Quantitative absolute continuity of planar measures with two independent Alberti representations

We study measures $\mu$ on the plane with two independent Alberti representations. It is known, due to Alberti, Cs\"ornyei, and Preiss, that such measures are absolutely continuous with respect to Lebesgue measure. The purpose of this paper is to quantify the result of A-C-P. Assuming that the representations of $\mu$ are bounded from above, in a natural way to be defined in the introduction, we prove that $\mu \in L^{2}$. If the representations are also bounded from below, we show that $\mu$ satisfies a reverse H\"older inequality with exponent $2$, and is consequently in $L^{2 + \epsilon}$ by Gehring's lemma. A substantial part of the paper is also devoted to showing that both results stated above are optimal.


INTRODUCTION
Before stating any results, we need to define a few key concepts.

Definition 1.1 (Cones and C-graphs).
A cone stands for a subset of R d of the form C " Cpe, θq " tx P R d : |x¨e| ě θ|x|u, where e P S d´1 and 0 ă θ ď 1. Given a cone C Ă R d , a C-graph is any set γ Ă R d such that x´y P C for all x, y P γ.
A C-graph γ is called maximal if the orthogonal projection π e : γ Ñ spanpeq is surjective. The family of all maximal C-graphs is denoted by Γ C . We record here that if γ is a Cpe, θqgraph with θ ą 0, then the orthogonal projection π e : γ Ñ spanpeq is a bilipschitz map. Also, if γ is maximal, then H 1 | γ is a 1-regular measure on γ. In other words, there exist constants 0 ă c ď C ă 8 such that cr ď H 1 pγ X Bpx, rqq ď Cr for all x P γ and r ą 0.
We say that two cones C 1 , C 2 are independent if they are angularly separated as follows: τ :" inft=px 1 , x 2 q : x 1 P C 1 z t0u and x 2 P C 2 z t0uu ą 0.
(1.2) Definition 1.3 (Alberti representations). Let C Ă R d be a cone. Let pΩ, Σ, Pq be a measure space with PpΩq ă 8, and let γ : Ω Ñ Γ C be a map such that for all Borel sets B Ă R d . Then, the formula νpBq :" ν pΩ,P,γq pBq :" ż Ω H 1 pB X γpωqq dPpωq (1.5) makes sense for all Borel sets B Ă R d , and evidently νpKq ă 8 for all compact sets K Ă R d . We extend the definition to all sets A Ă R d via the usual procedure of setting first ν˚pAq :" inftνpBq : A Ă B Borelu. This process yields a Radon measure ν˚which agrees with ν on Borel sets. In the sequel, we just write ν in place of ν˚.
If µ is another Radon measure on R d , we say that µ is representable by C-graphs if there is a triple pΩ, P, γq as above such that µ ! ν pΩ,P,γq ": ν. In this case, the quadruple pΩ, P, γ, dµ dν q is an Alberti representation of µ by C-graphs. The representation is ‚ bounded above (BoA) if dµ dν P L 8 pνq, ‚ bounded below (BoB) if p dµ dν q´1 P L 8 pνq. We also consider local versions of these properties: the representation is BoA (resp. BoB) on a Borel set B Ă R 2 if dµ dν P L 8 pB, νq (resp. p dµ dν q´1 P L 8 pB, νq). Two Alberti representations of µ by C 1 -and C 2 -graphs are independent, if the cones C 1 , C 2 are independent in the sense (1.2).
