kernel K ǫ px, yq remains unchanged if replace B by B´L, where Lpxq " pB 1 p0q, B 1 p0qxq is a 0-tame-affine map. Next, already using that B 1 p0q " 0, it is easy to show that there exists a 1-Lipschitz functionB 1 with sptB 1 Ă Bp0, 10q which agrees with B 1 on Bp0, 3q. Since only the values of B 1 on Bp0, 3q appear in (4.23), we may replace B 1 byB 1 without changing the value of (4.23). We will only use the tameness condition 9 B 2 pzq " B 1 pzq for z P Bp0, 3q (see (4.28)), and this now remains valid with B 1 instead. Alternatively, we could redefine B 2 on R so that 9 B 2 "B 1 on R, and hence acquire a new 1-tame functionB : R Ñ R 2 satisfying (4.24), but this is a little overkill.
To prove (4.23), we start as in the proof of [10, Theorem 10, p. 58], and fix an auxiliary function η P C 8 pRq satisfying Then, for x, y P R with x ‰ y fixed, we note that "ˆ8 0 ηprq dr r " 1.
In particular, for x P Bp0, 1q (as in (4.23)) fixed, we may write Let us point out that the integrals above are absolutely convergent, because, first, a necessary condition for ηp|x´y|{sqK ǫ px, yq ‰ 0 is ǫ{2 ď |x´y| ă s, so the integral over s ď ǫ{2 contributes zero. Second, observe that if s ą 16, then s´1|x´y| ă 1 4 for all pairs x P Bp0, 1q and y P spt b Ă Bp0, 3q, so the integral over s ą 16 also contributes zero. Also, the integration over s P pǫ{2, 4ǫq Y p1, 16q only yields an absolute constant, so we have reduced (4.23) to showing that  The lower bound "4ǫ" is convenient, because whenever |x´y|{s P spt η and s ě 4ǫ, we have |x´y| ě s{4 ě ǫ, and hence ψ ǫ px´yq " 1 by (4.22). Consequently, the value of (4. 26) does not change if -and when -we replace K ǫ by K. Finally, since sptr1´bs Ă RzBp0, 2q, we have |x´y| ě 1 for all x P Bp0, 1q and y P sptr1´bs. Consequently ηp|x´y|{sq " 0 whenever s P r0, 1s, x P Bp0, 1q, and y P sptr1´bs, and it follows that To prove (4.27), fix x P Bp0, 1q. Recall the exponents m, n ě 0 from the definition of the kernel K. The case n ě 2 turns out to be easy, see the Section 4.5, and the case n " 0 is the case of "standard" Calderón commutators. So, the case n " 1 contains the main news. To proceed with this expression, we first use the tameness condition 9 B 2 " B 1 to write It is easy to check that the right hand side on (4.28) vanishes if B 1 is affine. In particular,ˇˇ where B x,s pyq " ay`b is an affine map minimising the β-number (introduced in (4.15)) of B 1 in Bpx, sq. Therefore, we havěˇˇˇˇB 2 pxq´B 2 pyq´1 2 rB 1 pxq`B 1 pyqspx´yq px´yq 2ˇ β B 1 pBpx, sqq  4.6. The case n P t0, 1u. We then consider the case n P t0, 1u and m ě 0. We view n P t0, 1u as "fixed", and write K m px, yq :" 1 x´y " Apxq´Apyq x´y ı m " (4.32) Let ψ P C 8 pRq be a "standard bump function" as in (4.18). Then, as in Lemma 4.17, we consider the linear maps L x,s pyq :" pA 1˚ψ s qpxqy ": P s pA 1 qpxqy, s P p0, 1q.
The plan is to reduce the treatment of the kernel (4.32) to the case m " 0. To accomplish this, assume that initially m ě 1. Then, for x P Bp0, 1q and s P p0, 1q fixed, we write " Apxq´Apyq x´y Apxq´Apyq x´y  m´1 P s pA 1 qpxq.
Here, for y P Bpx, sq,ˇˇˇˇ" Apxq´Apyq x´y If still m´1 ě 0, we repeat the same procedure as in (4.33), separating one power of pApxq´Apyqq{px´yq from K m´1 , adding and subtracting L x,s px´yq, and then repeating the estimates (4.34)-(4.35). This operation yields two terms, one "error" term dominated, as before, by 1 (also using that }P s pA 1 q} L 8 ď 1), and then the "main" term Comparing (4.36) and (4.37), we note that if j ě 1, we can reduce the study of K j to the study of K j´1 at the cost of (1) committing an additive error of magnitude 1, and (2) replacing P s pA 1 qpxq j by P s pA 1 qpxq j`1 in (4.37). After repeating these steps m times, we see that (4.27 in the case n " 1. In the latter case we already plugged in (4.28). The case n " 1 is, of course, the "main case"; in fact, after a few changes of variables, we will reduce the treatment of (4.39) to the expression on line (4.45) below, which is more general than (4.38) (taking B 1 pxq " x). So, we can, and will, ignore the case n " 0. To proceed estimating the expression in (4.39), we concentrate for the moment on the three innermost integrals. We make the change-of-variables r Þ Ñ uy`p1´uqx in the r-integration, and then use Fubini's theorem, to find the expressioňˇˇˇˆ1 Finally, we briefly remark that the term on line (4.44) can be handled in the same way, first re-introducing the term B 1 pxq inside the y-integration. Then, in place of (4.46), one ends up with the expression which is easily bounded by m, as above. This completes the proof of the first estimate in (4.23), and consequently the proof of the theorem.

THE EXPONENTIAL KERNEL RETURNS
In Theorem 4.10, we showed that if A : R Ñ R is 1-Lipschitz, and B : R Ñ R 2 is 1tame, then K A,B is a CZ-kernel. In this section, we prove Theorem 4.8, which stated that }K A,B } C.Z. ď polypM, N q whenever A : R Ñ R is M -Lipschitz, and B : R Ñ R 2 is Ntame. The result will be reduced to the case M " 1 " N via the corona decompositions for Lipschitz functions and tame maps from Section 3.1.1. In fact, this manner of reasoning works, without extra effort, in slightly higher generality. Let us fix, for the entire section, an SK k : RˆR z △ Ñ R such that }k} α,strong ď 1, α P p0, 1s. We also assume that kpx, yq " 0, |x´y| ď ǫ, for some fixed ǫ ą 0. Then, let us (re-)define where A : R Ñ R is Lipschitz, and B : R Ñ R 2 is tame. The main point here is that the "homogeneity" of the specific kernel kpx, yq " px´yq´1 is not needed in this section. For M, N ě 1, and the fixed kernel k, we define ℘ k pM, N q :" ℘pM, N q :" supt}K A,B } C.Z. : A is M -Lipschitz and B is N -tameu. Thus, Theorem 4.10 implies that ℘ 1{px´yq p1, 1q ă 8. Without additional requirements on k, this is certainly not true, so we assume it a priori in this section: C 0 pkq :" ℘p1, 1q ă 8. where C M,N :" C maxtM, N 2 , ℘pM, N qu.
