Dimensional estimates and rectifiability for measures satisfying linear PDE constraints

We establish the rectifiability of measures satisfying a linear PDE constraint. The obtained rectifiability dimensions are optimal for many usual PDE operators, including all first-order systems and all second-order scalar operators. In particular, our general theorem provides a new proof of the rectifiability results for functions of bounded variations (BV) and functions of bounded deformation (BD). For divergence-free tensors we obtain refinements and new proofs of several known results on the rectifiability of varifolds and defect measures.

An R m -valued Radon measure μ ∈ M(U ; R m ) defined on an open set U ⊂ R d is said to be A-free if Aμ = 0 in the sense of distributions on U. (1.1) The Lebesgue-Radon-Nikodým theorem implies that where g ∈ L 1 (U ; R m ), |μ| s is the singular part of the total variation measure |μ| with respect to the d-dimensional Lebesgue measure L d , and dμ d|μ| (x) := lim is the polar of μ, which exists and belongs to S m−1 for |μ|-almost every x ∈ U . In [DR16] it was shown that for any A-free measure there is a strong constraint on the directions of the polar at singular points: Theorem 1.1 ([DR16, Theorem 1.1]). Let U ⊂ R d be an open set, let A be a kthorder linear constant-coefficient differential operator as above, and let μ ∈ M(U ; R m ) be an A-free Radon measure on U with values in R m . Then, where Λ A is the wave cone associated to A, namely It has been shown in [DR16], see also [D16,DR18] for recent surveys and [Rin18, Chapter 10] for further explanation, that by suitably choosing the operator A, the study of the singular part of A-free measures has several consequences in the calculus of variations and in geometric measure theory. In particular, we recall the following: • If A = curl, the above theorem gives a new proof of Alberti's rank-one theorem [Alb93] (see also [MV16] for a different proof based on a geometrical argument). • If A = div, combining Theorem 1.1 with the result of [AM16], one obtains the weak converse of Rademacher's theorem (see [DMR17,GP16,KM18] for other consequences in metric geometry).
The main results of this paper is to show how Theorem 1.1 can be improved by further constraining the direction of the polars on "lower dimensional parts" of the measure μ and to establish some consequences of this fact concerning dimensional estimates and rectifiability of A-free measures. To this end let us define a hierarchy of wave cones as follows: Definition 1.2. ( -wave cone) Let Gr( , d) be the Grassmannian of -planes in R d . For = 1, . . . , d we define the -dimensional wave cone as Equivalently, Λ A can be defined by the following analytical property: where (A π) is the partial differential operator