Representations of this kind first appeared in Alberti's paper [1] on the rank-1 theorem for BV -functions. It has been known for some time that planar measures with two independent Alberti representations are absolutely continuous with respect to Lebesgue measure; this fact is due to Alberti, Csörnyei, and Preiss, see [2,Proposition 8.6], but a closely related result is already contained in Alberti's original work, see [1,Lemma 3.3]. The argument in [2] is based on a decomposition result for null sets in the plane, [2, Theorem 3.1]. Inspecting the proof, the following statement can be easily deduced: if µ is a planar measure with two independent BoA representations, then µ P L 2,8 . The proof of [1, Lemma 3.3], however, seems to point towards µ P L 2 , and the first statement of Theorem 1.6 below asserts that this is the case. Our argument is short and very elementary, see Section 2.1. The main work in the present paper concerns measures with two independent representations which are both BoA and BoB. In this case, Theorem 1.6 asserts an ǫ-improvement over the L 2 -integrability. Theorem 1.6. Let µ be a Radon measure on R 2 with two independent Alberti representations. If both representations are BoA, then µ P L 2 pR 2 q. If both representations are BoA and BoB on Bp2q, then there exists a constant C ě 1 such that µ satisfies the reverse Hölder inequalitỹ As a consequence, µ P L 2`ǫ pBp 1 2 qq for some ǫ ą 0. The final conclusion follows easily from Gehring's lemma, see [5, Lemma 2].
1.1. Sharpness of the main theorem. We now discuss the sharpness of Theorem 1.6. For illustrative purposes, we make one more definition. Let µ be a Radon measure on R 2 . We say that pΩ, P, γ, dµ dν q is an axis-parallel representation of µ if Ω " R, and γ : Ω Ñ PpR 2 q is one of the two maps γ x pωq " tωuˆR or γ y pωq " Rˆtωu. Note that two axisparallel representations pR, P 1 , γ 1 , dµ dν 1 q and pR, P 2 , γ 2 , dµ dν 2 q are independent if and only if tγ 1 , γ 2 u " tγ x , γ y u.
The following example shows that two independent BoA representations -even axis parallel ones -do not guarantee anything more than L 2 : Example 1.8. Fix r ą 0 and consider the measure µ r " 1 r¨1 r0,rs 2 . Note that }µ r } L p " r 2´p for p ě 1, so µ r P L p uniformly in r ą 0 if and only if p ď 2. On the other hand, consider the probability P :" 1 r¨L 1 | r0,rs on Ω " R, and the maps γ 1 :" γ x and γ 2 :" γ y , as above. Writing ν j :" ν pΩ,P,γ j q for j P t1, 2u, it is easy to check that So, µ r has two independent axis-parallel BoA representations with constants uniformly bounded in r ą 0. After this, it is not difficult to produce a single measure µ with two independent axisparallel BoA representations which is not in L p for any p ą 2: simply place disjoint copies of c j µ r j along the diagonal tpx, yq : x " yu, where ř c j " 1 and r j Ñ 0 rapidly.
The situation where both representations are (locally) both BoA and BoB is more interesting. We start by recording the following simple proposition, which shows that Theorem 1.6 is far from sharp for axis-parallel representations: Proposition 1.9. Let µ be a finite Radon measure on R 2 with two independent axis-parallel representations, both of which are BoA and BoB on r0, 1q 2 . Then there exist constants 0 ă c ď C ă 8, depending only on the BoA and BoB constants, such that µ| r0,1q 2 " f dL 2 | r0,1q 2 , where 0 ă c ď f pxq ď C ă 8 for L 2 almost every x P r0, 1q 2 .
We give the easy details in the appendix. In the light of the proposition, the following theorem is perhaps a little surprising: The representations are, of course, not axis-parallel. For a picture, see Figure 4. Since this shows that L 2`ǫ -integrability claimed in Theorem 1.6 is sharp.
Remark 1.12. The localisation in Theorem 1.10 is necessary: for 0 ă α ă 1, the weight µ " |x|´α dx has no BoA representations in the sense of Definition 1.3, where we require that PpΩq ă 8. Indeed, let C " Cpe, θq be an arbitrary cone, and assume that pΩ, P, γ, dµ dν q is an Alberti representation of µ by C-graphs. Let e 1 K e, and let T be a strip of width 1 around spanpe 1 q. Then µpT q " 8. However, H 1 pγ X T q θ 1 for all γ P Γ C , and hence νpT q θ PpΩq ă 8. This implies that dµ dν R L 8 pνq.