Let us quickly deduce Theorem 5.4 from Theorem 5.6.
Proof of Theorem 5.4 assuming Theorem 5.6. Let C 1 :" maxtC 0 pkq, 2 log 2 C, 2u. Assume that we already have (5.5) with constant "C 1 " for some M " N P 2 N , that is, ℘pN, N q ď C 1 N C 1 . This is true for M " 1 " N by (5.3). From two applications of (5.7), the inductive hypothesis, and noting that 2 C 1 ě C 2 , we find that This completes the proof.
For the remainder of the section, we will view the Hölder continuity parameter α P p0, 1s as "fixed", so any "absolute constants" are actually allowed to depend on α.

Proof of Theorem 5.6: getting started.
We begin the proof of Theorem 5.6. The argument is based on ideas from Semmes' paper [45], although our setting allows for some simplifications. We fix an M -Lipschitz function A : R Ñ R, and an N -tame map B " pB 1 , B 2 q : R Ñ R 2 , with M, N P 2 N . Write T f pxq :"ˆK A,B px, yqf pyq dy, which is well-defined for e.g. f P L 2 pRq due to (5.1). In the sequel, we abbreviate Kpx, yq :" K A,B px, yq. The plan will be to show that for any dyadic interval Q 0 P D, the T 1 testing condition Fix b P C 8 pRq, as in (5.8). Now, (5.8) is actually composed of two distinct inequalities: we will mostly concentrate on proving the inequality that is, the one where the "tameness constant" is reduced by a factor of 2. The argument for the other inequality in (5.8) is virtually the same, and we will indicate the small differences in Section 5.4.6. To show (5.9), we start by applying the tame corona decomposition, Theorem 3.15 -or more precisely its Corollary 3.22 -to the N -tame function B, with parameter η " 1 2 . The result is a decomposition D " B 9 YG, as explained in the statement of Theorem 3.15, a collection F of trees T Ă D, and for each tree a function of the form Ψ T " ψ T`LT , where ψ T is pN {2q-tame, L T is tame-linear, and the good approximation property (3.23) holds. To recap: d π pBpsq, Ψ T psqq ď 1 2 N |Q|, s P 2Q, Q P T P F. To benefit from the decomposition D " B 9 YG, we will now decompose the operator T in an analogous manner. For j P Z, we first define the operator T j by T j f pxq :"ˆt y:2´jď|x´y|ď2´j`1u Kpx, yqf pyq dy.
Then, we set and write We begin by disposing of the first sum. Note that for Q P D j , we have using that |Kpx, yq| ď |x´y|´1 and }b} L 8 ď 1. Therefore, for g P L 8 pQ 0 q with }g} L 8 pQ 0 q " 1, we haveˇˇˇˇˇˆQ The implicit constants only depend on the Carleson packing constant of the family B. This is better than what we need for (5.9). We then concentrate on the second sum in (5.11). We claim that for individual trees T P F, we have the estimate as we will next check, and hence complete the proof of (5.9). Assume then for the moment that (5.12) holds, and writê where F 0 " tT P F : QpT q Ă Q 0 u. The second term in (5.14) is straightforward to estimate, so we start from there. If T P F z F 0 is tree satisfyinĝ then Q 0 Ă QpT q, since T T pbq is supported on QpT q. In addition, there exists Q P T and x P Q 0 such that Kpx, yqbpyq dy ‰ 0. (5.16) Hence x P Q X Q 0 , so either Q Ă Q 0 , or Q 0 Ă Q. In the second case, (5.16) forces |Q| |Q 0 |, because spt b Ă 3Q 0 . In the first case, since Q 0 Ă QpT q, there anyway exists a parent Q 1 P T of Q such that Q 0 Ă Q 1 and |Q 1 | " |Q 0 |. We conclude that whenever (5.15) holds for some T P F z F 0 , there exists Q P T with Q 0 Ă Q and |Q| |Q 0 |. But since the trees T P F are disjoint, this implies that (5.15) can only occur for boundedly many T P F z F 0 . Hence, the second sum in (5.14) is bounded by a constant timeŝ as desired. To estimate the first sum in (5.14), we use the Carleson packing condition for the top intervals QpT q with T P F 0 . Recalling that }b} L 8 ď 1 and spt b Ă 3Q 0 , and also observing that T T pbq " 1 QpT q T T p1 5QpT q bq, T P F, we estimate as follows: The implicit constants only depend on the Carleson packing constant of the top intervals QpT q, T P F. We have now reduced (5.13) to proving (5.12).
To prove (5.12), fix T P F and f P L 2 pRq, and write j 0 for the generation of QpT q, that is, QpT q P D j 0 . Note that Kpx, yqf pyq dy where ρ " 2´j 0`1 " 2|QpT q|, and hpxq :" inft|Q| : x P Q P T u, for x P QpT q. and d : R Ñ R is the 1-Lipschitz function dpxq " inft|Q|`distpx, Qq : Q P T u, x P R. (5.20) By "replacement", we mean that }T T } L 2 ÑL 2 }T T } L 2 ÑL 2`maxtM, N u, so it will suffice to prove (5.12) forT T in place of T T . Let us now see carefully how to dominate T T byT T .
Lemma 5.21. If x, y P R with x P QpT q and |x´y| ě hpxq, then |x´y| ě Dpx, yq.
So, at least T T is dominated byTT . But since D, ρ are 1 2 -Lipschitz functions (ρ being a 0-Lipschitz function), we find from Lemma 2.8 that K D,ρ is a GSK with }K D,ρ } }K} maxtM, N u, (5.24) and hence Cotlar's inequality (2.20) applies: Combining this inequality with (5.23) and (5.24), we infer that as desired. Consequently, (5.12) will follow (with a slightly worse constant) once we manage to establish that To simplify notation a little bit, we will, from now on, write "T T " in place of "T T " for the operator associated to the Dpx, yq-truncation. This should cause no confusion, because there will be no further reference to the original operator T T .

Applying the corona decomposition.