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with p π the orthogonal projection onto π. Note that, by the very definition of Λ A , we have the following inclusions: To state our main theorem, we also recall the definition of the integral-geometric measure, see [Mat95, Section 5.14]: Let ∈ {0, . . . , d}. For a Borel set E ⊂ R d , the -dimensional integral-geometric (outer) measure is where γ ,d is the unique O(d)-invariant probability measure on Gr( , d) and H is the -dimensional Hausdorff measure (normalized as in [Mat95] such that H (B 1 ) = 2 , where B 1 is the -dimensional unit ball).
Our main result establishes that the polar of an A-free measure is constrained to lie in a smaller cone on I -null sets: Note that, by taking = d, Theorem 1.3 recovers Theorem 1.1. As a corollary we obtain the following dimensional estimates on A-free measures; see also [Arr18] for a different proof of (1.4) in the case of first-order systems. The results above and (1.3) entail that the smaller the dimension of an A-free measure μ is, the more its polar is constrained at singular points. Let us also remark that the 1-dimensional wave cone Λ 1 A has been implicitly introduced by van Schaftigen in [Van13]. There, the author calls a (homogeneous) oparator A cocanceling provided that Λ 1 A = {0}. Moreover, it is shown that the cocanceling condition is equivalent to the property Thus, the conclusion of Theorem 1.3 improves upon the dimensional estimates for A-free measures with A cocanceling. The use of the integral-geometric measure, besides being natural in the proof, allows one to use the Besicovitch-Federer rectifiability criterion to deduce the following rectifiability result. Recall that for a positive measure σ ∈ M + (U ) the upper -dimensional density is defined as Theorem 1.5 (Rectifiability). Let A and μ be as in Theorem 1.3, and assume that where λ : R → S m−1 is H -measurable; for H -almost every x 0 ∈ R (or, equivalently, for |μ|-almost every x 0 ∈ R), Theorem 1.5 contains the classical rectfiability result for the jump part of the gradient of a BV function, see [AFP00], and the analogous result for BD, see [Koh79,ACD97]. By choosing A = div we also recover and (in some cases slightly generalize) several known rectifiability criteria, such as Allard's rectifiability theorem for varifolds [All72], its recent extensions to anisotropic energies [DDG18], the rectifiability of generalized varifolds established in [AS97], and the rectifiability of various defect measures in the spirit of [Lin99], see also [Mos03]. We refer the reader to Sect. 3 for some of these statements.
It is worth noting that, with the exception of the BD-rectifiability result in [ACD97, Proposition 3.5], none of the above rectifiability criteria rely on the Besicovitch-Federer theorem and their proofs are based on more standard blowup techniques. However, in the generality of Theorem 1.5 a blow-up proof seems hard to obtain. Indeed, roughly, a blow-up argument follows two steps: • By some measure-theoretic arguments one shows that, up to a subsequence, for some positive measure σ and some fixed vector λ. • One exploits this information together with the A k -freeness of λσ, where A k is the principal part of A, to deduce that σ is translation-invariant along the directions in an -dimensional plane π and thus σ = H π. In this step one usually uses that π is uniquely determined by λ and A.
However, assuming that σ = H π, the only information one can get is see Lemma 2.3, and this does not uniquely determine π in general. Let us now briefly discuss the optimality of our results. First note that (1.6) and (1.7) are true whenever an A-free measure μ has a non-trivial part concentrated on an -rectifiable set R, see Lemma 2.3 below.
In particular, defining for = 0, . . . , d − 1 the cone we have that and Hence, setting * the above discussion yields that if μ has a non-trivial -rectifiable part, then necessarily and this bound is sharp for homogeneous operators since if λ ∈ ξ∈π ⊥ ker A k (ξ)\{0} for some -plane π, then λH π is an A k -free measure.
Recalling the definition of A in (1.5), this discussion together with Corollary 1.4 and (1.8) can then be summarized for homogeneous operators A as For first-order operators it is not hard to check that A = * A (by the linearity of ξ → A k (ξ)). The same is true for second-order scalar operators (n = 1) by reducing the polynomial to canonical form (which makes A k (ξ) linear in ξ 2 1 , . . . , ξ 2 d ). Hence, the above inequality for such homogeneous operators becomes an equality and our theorem is sharp.
On the other hand, it is easy to build examples where A < * A . For instance, one can easily check that for the 3rd-order scalar operator defined on C ∞ (R 3 ) by } is a ruled surface (and hence it contains lines) but it does not contain planes. Moreover, let A be the 6 th -order operator acting on maps from R 3 to R 2 with symbol For this operator we still have A = 1 < 2 = * A , but A additionally satisfies Murat's constant rank condition [Mur81].
Let us remark that in the case A < * A , Theorem 1.5 implies that if μ is an A-free measure, then |μ| {θ * A (|μ|) > 0}) = 0. Hence μ is "more diffuse" than an A -dimensional measure. Furthermore, μ cannot sit on rectifiable sets of any (integer) dimension ∈ [ A , * A ). It seems thus reasonable to expect that its dimension should be larger than A . In particular, one might conjecture the following improvement of Corollary 1.4: Conjecture 1.6. Let μ be A-free and let * A be the rectifiability dimension defined in (1.8). Then, We note that the same conjecture has also been advanced by Raita in [Rai18 Recently, a related (dual) notion of " -canceling" operators has been introduced in [SV18]. We conclude this introduction by remarking that the above results can be used to provide dimensional estimates and rectifiability results for measures whose decomposability bundle, defined in [AM16], has dimension at least . Namely, in this case the measure is absolutely continuous with respect to I and the set where the upper -dimensional density is positive, is rectifiable, compare with [Bat17, Theorem 2.19] and with [AMS]. However, since by its very definition the dimension of the decomposability bundle is stable under projections, in this setting one can directly rely on [DR16, Corollary 1.12]. This is essentially the strategy followed in the cited references.