Notation 1.13. For A, B ą 0, the notation A B will signify that there exists a constant C ě 1 such that A ď CB. This is typically used in a context where one or both of A, B are functions of some variable "x": then Apxq Bpxq means that Apxq ď CBpxq for some constant C ě 1 independent of x. Sometimes it is worth emphasising that the constant C depends on some parameter "p", and we will signal this by writing A p B.

1.2.
Higher dimensions, and connections to PDEs. The problems discussed above have natural -but harder -generalisations to higher dimensions. A collection of d cones With this definition in mind, one can discuss Radon measures on R d with d independent Alberti representations. It follows from the recent breakthrough work of De Philippis and Rindler [4] that such measures are absolutely continuous with respect to Lebesgue measure. It is tempting to ask for more quantitative statements, similar to the ones in Theorem 1.6. Such statement do not appear to easily follow from the strategy in [4].

Question 1.
If µ is a Radon measure on R d with d independent BoA representations, then is µ P L p for some p ą 1?
In the case of independent axis-parallel representations, µ P L d{pd´1q , see the next paragraph. This is the best exponent, as can be seen by a variant of Example 1.8. In general, we do not know how to prove µ P L p for any p ą 1. Some results of this nature will likely follow from work in progress recently announced by Csörnyei and Jones.
Question 1 is closely connected with the analogue of the multilinear Kakeya problem for thin neighbourhoods of C-graphs. A near-optimal result on this variant of the multilinear Kakeya problem is contained in the paper [6] of Guth, see [6,Theorem 7]. We discuss this connection explicitly in [3, Section 5]. It seems that the "S ǫ -factor" in [6, Theorem 7] makes it inapplicable to Question 1, and it does not even imply the qualitative absolute continuity of µ established in [4]. On the other hand, the analogue of [6,Theorem 7] without the S ǫ -factor would imply a positive answer to Question 1 with p " d{pd´1q, see the proof of [3, Lemma 5.2]. We do not know if this is a plausible strategy, but it certainly works for the axis-parallel case: the analogue of [6, Theorem 7] for neighbourhoods of axis-parallel lines is simply the classical Loomis-Whitney inequality (see [7] or [6, Theorem 3]), where no S ǫ -factor appears.
As mentioned above, the main results in this paper, and Question 1, are related to the recent work of De Philippis and Rindler [4] on A-free measures. Introducing the notation of [4] would be a long detour, but let us briefly explain some connections, assuming familiarity with the terminology of [4].
The qualitative absolute continuity result, mentioned above Question 1, follows from [4, Corollary 1.12] after realising that, for each Alberti representation of µ, (1.5) may be used to construct a normal 1-current The independence of the representations translates into the statement (1.14) One may view the d-tuple of normal currents T " pT 1 , . . . , T d q as an R dˆd -valued measure T " T}T}, where | T| " 1, and }T} is a finite positive measure. Since each T i is normal, div T is also a finite measure, and this is the key point relating our situation with the work of De Philippis and Rindler. If the Alberti representations of µ are BoA, then dµ{d}T} P L 8 p}T}q, and µ P L 2 pR 2 q by Theorem 1.6. As far as we know, PDE methods do not yield the same conclusion. However, if in addition the Jacobian of T is uniformly bounded from below }T} almost everywhere, PDE methods look more promising. We formulate the following question, which is parallel to Question 1: Let T " T}T} be a finite R dˆd -valued measure, whose divergence is also a finite (signed) measure such that the Jacobian of T is uniformly bounded from below in absolute value }T} a.e. Is it true that }T} P L p pR d q for some p ą 1?
1.3. Acknowledgements. We would like to thank Vesa Julin for many useful conversations on the topics of the paper. We also thank the anonymous referee for helpful suggestions leading to Question 2.

PROOF OF THE MAIN THEOREM
We prove Theorem 1.6 in two parts, first considering representations which are only BoA, and then representations which are both BoA and BoB at the same time.