To prove (5.25), we recall the functions Ψ T :" Ψ " ψ T`LT ": ψ`L associated to the fixed tree T , where ψ " pψ 1 , ψ 2 q : R Ñ R 2 is pN {2q-tame, and L " pL, P q :" R Ñ R 2 is 2N -tame-linear. We recall from (3.23) that To be accurate, (3.23) only gives (5.26) for s P 2Q, but enlarging the constant from "2" (or anything ą 1) to "11" is a standard trick, see e.g. the argument on [14, p. 20]. Alternatively, one could just prove (3.23) directly with constant "11". To establish the good L 2 -bound for T T , we want to compare it to a suitable operator T Ψ associated to the kernel The reader should protest that the right hand side of (5.27) is, in fact, the kernel of K A,ψ instead of K A,Ψ . Have we forgotten about the tame-linear part L " pL, P q altogether? No: recalling that L is linear, and 9 P " L, one easily checks that P pxq´P pyq´1 2 rLpxq`Lpyqspx´yq " 0. In other words, This is crucial: the kernel K A,Ψ approximates K A,B well (using information from the corona decomposition, as we will soon see), while K A,ψ is a kernel associated to an pN {2qtame function ψ. On the other hand, Ψ can be, at worst, 2N -tame, so without knowing (5.28), the kernel K A,Ψ would be no better than K A,B ! Now, we abbreviateK px, yq :" K A,ψ px, yq " K A,Ψ px, yq, and define the operator T Ψ with the same Dpx, yq-truncation as in the definition of T T : To prove (5.25), we will establish that The second inequality in (5.30)  We used that Dpx, yq ď dpxq{2`|x´y|{4, so |x´y| ď Dpx, yq implies that |x´y| ď dpxq. Now, it follows from (5.31) and Cotlar's inequality that Here }T A,Ψ f } L 2 " }T A,ψ f } L 2 ď ℘pM, N {2q}f } L 2 by (5.28) and the definition of ℘pM, N {2q, while }K} maxtM, N u. This completes the proof of the second part of (5.30), and the rest of the section is devoted to establishing the first part.

A Whitney decomposition.
Recall that dpxq " inftdistpx, Qq`|Q| : Q P T u, so d is 1-Lipschitz, and well-defined on R. However, the set is a compact subset of QpT q. It follows easily from (5.26) that Ψpsq " Bpsq, s P E. (5.32) In this short section, we perform a Whitney type decomposition of R z E. Fix x P R z E. These intervals are disjoint and cover R z E, and we will denote them S. We first observe that |S| ď dpyq ď 4|S|, y P S P S. (5.34) Indeed, the lower bound is immediate from the definition (5.33). To see the upper bound, note that by the maximality of S P S there exists y 1 in the parent p S of S with dpy 1 q ă | p S| " 2|S|, whence dpyq ď dpy 1 q`| p S| ď 4|S|, as claimed. We next observe that S P S and S Ă 11QpT q ùñ d π pBpsq, Ψpsqq N |S|, s P S. (5.35) Indeed, fix x P S and, based on (5.34), find Q P T with dpx, Qq`|Q| ď 5|S|. Then, let Q 1 P T be the minimal ancestor of Q in T with S Ă 11Q 1 (this exists because S Ă 11QpT q).
It is easy to check that |Q 1 | " |S|, and now (5.35) follows from (5.26) applied to s P 11Q 1 .

Comparing T T and T Ψ .
Recall that T T and T Ψ are the operators defined in (5.18) and (5.29), respectively. To prove the first inequality in (5.30), that is, we fix f, g P L 2 pRq. It suffices to show thaťˇˇˇˆp Since T T pf q " 1 QpT q T T pf 1 5QpT q q and T Ψ pf q " 1 QpT q T Ψ pf 1 5QpT q q, which follows from the upper ρ-truncation in (5.18) and (5.29) (recall: ρ " 2|QpT q|), it moreover suffices to prove (5.36) for f, g satisfying To estimate the difference in (5.36), we introduce the following auxiliary notation. If x P E, we define Spxq " txu, and otherwise Spxq is the unique element in S containing x. If h : R Ñ R is a function, and x P R, we then define h ěx pyq :" hpyq1 t|Spyq|ě|Spxq|u pyq and h ąx pyq :" hpyq1 t|Spyq|ą|Spxq|u pyq.
The functions h ďx and h ăx are defined similarly, swapping the inequalities. Note that h ąx | E " 0 for any x P R, and h ăx " 0 whenever x P E. With this notation, we havê where further Kpx, yqgpxq1 t|Spxq|ąSpyq|u pxq dx ff dy "ˆpT t T g ąy qpyqf pyq dy.
The same calculation works if "T " is replaced with "Ψ". Consequently, We will only estimate the term on line (5.37), since the argument for the second term is virtually the same. This is actually a reason why we introduced the "symmetric" Dpx, yqtruncation: to make the term on line (5.38) look as similar to (5.37) as possible.
However, when x, y P E, as in the second integration, then Bpxq " Ψpxq and Bpyq " Ψpyq by (5.32), so Kpx, yq "Kpx, yq. Consequently, the second integral contributes nothing, and (5.39) is indeed true even when x P E.
We will now write "I x pSq" for the individual terms in (5.39), with |S| ě |Spxq|. Note that intervals S P S with S X 5QpT q " H contribute nothing to (5.39), so they can be discarded. But if S X 5QpT q ‰ H, then dpyq ď distpy, QpT qq`|QpT q| ď 3|QpT q| for some y P S. This implies by (5.34) that |S| ď 3|QpT q|, and consequently, S Ă 11QpT q.
(5. 40) In fact this inclusion explains our choice of the constant "11" in (5.26). We proceed to estimate the pieces I x pSq in a manner adapted from [45], eventually proving the following claim: the intervals S P S with |S| ě |Spxq| and S Ă 5QpT q can be split into two groups G 1 pxq and G 2 pxq, where |I x pSq| maxtM, N 2 u|S| distpx, Sq 2`| S| 2ˆS |f pyq| dy, S P G 1 pxq, (5.41) and ÿ The estimate (for (5.37)) concerning group G 2 pxq is straightforward: Before proceeding with the proofs of (5.41)-(5.42), let us briefly see that the estimate (5.41) leads to essentially the same conclusion (up to multiplication by maxtM, N 2 u): Lemma 5.43. Let 1 ă p ă 8, and 1{p`1{q " 1. Then, for g P L p and f P L q , we havê Proof. We start by rewriting and estimating the left hand side as follows: Since the intervals in S are disjoint, the second factor is evidently controlled by }M f } L q p }f } L q . The first factor is also dominated by the maximal function, since for S P S fixed, as desired.
This allows us to conclude the estimate for (5.37) (but see Section 5.4.5 for a final "wrapup" of the whole argument). We then begin to verify the estimates (5.41)-(5.42). We fix x P 5QpT q and S P S with |S| ě |Spxq| and S Ă 11QpT q. Since Spxq X 5QpT q ‰ H, the argument above (5.40) also yields Spxq Ă 11QpT q. (5.45)
Combining these estimates, and recalling that |x´y 0 | ě distpSpxq, Sq ě 2|S|, we infer that |Kpx, y 0 q´Kpx, y 0 q| Combining (5.47) and the estimate above, we conclude that This matches the estimate in (5.41), so in this case S P G 1 pxq.