Proofs
The proof of Theorem 1.3 is a combination of ideas from [DR16] and [DDG18]. We start with the following lemma.
Lemma 2.1. Let B be a homogeneous kth-order linear constant-coefficient operator on R , Let {ν j } ⊂ M(B 1 ; R m ), where B 1 ⊂ R is the unit ball in R , be a uniformly norm-bounded sequence of Radon measures satisfying the following assumptions: (a1) Bλ is elliptic for some λ ∈ R m , that is, Then, up to taking a subsequence, there exists θ ∈ L 1 (B 1 ) such that Proof. The proof is a straightforward modification of the main step of the proof of [DR16, Theorem 1.1], see also [All86] and [Rin18, Chapter 10]. We give it here in terse form for the sake of completeness.
Passing to a subsequence we may assume that |ν j | * σ in C ∞ c (B 1 ) * for some positive measure σ ∈ M + (B 1 ). We must show that σ = θL d and that (2.1) holds. Fix t < 1 and two smooth cut-off functions 0 ≤ χ ≤χ ≤ 1 with χ = 1 on B t , 646 A. ARROYO-RABASA ET AL. GAFÃ χ = 1 on spt(χ), and spt(χ) ⊂ spt(χ) ⊂ B 1 . Let (ϕ ε ) ε>0 be a family of smooth approximations of the identity. Choose j ↓ 0 with 0 < j < 1 − t for all j, such that We will show that the sequence is pre-compact in L 1 (B 1 ), which proves the lemma. For every j we set f j := Bν j and compute Note that the commutator [B, χ] := B • χ − χ • B is a differential operator of order at most k − 1 with smooth coefficients. Taking the Fourier transform (which we denote by F or by the hat " "), multiplying by [B(ξ)λ] * , and adding u j (ξ), we obtain Hence, with the pseudo-differential operators T 0 , . . . , T 3 defined as follows: We see that, in the language of pseudo-differential operators (see for instance [Ste93, Chapter VI]): (i) the symbol for T 0 is a Hörmander-Mihlin multiplier (i.e. a pseudo-differential operator with smooth symbol of order 0) since, due to (a1), |B(ξ)λ| ≥ c|ξ| k for some c > 0 and all ξ ∈ R ; (ii) T 1 is a pseudo-differential operator with smooth symbol of order −k; (iii) T 2 is a pseudo-differential operator with smooth symbol of order −1; GAFA DIMENSIONAL ESTIMATES AND RECTIFIABILITY 647 (iv) T 3 is a pseudo-differential operator with smooth symbol of order −2k.
By the classical theory of Fourier multipliers and pseudo-differential operators we then get the following: (I) T 0 is bounded from L 1 to L 1,∞ (weak-L 1 ), see e.g. [Gra14, Theorem 6.2.7].
Owing to (a3), it follows that for j → ∞ we obtain Thus, Hence, passing to a subsequence, we may assume that loc and T 0 [V j ] → 0 in measure. Since furthermore u j ≥ 0, we can apply Lemma 2.2 below and deduce that T 0 [V j ] → 0 strongly in L 1 . This concludes the proof.
The following is Lemma 2.2 in [DR16], we report here its straightforward proof for the sake of completeness.
(iii) the family of negative parts {f − j } is equiintegrable.

Proof.
Let ϕ ∈ C ∞ c (B 1 ; [0, 1]). Then, The first term on the right-hand side vanishes as j → ∞ by assumption (i). Vitali's convergence theorem in conjunction with assumptions (ii) and (iii) further gives that the second term also tends to zero in the limit.
Proof of Theorem 1.3. Let E be such that I (E) = 0 and let us define By contradiction, let us suppose that |μ|(F ) > 0. Note that, by the very definition of F , for all x ∈ F there exists an -dimensional planeπ x ⊂ R d such that it holds that By continuity, the same is true for all planes π in a neighbourhood ofπ x . In particular, since by assumption I (F ) = 0, for every x ∈ E there is an -dimensional plane π x such that Since we assume |μ|(F ) > 0, by standard measure-theoretic arguments (see the proof of [DR16, Theorem 1.1] for details), we can find a point x 0 ∈ F , an -dimensional plane π 0 , and a sequence of radii r j ↓ 0 with the following properties: (b1) λ := dμ d|μ| (x 0 ) exists, belongs to S m−1 , and satisfies (b3) for the following convergence holds: for some σ ∈ M + (B 2 ) with σ B 1/2 = 0. Here, T x0,rj (x) := x − x 0 r j .

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After a rotation we may assume that π 0 = R × {0}. We shall use the coordinates (y, z) ∈ R × R d− and we will denote by p the orthogonal projection onto R . Note that where A k is the kth-order homogeneous part of A, i.e., and R j contains all derivatives of μ j of order at most k − 1. Thus, Note that B is a homogeneous constant-coefficient linear differential operator such that for any ψ ∈ C ∞ (R ), We consider the (localized) sequence of measures where χ(y, z) =χ(z) for some cut-off functionχ ∈ C ∞ c (B d− 1 ; [0, 1]) satisfying χ ≡ 1 on B d− 1/2 . Our goal is to apply Lemma 2.1 to the sequence {ν j } ⊂ M(B 1 ; R m ), from where we will reach a contradiction. We must first check that {ν j } satisfies the assumptions of Lemma 2.1. Since Consequently, assumption (a3) in Lemma 2.1 is then satisfied for {ν j }.
Hence, by (2.8), we can assume that R ⊂ {θ * (|μ|) < +∞}. In particular, by [Mat95, Theorem 6.9] again, H R is σ-finite and the Radon-Nikodým theorem implies μ R = f H R with f ∈ L 1 (R, H ; R m ) such that |f | > 0 (H R)-almost everywhere. A standard blow-up argument then gives (2.9) and (2.10). Choosing a point such that the conclusion of (2.10) holds true and blowing up around that point, one deduces that the measureμ is A k -free, where A k is the k-homogeneous part of A. Since H (T x0 R) is a tempered distribution, by taking the Fourier transform of the equation A kμ = 0, we obtain which implies that A k (ξ)λ(x 0 ) = 0 for all ξ ∈ (T x0 R) ⊥ . This concludes the proof.