Proof. It suffices to show that the restriction of µ to any dyadic square Q 0 Ă R 2 is in L 2 , with norm bounded (independently of Q 0 ) as in (2.2). For notational simplicity, we assume that Q 0 " r0, 1q 2 . Let D n :" D n pr0, 1q 2 q, n P N, be the family of dyadic subsquares of r0, 1q 2 of side-length 2´n. Fix n P N, pick Q P D n , and write ΩpQq :" tpω 1 , ω 2 q P Ω 1ˆΩ2 : H 1 pQ X γpω 1 qq ą 0 and H 1 pQ X γpω 2 qq ą 0u.
Note that tω P Ω j : H 1 pγpωq X Qq ą 0u P Σ j for j P t1, 2u by (1.4), so ΩpQq lies in the σ-algebra generated by Σ 1ˆΣ2 . We start by showing that To prove (2.3), it suffices to fix a pair pγ 1 , γ 2 q P Γ 1ˆΓ2 , where Γ j :" Γ C j , and show that there are τ 1 squares Q P D n with γ 1 X Q ‰ H ‰ γ 2 X Q. So, fix pγ 1 , γ 2 q P Γ 1ˆΓ2 , and assume that there is at least one square Q such that γ 1 X Q ‰ H ‰ γ 2 X Q, see Figure 1.
To simplify some numerics, assume that Q " r0, 2´nq 2 . Pick x 1 P γ 1 X Q and x 2 P γ 2 X Q, and note that γ 1 Ă x 1`C1 and γ 2 Ă x 2`C2 , since γ 1 P Γ C 1 and γ 2 P Γ C 2 . It follows that whenever Q 1 P D n is another square with γ 1 X Q 1 ‰ H ‰ γ 2 X Q 1 , we can find points The curves γ 1 , γ 2 and the square Q.
This completes the proof.
In the next corollary, we write Apx, r, Rq :" Bpx, Rq z Bpx, rq for x P R 2 and 0 ă r ă R ă 8. Also, if C " Cpe, θq Ă R 2 is a cone, we write C`:" tx P R 2 : x¨e ě θ|x|u and C´:" tx P R 2 : x¨e ď´θ|x|u for the corresponding "one-sided" cones. Corollary 2.7. Let C 1 , C 2 Ă R 2 be two cones with mint=px 1 , x 2 q : x 1 P C 1 z t0u and x 2 P C 2 z t0uu ě τ ą 0.
Then, the following holds for ǫ :" mint|τ {100| 4 , 10´4u, and for any x P R 2 , r ą 0, and n P BBpx, rq. There exists j P t1, 2u and a sign ‹ P t´,`u (depending only on x, n) such that Apy, r ? ǫ, 2r ? ǫq X ry`C ‹ j s Ă Bpx, rq, y P Bpn, ǫrq. (2.8) The statement is best illustrated by a picture, see Figure 2.
Proof of Corollary 2.7. After rescaling, translation, and rotation, we may assume that x " 0, r " 1, and n " n " p0, 1q. (2.9) Write π y px, yq :" y. We start by noting that π y pC j X S 1 q X r´τ 100 , τ 100 s " H (2.10) for either j " 1 or j " 2. If this were not the case, we could find x 1 P C 1 X S 1 and x 2 P C 2 X S 1 such that |sin =px j , p1, 0qq| " |π y px j q| ď τ {100 for j P t1, 2u. Then either =px 1 , x 2 q ă τ or =px 1 ,´x 2 q ă τ . Both contradict the definition of τ , given that also x 2 P C 2 . This proves (2.10). Fix j P t1, 2u such that (2.10) holds, and write, for ‹ P t´,`u, Then Jj "´Jj , and consequently π y pJj q "´π y pJj q. It follows from this, (2.10), and the fact that π y pJ ‹ j q is an interval, that either π y pvq ă´τ {100 for all v P Jj or π y pvq ă´τ {100 for all v P Jj . We pick ‹ P t´,`u such that this conclusion holds. In other words, the y-coordinate of every point v P C ‹ j X S 1 is ă´τ {100. It follows from the previous lemma, and the choice of ǫ, that which is equivalent to (2.8) (recalling (2.9)).