This is the estimate desired in (5.42), so we can include all S P S with inf yPS r|x´y|D px, yqs ă 0 to the collection G 2 pxq. Finally, assume that the second option in (5.51) is realised, and pick y 0 P S accordingly. If |S| ď ρ{2, then inf yPS |x´y| ě ρ{2 by the triangle inequality. But even if |S| ě ρ{2, we have inf yPS |x´y| " distpx, Sq ě 2|S| ě ρ by the case assumption. So, ÿ SĂ11QpT q sup y 0 PS |x´y 0 |ąρ |I x pSq| ρ´1ˆ5 QpT q |f pyq| dy M f pxq, which is the same estimate as in (5.52). The proof of this -final -case is complete. 5.4.5. Summary. We have now proven that all the intervals S P S with |S| ě |Spxq| and S Ă 11QpT q, for x P 5QpT q, can be split into the groups G 1 pxq and G 2 pxq so that (5.41)-(5.42) hold. As we saw directly under (5.41)-(5.42), we can then conclude the estimatê Repeating rather verbatim the same argument, we could also show that and consequently the splitting in (5.37) shows thaťˇˇˇˆp Since f, g P L 2 pRq were arbitrary functions, this allows us to conclude the first inequality in (5.30), namely that }T T } L 2 ÑL 2 }T Ψ } L 2 ÑL 2`maxtM, N 2 u. Since we already established the second inequality in (5.30), we may then infer (5.25), which then implies (5.12), and finally (5.9) (one of the two inequalities in (5.8)). 5.4.6. The second inequality in (5.8). As we explained above, we have now established one of the two inequalities claimed in (5.8). We still need to establish the second: (5.53) As we noted below (5.9), the first step is to apply Theorem 3.20 to the M -Lipschitz function A at level M {2, and then decompose the operator T with respect to the ensuing families of intervals B and tT u T PF , as in (5.11). For each tree T P F, the corona decomposition yields an pM {2q-Lipschitz function ψ T : R Ñ R, and a linear map L T : R Ñ R. However, the proof presented above makes no explicit reference to these "approximating" functions before the introduction of the kernel K A,Ψ in (5.27). So, the argument is literally the same until that point. In proving (5.53), the relevant "approximating" kernel isK px, yq " kpx, yq expˆ2πi " pψ`Lqpxq´pψ`Lqpyq x´y`B 2 pxq´B 2 pyq´1 2 rB 1 pxq`B 2 pyqspx´yq px´yq 2

˙,
because |Apxq´pψ`Lqpxq| is the quantity controlled by the corona information for x P 2Q and Q P T , recall the estimates in Section 5.4.2. As before, the crux of the proof is to prove the analogue of (5.30), namely Here T T is precisely the same object as in the previous sections, and T Ψ f pxq "ˆt y:Dpx,yqď|x´y|ďρuK px, yqf pyq dy.
The proof of the first inequality in (5.54) is virtually the same as above: the formula of the kernelK only plays a role in Section 5.4.2, and the upper bound for |Apxq´pψ`Lqpxq|, coming from the corona decomposition, is exactly of the form applicable in (5.49). So, one can conclude (5.50), in fact with constant "maxtM, N u" in place of "maxtM, N 2 u". The proof of the second inequality in (5.54) contains the only essential, albeit easy, difference in the proofs. Namely, recall from the discussion around (5.28) that the equation K A,Ψ " K A,ψ was crucially important. Now, the same is not true, but we have something comparable, and good enough. Namely, if Lpxq " cx, we have K ψ`L,B px, yq " e 2πic K ψ,B px, yq, x, y P R, x ‰ y.
Thus, even though ψ`L is not pM {2q-Lipschitz, the L 2 Ñ L 2 operator norm of T ψ`L,B f pxq "ˆK ψ`L,B px, yqf pyq dy " e 2πicˆK ψ,B px, yqf pyq dy is bounded from above by ℘pM {2, N q. This fact (in combination with Cotlar's inequality, as discussed after (5.30)) allows us to conclude the second inequality in (5.54). This completes the proof of (5.53), and hence the proof of (5.8) and of Theorem 5.6.

REGULAR CURVES AND BIG PIECES OF INTRINSIC LIPSCHITZ GRAPHS
In this section, we prove Theorem 1.6, which states that certain SKs in H are CZ kernels for (Hausdorff measures on) regular curves. The plan is to reduce the assertion to its special case concerning intrinsic Lipschitz graphs, Theorem 1.7, through the observation that regular curves have big pieces of intrinsic Lipschitz graphs (Theorem 6.42). Further, the transition from "intrinsic Lipschitz graphs" to sets with "big pieces of intrinsic Lipschitz graphs" is based on an abstract argument, originally due to David [16,17] in R n . We will record a version of this argument in all proper metric spaces pX, dq, see Theorem 6.3 below, although the case X " H suffices for our application.
6.1. David's big piece theorem in metric spaces. Definition 6.1 (Regular measures). Let pX, dq be a metric space, and let k ą 0. We write Σ k for the class of k-regular measures on X, that is, Borel regular measures µ on X with the property that there exists a finite constant C ě 1 such that C´1r k ď µpBpx, rqq ď Cr k , x P spt µ, r ą 0.
The smallest constant C ě 1 such that (6.2) holds will be denoted reg k pµq, or just regpµq.
If µ P Σ k , then spt µ is a k-regular set and, since the lower bound is required to hold for arbitrarily large r ą 0, it follows that diampX, dq ě diampspt µq " 8. This is a matter of technical convenience. Anyway, our focus will be on 1-regular curves in the metric space X " H, and every such curve is contained in an unbounded 1-regular curve. Theorem 6.3. Let pX, dq be a proper metric space, and let k ą 0. Let K : XˆX z △ Ñ C be a k-GSK, and assume that µ P Σ k has the following properties. There exist constants 0 ă θ ă 1, C ě 1 and, for each 1 ă p ă 8, a finite constant A p ě 0 such that the following is true. For every closed ball B centred on spt µ, there exists a Borel regular measure σ on X, and a compact set E Ă B X spt µ, such that (1) σ P Σ k with regpσq ď C, (2) µpEq ě θµpBq, (3) µpA X Eq ď σpAq for all A Ă X, (4) }Tσ f } L p pσq ď A p }f } L p pσq for f P C c pXq.
In contrast, we consider k-GSKs, and associated operators T˚. In this generality, we do not know if T˚f is lower semicontinuous, which causes minor technical trouble in the proof of Lemma 6.25. ‚ At one point of the original proof, David seems to refer to the Besicovitch covering theorem, which is not available in metric spaces. However, it turns out that the 5r-covering theorem suffices, see Lemma 6.8. Often, when arguments follow [17,Proposition 4 bis.] verbatim, we will omit details. 6.1.1. Proof of Theorem 6.3. The version of the "good λ inequalities" which we use in the proof of Theorem 6.3 is borrowed from [18, III, Lemma 3.1]: Proposition 6.5. Let pX, µq be a measure space, and let 1 ă p ă 8. Let u : X Ñ r0,`8s be a µ measurable function that agrees with an L p pµq function outside a set of finite µ measure, and let v : X Ñ r0,`8s be an L p pµq function. Assume that there exists a constant 0 ă ν ă 1 such that, for all ε ą 0, there is a constant γ ą 0 so that, for all λ ą 0, µptx P X : upxq ą λ`ελ and vpxq ď γλuq ď p1´νqµptx P X : upxq ą λuq. (6.6) Then u P L p pµq with }u} L p pµq ď Cpp, ε, ν, γq}v} L p pµq .
A proof for the case X " R and µ " L 1 is included below [16, Lemme 12] (we do not need an explicit expression of Cpp, ε, ν, γq for our purposes). The version for an arbitrary measure space pX, µq is proven in the same way (David leaves this as an exercise in [18]).
The proof of Theorem 6.3 follows by applying Proposition 6.5 for given f P C c pXq and 1 ă p ă 8 to the functions u :" Tμ f and v :" where M µ,k is the radial maximal function of order k (see Section 2.2). For µ P Σ k , we will abbreviate M µ :" M µ,k . In order to employ Proposition 6.5, we want to show that u agrees with an L p pµq function outside a compact set, namely outside a closed ball Bpx˚, 2Rq, where x˚P X, and R ą 0 is so large that sptf Ď Bpx˚, 2Rq. Moreover, we have to verify that u and v P L p pµq satisfy (6.6). This will yield Theorem 6.3 since }v} L p pµq ď Cpp, regpµqq }f } L p pµq . We start with some preliminaries. Whenever µ P Σ k , the triple pspt µ, µ, dq a doubling metric measure space, and M µ is bounded on L p pµq for 1 ă p ă 8. We need a more general version of this result that involves two distinct measures in Σ k with potentially distinct, even disjoint, supports. David states this in [18,Lemma 2.2,p. 58], and writes that the proof is easy, and based on the Besicovitch covering theorem. This tool is not available in our generality, but, in fact, the 5r-covering theorem is good enough. Lemma 6.8. Assume that pX, dq is a proper metric space and k ą 0. Let µ, σ P Σ k , and 1 ă p ă 8. Then, there exists a constant 0 ă C ă 8, depending only on p and regpµq, regpσq, such that Proof. Lemma 6.8 is proved in the same way as [16,Proposition 4], using Marcinkiewicz interpolation. One has to show that M µ maps L 8 pµq into L 8 pσq, which is clear (only using µ P Σ k ), and that it also maps L 1 pµq into L 1,8 pσq: This follows from the "standard" proof, and only uses that σ P Σ k , but to convince the reader that no Besicovitch covering theorem is needed, let us record the details. Fix f P L 1 pµq, and consider the ball family B :" # Bpx, rq Ă X : x P spt σ and 1 r kˆB px,rq |f | dµ ą λ + .
Since f P L 1 pµq, the radii of the balls in B are uniformly bounded. Second, B is a cover for the set E " tx P spt σ : M µ f pxq ą λu, which has the same σ-measure as the left hand side of (6.9). Using the 5r-covering theorem, we extract a countable disjoint subfamily Finally, as claimed.
Lemma 6.8 yields a "two-measure statement" for SIOs, Proposition 6.16 below. We follow closely David's proof of [17,Proposition 2] and deduce Proposition 6.16 from two auxiliary lemmas. Lemma 6.10. Let pX, dq be a proper metric space, k ą 0, and let K : XˆX z△ Ñ C a k-GSK. Assume that σ P Σ k . Then there exists a constant C ą 0, depending only on k, }K}, and regpσq, such that The main point is that we can take x 0 P X z spt σ.
Proof. One first shows that there exists a constant C 0 ą 0, depending only on k and }K}, such that for all ε ą 0 and x 0 P X, one has |T σ,ε f px 0 q| ď Tσ f pxq`C 0 M σ f px 0 q, x P Bpx 0 , ε{2q. (6.12) This can be done as in the proof of [17,Lemme 4].
The next lemma is a Cotlar-type inequality. Such inequalities are available in very general settings, cf. [47,I.7.3,Proposition 2], [34, p.56], [9, p.606], and [41], but we are not aware of one that would be precisely in the desired form for our purposes. In particular, we have to deal simultaneously with two measures µ and σ in a metric space pX, dq. Lemma 6.14. Let pX, dq be a proper metric space, k ą 0, and µ P Σ k . LetK : XˆX z△ Ñ H be a bounded k-GSK, and let T be the operator induced by pK, µq. Let σ P Σ k with regularity constant C 0 ě 1, and assume, for some 1 ă s ă 8, that Then, there exists a constant C " CpA, C 0 , k, }K}, sq 3 such that Proof. The proof is verbatim the same as for [17,Lemme 5].
Proposition 6.16. Let pX, dq be a proper metric space, k ą 0, let K : XˆX z △ Ñ C be a k-GSK, and let σ P Σ k . Assume that, for all 1 ă p ă 8, there is a constant C p ě 1 such that Then for all 1 ă p ă 8 and µ P Σ k , there is a constant C 1 p ě 1 such that for all f P C c pXq, p depend only on p, C p , k, }K}, and regpµq, regpσq. Proof. Part (1) is a straightforward consequence Lemmas 6.10 and 6.8.
Part (2) is proved by duality. Fix µ P Σ k , 1 ă p ă 8, and let q " p{pp´1q. From the first part of the lemma, we know that the operators T σ,ε are uniformly bounded L q pσq Ñ L q pµq. Now we define K t px, yq :" Kpy, xq, and let T t µ,ε be the (adjoint) ǫ-SIO induced by pK t ǫ , µq. Then, sup εą0 }T t µ,ε } L p pµqÑL p pσq ď C p .
As an intermediate step towards (2), we wish to deduce from Lemma 6.14 the corresponding bound for the maximal SIO T t,μ . A small technical issue is that K is not necessarily a bounded GSK, as required in the hypothesis (to even make sense of T ). To remedy this, fix ǫ ą 0, and note that K t ǫ is a bounded GSK, with GSK constants independent of ǫ, by Lemma 2.8. Consequently, Lemma 6.14, applied with K t ǫ and s :" for f P C c pXq. Here T t,μ ,ǫ is the maximal SIO associated to K t ǫ , and we also used the L p pµq Ñ L p pσq and L s pµq Ñ L s pσq boundedness of M µ from Lemma 6.8, and the L p pσq Ñ L p pσq boundedness of M σ . To proceed, we note that (6.18) and monotone convergence yield This almost looks like (2), except that it concerns T t in place of T . However, applying (6.19) to µ :" σ, we conclude that also K t satisfies (6.17). Hence, we can re-run the whole argument with K t ! But since pK t q t " K, this time we end up with (2).
Let us continue with the proof of Theorem 6.3. Fix µ P Σ k as in the statement, fix 1 ă p ă 8, and let f P C c pXq. Our task is to show that }Tμ f } L p pµq ď C p }f } L p pµq , f P C c pXq. (6.20) This will follow from Proposition 6.5 ("good λ inequality") applied to u :" Tμ f and v :" The rest of the proof consists of explaining how Proposition 6.16 can be used to verify that the assumptions of Proposition 6.5 are fulfilled.
Lemma 6.22. Let pX, dq be a proper metric space, k ą 0, and let K : XˆX z △ Ñ C be a k-GSK. Let µ P Σ k , f P C c pXq and 1 ă p ă 8. Then u :" Tμ f is a Borel function on pX, dq and it agrees with an L p pµq function outside a ball, hence outside a set of finite µ measure.