For the rest of the section, we assume that µ is a Radon measure on R 2 with µpBp1qq ą 0, and that µ has two independent Alberti representations which are both BoA and BoB on Bp2q. Thus, there exists a constant C ě 1 such that C´1νpAq ď µpAq ď CνpAq (2.11) for all Borel sets A Ă Bp2q. By Theorem 2.1, we already know that µ P L 2 pBp1qq. We next aim to show that Bp1q Ă spt µ, and µ is a doubling weight on Bp1q in the following sense: Bpx, rq Ă Bp1q ùñ µpBpx, 3 2 rqq C,τ µpBpx, rqq. (2.12) After this, it will be easy to complete the proof of the reverse Hölder inequality (1.7).

Lemma 2.13.
Let µ be a measure as above. Then µ is doubling on Bp1q in the sense of (2.12), where the constants only depend on C from (2.11) and τ from (1.2). In particular, Bp1q Ă spt µ.
Recall the half-cones C ‹ j , ‹ P t´,`u, defined above Corollary 2.7. By Corollary 2.7, there exist choices of j P t1, 2u and ‹ P t´,`u, depending only on x and x i P BBpx, rq (i.e. the centre of B), such that Define Ω j pBq :" tω P Ω j : H 1 pB X γ j pωqq ą 0u P Σ j . We observe that if ω P Ω j pBq, then H 1 pG X γ j pωqq " ǫ r. Indeed, if ω P Ω j pBq, then certainly γpωq contains a point y P B and then one half of the graph γ j pωq is contained in y`C ‹ j . This half intersects Apyq in length " ǫ r, and the intersection is contained in G by definition. It follows that ν j pGq ě ż Ω j pBq H 1 pG X γ j pωqq dP j pωq ǫ r¨P j pΩ j pBqq ż Ω j pBq H 1 pB X γ j pωqq dP j pωq " ν j pBq.
Since G Ă Bpx, rq, this yields (2.15) and completes the proof.
We can now complete the proof of the reverse Hölder inequality (1.7).
Concluding the proof of Theorem 1.6. Fix a ball B :" Bpx, rq Ă Bp1q, and consider the restrictions of the measures P 1 , P 2 to the sets Ω j pBq :" tω P Ω j : H 1 pB X γ j pωqq ą 0u P Σ j , j P t1, 2u.

SHARPNESS OF THE REVERSE HÖLDER EXPONENT
The purpose of this section is to prove Theorem 1.10. The statement is repeated below: has two independent Alberti representations which are both BoA and BoB on r´1, 1s 2 .
Remark 3.3. It may be worth pointing out that, in the construction below, the BoA and BoB constants stay uniformly bounded for α P p0, 1q. However, the independence constant of the two representations (that is, the constant "τ " from (1.2)) tends to zero as α Õ 1. In this section, the constants hidden in the """ and " " notation will not depend on α.