Proof. First we note that Tμ f pxq " sup εPQXp0,`8q |T µ,ε f pxq|. (6.23) Indeed, for every ε P p0,`8q, there exists a sequence pε j q jPN Ă Q with ε j OE ε as j Ñ 8, and it follows that Since T µ,ε f is a Borel function for every ε ą 0, we deduce from (6.23) that u is a Borel function.
Regarding the second claim, if spt f Ă Bpx 0 , Rq, the "size" condition for K alone implies that Tμ f pxq M µ,k f pxq for x P X z Bpx 0 , 2Rq. Now, the claim follows from the L p pµq-boundedness of M µ,k . Lemma 6.24. Let pX, dq be a proper metric space, k ą 0, µ P Σ k , f P C c pXq, and 1 ă p ă 8. Proof. This follows from the boundedness of M µ,k on L p pµq and L ? p pµq.
Lemma 6.25. Assume that pX, dq, k ą 0, K : XˆX z △ Ñ C, and µ P Σ k are as in Theorem 6.3. Then there exists ν P p0, 1q, depending only on regpµq and the parameter θ ą 0, such that the following holds. Let 1 ă p ă 8, f P C c pXq, and define the functions u and v as in Lemmas 6.22 and 6.24. Then, for all ε ą 0, there is γ " γpǫq ą 0 such that µptx P X : upxq ą λ`ελ and vpxq ď γλuq ď p1´νqµptx P X : upxq ą λuq (6.26) for λ ą 0. The choice of γ is also allowed to depend on p, and the "data" of Theorem 6.3.
We start by constructing a cover for Ω. Since f P C c pXq, it follows from the "size" estimate |Kpx, yq| dpx, yq´k, and from µ P Σ k , that Tμ f pxq Ñ 0 as distpx, spt f q Ñ 8. Hence Ω is a bounded set. On the other hand, for µ almost every x P Ω, lim jÑ8 µpBpx, 2´jq X Ωq µpBpx, 2´jqq " 1, (6.27) by Lebesgue differentiation in the doubling metric measure space pspt µ, µ, dq. Combining (6.27) and the fact that Ω is bounded, it follows that for µ almost every x P Ω, there exists a maximal dyadic radius r x " 2´j x Ω,µ 1, with j x P Z, such that µpBpx, r x q X Ωq µpBpx, r x qq ě 1´θ 2 . (6.28) In particular, since the reverse inequality already holds for 2r x , we can find We then apply the 5r-covering theorem to find a disjoint family tBpx i , r i qu iPN Ă tBpx, r x q : x P Ωu with the property that µ almost all of Ω is contained in ď iPN Bpx i , 5r i q.
We write B i :" Bpx i , r i q, 5B i :" Bpx i , 5r i q, and a i :" a x i . In order to prove (6.26), it suffices to show that µ prB i X Ωs z Aq µpB i q ą θ 4 , i P N. and consequently µpAq ď p1´νqµpΩq for some ν " νpregpµq, θq ą 0, as desired. We then prove that (6.30) holds if γ " γpǫq ą 0 is chosen small enough (recall that A " A λ,ǫ,γ ). For now, let γ ą 0 be arbitrary, and fix B i . Note that (6.30) is clear if νpxq ą γλ for all x P B i (then rB i X Ωs z A " B i X Ω, which has density ě 1´θ{2 ě θ{2), so we may assume that there exists a point ξ i P B i with Now, we decompose f " f 1`f2 , where f 1 " f φ, and φ P C c pXq satisfies upxq ď Tμ f 1 pxq`Tμ f 2 pxq, x P B i , (6.32) and we will check in a moment that if γ " γpǫq ą 0 is small enough. Thus, (6.32)-(6.33) imply that tx P B i : Tμ f 1 pxq ď ε λ 2 u Ď B i z A, and the proof of (6.30) has been reduced to showing that µptx P B i X Ω : Tμ f 1 pxq ď ε λ 2 uq ě θ 4 µpB i q. (6.34) Before tackling (6.34), we verify (6.33). In fact, (6.33) follows from the chain by choosing γ small enough so that Cγ ď ε{2. The second inequality in (6.35) follows from the choices of a i P Ω c and ξ i in (6.31). The first inequality can be obtained by writing R i :" 10r i , and decomposing The first term is bounded by Tμ f pa i q, as desired. The three latter ones are bounded by M µ f pξ i q, using the GSK bounds of K, and recalling that x, a i , ξ i P 2B i Ă Bpξ i , R i {2q, and that φ| Bpξ i ,R i q " 1. Similar, but slightly messier, estimates also work for T µ,δ , δ ą 0, in place of T µ , so (6.35) has been confirmed.
We are now in possession of all ingredients necessary for the proof of Theorem 6.3.
Proof of Theorem 6.3. Lemmas 6.22, 6.24, and 6.25 show that Proposition 6.5 can be applied to the functions u and v as defined in (6.21). This establishes (6.20).

Regular curves and BPiLG.
Recall that a closed set E in H is 1-regular if there exists a finite constant C ě 1 such that C´1r ď H 1 pBpp, rq X Eq ď Cr, for all p P E, 0 ă r ď diamE. (6.40) The smallest constant C ě 1 such that (6.40) holds will be denoted regpEq.
Recall further that a regular curve in H is a closed 1-regular subset of H which has a Lipschitz parametrisation by an interval I Ă R. In this section, we will use the letter "γ" for both the set, and the Lipschitz map I Ñ γ. A compact regular curve is a regular curve parametrised by a compact interval I Ă R. Definition 6.41 (Big pieces of intrinsic Lipschitz graphs). A closed 1-regular set E Ă H has big pieces of intrinsic Lipschitz graphs (over horizontal subgroups) (BPiLG) if there exist constants c, L ą 0 such that for all p P E and all 0 ă r ď diampEq there is an intrinsic L-Lipschitz graph Γ Ă H over some horizontal subgroup such that H 1 pEXΓXBpp, rqq ě cr.
In this section, we prove the following: Theorem 6.42. Every regular curve in H has BPiLG.
A short proof for the fact that regular curves in R n have big pieces of 1-dimensional Lipschitz graphs can be found in [18,III.4]. It is based on the rising sun lemma, and we did not find a way to adapt it to intrinsic Lipschitz graphs. Instead, we follow [20].
The proof of Theorem 6.42 employs a system D of dyadic cubes on a closed 1-regular set E Ă H, see [6, Section 3.0.1] for a more thorough introduction. These are Borel subsets of E with the following properties: ‚ D " Y j D j , j P Z, where each D j is a partition of E. ‚ There exist 0 ă c 0 ă C 0 ă 8, depending on regpEq, such that diampQq ď C 0 ℓpQq for Q P D j , where ℓpQq :" 2´j. For every Q P D j , there exists a "midpoint" z Q P Q such that E X Bpz Q , c 0 ℓpQqq Ă Q. With this notation, we write B Q :" Bpz Q , 2C 0 ℓpQqq, so that Q Ă B Q (with room to spare). For Q P D, we define the horizontal β-number where the infimum is taken over the horizontal lines familiar from Definition 3.37, L :" tp¨V : p P H, V is a horizontal subgroupu.
These numbers, notably their summability on horizontal curves, has been investigated extensively, see for example [35,37] and the discussion in the introduction. Given a system D of dyadic cubes on a closed 1-regular set E, we introduce the following subclass of good cubes in D: Definition 6.43. Let E Ă H be a closed 1-regular set with a system D of dyadic cubes. Given 0 ă c, ε ă 1 and a horizontal subgroup V, we say that Q P D is pc, ε, Vq-good if Here π V is the horizontal projection introduced in Definition 3.38. Recall also the cones C V pαq from Section 3.3. The next lemma shows that pc, ε, Vq good cubes Q P D look like intrinsic Lipschitz graphs over V at scale ℓpQq. Lemma 6.44. Let E Ă H be a closed 1-regular set with a system D of dyadic cubes. Then for all c ą 0 and M ě 2C 0 ě 1, there exists α, ε ą 0, depending only on c and M , such that the following holds. If Q P D is a pc, ε, Vq-good cube, then p P Q, q P B Q X E and dpp, qq ě ℓpQq{M ùñ p´1¨q R C V pαq. (6.45) Proof. Using rotations around the t-axis, we may, without loss of generality, suppose that V " tpx, 0, 0q : x P Ru. Now, fix c ą 0 and M ě 2C 0 . We also fix arbitrary ε, α ą 0 at this point, and we fix a cube Q P D such that Definition 6.43 (2) is satisfied, that is, βpQq ď ε. The plan is to show that if (6.45) fails for some p P Q and q P B Q with dpp, qq ě ℓpQq{M , and if α, ε ą 0 are small enough, then Q cannot be a pc, ε, Vq-good cube, that is, H 1 pπ V pQqq ă cH 1 pQq. Since the constants in Definition 6.43 are invariant under left translations and dilations, we may arrange that We write in coordinates q " px, y, tq, so that q P C V pαq ðñ }px, 0, 0q} ď α › ›`0 , y, t´x y 2˘› › . (6.47) If α " α M ą 0 is sufficiently small, this implies, together with (6.46), that }p0, y, tq} " M 1.
Next we will use βpQq ď ε to infer that t is small, and hence q lies close to tp0, y, 0q : y P Ru. But since Q lies close to the segment rp, qs " r0, qs, again by βpQq ď ǫ, and π V ptp0, y, 0q : y P Ruq " t0u, this will eventually show that H 1 pπ V pQqq ă cH 1 pQq. We turn to the details. To deduce more precise information about the coordinates of the point q, we use the assumption βpQq ď ε, which ensures the existence of a horizontal line ℓ " p 0¨V 1 with the property that distpq 1 , ℓq ď 2ε, q 1 P B Q X E. Thus there exist pa, bq P R 2 , a 2`b2 " 1, p 0 " px 0 , y 0 , t 0 q P H, and s P R, such that max tdpq, p 0¨p as, bs, 0qq, dp0, p 0 qu ď 2ε. (6.50) Triangle inequality, (6.46), (6.50), and left-invariance of the metric d yield M´1´4ε ď dpp 0¨p as, bs, 0q, p 0 q " |s| ď M`4ε.
The right hand side gives an upper bound for H 1 pπ V pQqq which tends to zero if M is fixed, and α, ε Ñ 0. For sufficiently small α, ε ą 0, we arrive at H 1 pπ W pQqq ă c, and hence Q is not a pc, ε, Vq-good cube. The proof is complete.
The geometry of horizontal lines in H enters the proof of Theorem 6.42 only through Lemma 6.44. With this result in hand, intrinsic Lipschitz graphs over horizontal subgroups can be constructed inside regular curves by an abstract coding argument, due to Jones [31]. The construction requires to control the "bad" cubes of γ that violate the second condition in Definition 6.43. For that purpose we first recall the following lemma, which follows from [37,Theorem I], and the observation in [7, Proposition 3.1]. Lemma 6.53 (Weak geometric lemma (WGL)). Let γ Ă H be a compact regular curve, and let D be a system of dyadic cubes on γ. Then for every ε ą 0, we have ÿ βpQqąε,QĎQ 0 ℓpQq regpγq,ε ℓpQ 0 q, Q 0 P D. (6.54) In general, a closed 1-regular set E Ă H satisfying (6.54) is said to satisfy the WGL. This lemma is the only spot where we need compact regular curves; quite likely the WGL is true for all regular curves, but it has only been stated for compact ones in the literature. Theorem 6.55. Let E Ď H be a closed 1-regular set satisfying the WGL, let b ą 0, and let V Ă H be a horizontal subgroup. Then there exist L ě 1 and N P N, depending only on b, regpEq, and the WGL constants of E, such that the following holds: for every Q 0 P D, there exist intrinsic L-Lipschitz graphs Γ 1 , . . . , Γ N Ă H over V such that With the geometric result from Lemma 6.44 in hand, the proof of 6.55 only uses the 1-Lipschitz property of π V , and an abstract "coding argument", due to Jones [31], which has been applied to prove variants of Theorem 6.55 for k-regular sets in R d ([20, Theorem 2.11]) and for p2n`1q-regular sets in H n ([6, Theorem 3.9] or [22]) satisfying natural analogues of the WGL property. The argument, and the notation, is nearly verbatim the same as in the proof of [6, Theorem 3.9], so we refer there for details.
The conclusion of Theorem 6.55 is only meaningful if H 1 pπ V pQ 0 qq is relatively large. If γ Ă H is a regular curve, then Lemma 6.57 below ensures that for every Q 0 P D, there exists a horizontal subgroup V Ă H such that H 1 pπ V pQ 0 qq regpγq ℓpQ 0 q. (6.56) The enemy is the possibility Q 0 Ă γ "wraps tightly around a vertical line", so that it projects to a set of small H 1 measure on the xy-plane, and in particular on every horizontal subgroup V. Yet, heuristically, the regular curve γ simply cannot resemble a vertical line that much. This eventually gives the existence of V such that (6.56) holds.
Lemma 6.57. Let γ Ă H be a regular curve. Then γ has big horizontal projections, which means the following. There exists a constant c regpγq 1 such that such for all p 0 P γ and all 0 ă r ď diampγq, there is a horizontal subgroup V Ă H such that H 1 pπ V pγ X Bpp 0 , rqqq ě cr. (6.58) Proof of Lemma 6.57. Let γ Ă H be a regular curve parametrised by an interval I Ă R.