We have replaced Bp1q by r´1, 1s 2 for technical convenience; since Bp1q Ă r´1, 1s 2 , the result is technically stronger than Theorem 1.10. The two representations will be denoted by tΩ 1 , P 1 , γ 1 , dµ dν 1 u and tΩ 2 , P 2 , γ 2 , dµ dν 2 u. We will first construct one representation of µ restricted to r0, 1s 2 , as in Figure 3, and eventually extend that representation to r´1, 1s 2 , as on the left hand side of Figure 4. We set Ω 1 :" r´1, 1sˆt1u ": Ω 2 , and we let P " P j :" H 1 | Ω j . The main challenge is of course to construct the graphs γ j pωq, ω P Ω j . A key feature of f is that f pr, tq " f pt, rq for pr, tq P r´1, 1s 2 . Hence, as we will argue carefully later, it suffices to construct a single representation by C-graphs, where C is a cone around the y-axis, with opening angle strictly smaller than π{2; such a representation is depicted on the left hand side of Figure 3. We remark that, as the picture suggests, every C-graph associated to the representation can be expressed as a countable union of line segments. The second representation is eventually acquired by rotating the first representation by π{2, see the right hand side of Figure 4. Now we construct certain graphs γpωq for ω P Ω :" r0, 1sˆt1u Ă Ω 1 . The idea is that eventually γ 1 pωq X r0, 1s 2 " γpωq for ω P Ω. The graphs γpωq will be constructed so that γpωq X Ω " tωu, ω P Ω. The right idea to keep in mind is that the graph γpωq "starts from ω P Ω " r0, 1sˆt1u, travels downwards, and ends somewhere on r0, 1sˆt0u". We will ensure that r0, 1s 2 is foliated by the graphs γpωq, ω P Ω. Start by fixing a point p P Ω whose x-coordinate lies in p1{2, 1q, see Figure 3. The relationship between p and the exponent α in (3.2) will be specified under (3.6). Let I 0 :" rp0, 1q, ps Ă Ω and I 1 :" pp, p1, 1qs Ă Ω.  We can now specify the graphs γpωq with ω P I 1 . Each of them consists of two line segments: the first one connects I 1 to pp 1 2 , 1 2 q, p1, 1 2 qs, and the second one is vertical, connecting pp 1 2 , 1 2 q, p1, 1 2 qs to r1{2, 1sˆt0u, see Figure 3. We also require that the graphs γpωq foliate the yellow pentagon R 0 in Figure 3. This description still gives some freedom on how to choose the first segments, but if the choice is done in a natural way, we will find that |tω P I 1 : γpωq X B ‰ Hu| " diampBq (3.5) for all balls B Ă R 0 . Here, and in the sequel, |¨| stands for 1-dimensional Hausdorff measure. The implicit constant of course depends on the length of I 1 (and hence p, and eventually α).
We then move our attention to defining the graphs γpωq with ω P I 0 . Look again at Figure 3 and note the green trapezoidal regions, denoted by T j , j ě 0. To be precise, T 0 is the convex hull of I 0 Y rp0, 1 2 q, p 1 2 , 1 2 qs, and T j :" 2´jT 0 " t2´jpx, yq : px, yq P T 1 u, j ě 1.
(3.7) Now, we have defined the intersections of the curves γpωq with T 0 Y R 0 . In particular, the following set families are well-defined: ΓpT 0 q :" tγpωq X T 0 : ω P I 0 u and ΓpR 0 q :" tγpωq X R 0 : ω P I 1 u. The graphs in ΓpR 0 q are already complete in the sense that they connect Ω to r0, 1sˆt0u. The graphs in ΓpT 0 q are evidently not complete, and they need to be extended. To do this, we define R j :" 2´jR 0 for j ě 1, see Figure 3, and we define the set families ΓpR j q " 2´jΓpR 0 q, j ě 1. for j ě 1. In other words, the sets in ΓpR j q are obtained by rescaling the graphs in ΓpR 0 q so they fit inside, and foliate, R j . We note that the sets in ΓpR j q connect points in I j 1 to r0, 1sˆt0u for j ě 0.
Finally, we define the complete graphs γpωq, ω P I 0 as follows. Fix ω P I 0 , and note that γ 0 :" γpωq X T 0 has already been defined, and the intersection γ 0 X Ω 1 contains a single point z " σ 0 pωq, which lies in either I 1 0 or I 1 1 . If z P I 1 1 Ă R 1 , then there is a unique set γ 1 P ΓpR 1 q with z P γ 1 . Then we define γpωq X rT 0 Y R 1 s :" γ 0 Y γ 1 . In this case γpωq is now a complete graph, and the construction of γpωq terminates. Before proceeding with the case z P I 1 0 , we pause for a moment to record a useful observation. If B Ă R 1 is a ball, consider Ω 1 pBq :" tx P Ω 1 : x P γ and γ X B ‰ H for some γ P Γu. Since all the graphs γ P Γ entering R 1 can be written as γ 0 Y γ 1 with γ 0 terminating at I 1 1 and γ 1 P ΓpR 1 q, the set Ω 1 pBq can be rewritten as Ω 1 pBq " tx P I 1 1 : x P γ and γ X B ‰ H for some γ P ΓpR 1 qu. Then, recalling that I 1 1 " 2´1I 1 , and ΓpR 1 q " 2´1ΓpR 0 q, and noting that 2B Ă R 0 , we see that |Ω 1 pBq| " 1 2 |tω P I 1 : γpωq X 2B ‰ Hu " 1 2¨d iamp2Bq " diampBq, (3.8) using (3.5). The main point here is that the implicit constant is the same (absolute constant) as in (3.5). We remark that here 2B " t2x : x P Bu is the honest dilation of B (and not a ball with the same centre and twice the radius as B).
It is now clear how to proceed inductively, assuming that γpωq X rT 0 Y . . . Y T k s has already been defined for some k ě 1, and then considering separately the cases In the case γpωq X Ω k`1 Ă I k`1 1 , we extend γpωq to a complete graph contained in T 0 Y . . . Y T k Y R k`1 by concatenating γpωq X rT 1 Y . . . Y T k s with a set from ΓpR k`1 q. Arguing as in (3.8), we have in this case the following estimate for all balls B Ă R k`1 : |tx P Ω k`1 : x P γ and γ X B ‰ H for some γ P Γu| " diampBq, (3.10) where the implicit constant is the same as in (3.5). Indeed, the set on the left hand side of (3.10) is equal to a translate of 2´p k`1q tω P I 1 : γpωq X 2 k`1 B ‰ Hu.
In the case γpωq X Ω k`1 Ă I k`1 0 , we define the map σ k`1 : Ñ Ω k`2 as before: and connect the points z P I k`1 0 to σ k`1 pzq P Ω k`2 by line segments. Arguing as in (3.7) and (3.9), we find that Ptω P Ω : γpωq X I ‰ Hu " β´k¨|I|, k ě 0, I Ă Ω k Borel. (3.11) This completes the definition of the graphs in Γ. It is easy to check inductively that graphs in Γ foliate p0, 1s 2 . Moreover, the (partially defined) graph γp0, 1q never leaves t0uˆr0, 1s during the construction, so we can simply agree that p0, 0q is the endpoint of γp0, 1q, thus completing the foliation of r0, 1s 2 . The sets in Γ are clearly (non-maximal) C-graphs with respect to some cone of the form C " Cpp0, 1q, θq. As long as p ‰ p1, 1q, the opening angle of C is strictly smaller than π{2, or in other words θ ą sinp π 4 q. We then extend the graphs γpωq P Γ to maximal C-graphs γ 1 pωq, ω P Ω, as follows (see Figure 4 for an idea of what is happening). For ω P Ω P r0, 1sˆt1u, let γpωq Ă r0, 1s 2 be the graph constructed above, and let Xpx, yq :" px,´yq and Y px, yq :" p´x, yq be the reflections over the x-axis and y-axis, respectively. First concatenate γpωq with a vertical half-line starting from ω and travelling upwards. Denoting this "half-maximal" graph byγpωq, we let γ 1 pωq :"γpωq Y Xpγpωqq, ω P Ω.
Recall the measure µ " f dL 2 | r´1,1s 2 defined in (3.2). We will next show that µ P L 8 pν 1 q and ν 1 P L 8 pr´1, 1s 2 , µq. (3.12) In other words, the Alberti representation pΩ, P, γ, dµ dν 1 q of µ by C-graphs is both BoA and BoB on r´1, 1s 2 . Noting that f˝X " f˝Y " f , and Xpν 1 q " Y pν 1 q " ν 1 , it suffices to compare µ and ν 1 on r0, 1s 2 . Moreover, it suffices to show that the Radon-Nikodym derivative pdν 1 {dL 2 qpzq at L 2 almost every interior point z of one of the regions T k or R k is comparable to f pzq.