Write π : H Ñ R 2 for the projection map πpx, y, tq " px, yq. Fix a point p 0 P γ, and a radius 0 ă r ă κ diampγq for a suitable small absolute constant κ ą 0 (if diampγq " 8, there is no restriction for r ą 0). Consider then the projection γ π :" πpγq Ă R 2 , and write γ π psq :" πpγpsqq for s P I. Assume without loss of generality that p 0 " γp0q " 0. Since r ă κ diampγq, there exists another point p 1 " γps 1 q P γ with }p 1 } ě r{κ. We choose the smallest parameter s 1 ą 0 with this property, and we restrict attention to considering γ| r0,s 1 s and γ π | r0,s 1 s . We claim that if κ ą 0 was chosen small enough, depending on regpγq, then there exists a point s P r0, s 1 s with the property that |γ π psq| " r. (6.59) We only have to exclude the possibility that the projection γ π | r0,s 1 s stays inside the open disc U p0, rq. To see this, assume that (6.59) fails for all 0 ă s ď s 1 . We assume, for example, that the third component t 1 of p 1 " γps 1 q is strictly positive. Now comparing the conditions |πpp 1 q| " |γ π ps 1 q| ă r and }p 1 } ě r{κ in fact shows that ? t 1 r{κ, hence t 1 r 2 κ 2 . (6.60) To proceed, cover the box U p0, rqˆr0, t 1 s Ă H with boundedly overlapping balls of radius 2r centred on the t-axis or, equivalently, with vertical translates of the box U p0, 2rqr´4 r 2 , 4r 2 s. According to (6.60), the required number of such boxes is " t 1 {r 2 . Moreover, since γ| r0,s 1 s is a continuum satisfying |γ π psq| ă r for all s P r0, s 1 s, and γps 1 q " p 1 , it must in fact meet t 1 {r 2 of the slightly smaller boxes of the type U p0, rqˆr´r 2 , r 2 s. Finally, by the 1-regularity of γ, we have γ X rU p0, rqˆr´r 2 , r 2 ss ‰ H ùñ H 1 pγ X rU p0, 2rqˆr´4r 2 , 4r 2 ssq " r. Since also ? t 1 is much larger than r, we on the other hand observe that U p0, 2rqˆr0, t 1 s is covered by the single " ? t 1 -ball" B ? t 1 :" U p0, ? t 1 qˆr0, t 1 s. This gives us the two-sided estimate t 1 r " t 1 r 2 r H 1 pγ X rU p0, 2rqˆr0, t 1 ssq ď H 1 pγ X B ? t 1 q ? t 1 , hence t 1 r 2 . This violates (6.60) for κ ą 0 small enough, and the proof of (6.59) is complete. Now, we let s 0 P r0, s 1 s be the first parameter such that (6.59) holds, and we also recall that γpsq P Bp0, r{κq for all s P r0, s 1 s. Then t0, γ π ps 0 qu Ď γ π pr0, s 0 sq Ď πpBp0, r{κqq.
We then put the pieces together to prove Theorem 6.42.
Proof of Theorem 6.42. Let γ Ă H be a regular curve. Fix p P γ and 0 ă r ď diampγq. Start by choosing a compact regular curve γ 0 Ă γ with regpγ 0 q regpγq, which contains p, and satisfies diampγ 0 q ě r. Then γ 0 satisfies the WGL by Lemma 6.53, and, on the other hand, Lemma 6.57 gives a horizontal subgroup V Ă H such that H 1 pπ V pBpp, rq X γ 0 qq ě cr, where c regpγq 1 (to be precise, use the version (6.56) for a dyadic cube Q 0 Ă Bpp, rq X γ 0 with ℓpQ 0 q " r). Finally, apply Theorem 6.55 to γ 0 , with parameter b " c{2, and use the 1-Lipschitz property of π V to deduce that H 1 pγ 0 X Γ i q c{N for some 1 ď i ď N . Since N only depends on the WGL and 1-regularity constants of γ 0 (both of which are uniform), the proof is complete.

Singular integrals on regular curves.
It is now easy to put the pieces together to arrive at the main result, Theorem 1.6, which stated that good kernels are CZ kernels for regular curves in H.
Proof of Theorem 1.6. Let γ Ă H be a regular curve. Then γ is contained in an unbounded regular curve (attach horizontal half-lines if necessary). Since it suffices to prove the boundedness of any SIO on the extension, we may assume that diampγq " 8 to begin with. Therefore, µ :" H 1 | γ P Σ 1 in the sense of Definition 6.1. By Theorem 6.42, moreover, γ has BPiLG. This means that, for every ball B centred on γ, there exists an intrinsic Lipschitz graph Γ B with µpΓ B q ě θµpBq (with regpΓ B q uniformly bounded). By Proposition 3.56 (extension of intrinsic Lipschitz graphs), we may moreover arrange that Γ is unbounded, and σ B :" H 1 | Γ B P Σ 1 (with regpσ B q uniformly bounded from above). Now, let k : H z t0u Ñ C be a good kernel, and write Kpp, qq :" kpq´1¨pq. We already know, by Theorem 1.7 and Remark 2.17, that the maximal SIO Tσ B induced by pK, σ B q is bounded on L p pσ B q, 1 ă p ă 8, with constants independent of the choice of B. Therefore, the hypotheses of Theorem 6.3 are met for K and µ, and (6.4) implies that K is a CZ kernel for µ, as claimed in Theorem 1.6. To deduce Theorem A.1 from this statement, all we need to do is establish (A.2), that is, find an η-Lipschitz map ψ T : R Ñ R, and a 2-Lipschitz linear map L T : R Ñ R, such that (A.2) holds. We start by applying Theorem A.3 with a sufficiently small parameter η 1 ą 0, at least so small that 0 ă η 1 ă η{12. Then, fix T P F, and Q P T . Let Γ T " R θ ptpx, φ T pxqq : x P Ruq be a rotated η 1 -Lipschitz graph appearing in (A.4), that is, R θ px, yq " px cos θ´y sin θ, x sin θ`y cos θq, and φ T : R Ñ R is η 1 -Lipschitz. We first observe that, if η 1 ą 0 is small enough, then |tan θ| ď 2. Namely, the case tan θ " 2 and η 1 " 0 would imply, by (A.4), that φ| Q is affine with slope in t´2, 2u, contradicting the 1-Lipschitz assumption. The case of "small η 1 " requires a small additional argument, which we leave to the reader. Now, we claim that Γ T can be written as the graph of a function of the form ψ T`LT , where ψ T is η-Lipschitz, and L T pxq " x tan θ. To this end, we note that Γ T " tpzpxq, x sin θ`φ T pxq cos θq : x P Ru, where zpxq " x cos θ´φ T pxq sin θ. Here, |zpxq´zpx 1 q| ě r|cos θ|´η 1 |sin θ|s|x´x 1 | ě 1 4 |x´x 1 |, (A.5) taking η 1 ą 0 small enough, since |cos θ| ě 1{ ? 5. In particular, the change-of-variables x Þ Ñ zpxq is bijective, and it now suffices to find a η-Lipschitz ψ T : R Ñ R such that x sin θ`φ T pxq cos θ " ψ T pzpxqq`zpxq tan θ.