The structure ofA-free measures revisited

Abstract: We re ne a recent result on the structure of measures satisfying a linear partial di erential equation Aμ = σ, μ, σ are Radon measures, considering the measure μ(x) = g(x)dx + μus(x̃)(μs(x̄) + dx̄) where x = (x̃, x̄) ∈ Rk × Rd−k , μus is a uniformly singular measure in x̃0 and the measure μs is a singular measure. We proved that for μus-a.e. x̃0 the range of the Radon-Nykodim derivative f̃ (x̃0) = dμus d|μus| (x̃0) is in the set ∩ξ̃∈P̃KerA P̃(ξ ) and, if μs is di erent to zero, for μs-a.e. x̄0 the range of the Radon-Nykodim derivative f̄ (x̄0) = dμs d|μs| (x̄0) is in the set ∪ξ̄∈P̄KerA P̄(ξ ) where P̃ × P̄ = P is a manifold determined by the main symbol AP = AP̃ · AP̄ of the operatorA.


Introduction
In the paper, we consider a nite Radon measure µ = (µ , . . . , µm) de ned on R d satisfying the system of partial di erential equations Aµ The sum given above is taken over all terms from (1.2) whose order of derivative α is not dominated by any other multi-index from I j . As usual, ξ α = ξ α . . . ξ α d d for α = (α , . . . , α d ), and |α| = α + · · · + α d . Let us emphasize the fact that the equation (1.1) includes the case Aµ = σ, (1.4) where σ ∈ M(R d , R n ). Namely, regarding the equation (1.1) we may consider the measureμ = (µ, σ) ∈ M(R d , R m+n ) and the equationÃμ = (where 0'th-order term was added toÃ) which is equivalent to (1.4). We are interested in the range of the Radon-Nikodym derivativesf (x) = dµus d|µus| (x) andf (x) = dµs d|µs| (x) where the measures µus and µs are the parts of the measure (1.5) satisfying (1.1). The measure µs is a singular measure while µus is uniformly singular measure. Roughly speaking, we require that µs is singular with respect to every of the variables. For instance, such a condition is not ful lled by the measure µ = δ(x − x )dx since it is not singular with respect to x . However, if we introduce the change of variables z = x − x and z = x , then µ becomes δ(z )dz and we have the combination of the uniformly singular measure and a regular measure.
For instance, it is clear that the measure δ(x )δ(x ) satis es the latter condition with Eε = {( , )} and arbitrary α(ε) and β(ε) satisfying α(ε) ε → and ε β(ε) → . In general, a measure supported on the set whose Hausdorf dimension is less than n − is a candidate for the uniformly singular measure (see below for further explanations). It is not di cult to see that δ(x )dx is a singular measure on R , but not uniformly singular.
If we put k = i.e. m = d (implying that we do not have uniformly singular part: µ = g(x)dx + µs) then the problem is solved elegantly in [3] con rming the conjecture from [1] that for the k-th order operator A, the functionf (x) = dµs d|µs| (x) must take values in the wave cone Λ A = ∪ |ξ |= KerA k (ξ ) where A k (ξ ) is the sum of all symbols of order k (see [3] for details and thorough information concerning history and applications of this issue; in particular in the calculus of variations and geometric measure theory).
In the case when µus is nontrivial, we are able to prove a stronger result as announced in the abstract and formally introduced in Theorem 2. However, in the latter case, we have the measure of very special form which actually separates variables. This shortens space of measures that t into our considerations, but the space is far from trivial. For instance, consider the singular measure µ = δ(x − x )dx . This measure is not uniformly singular and it satis es the equation ∂x µ + ∂x µ = and after introducing the change of variables x − x = y and x = y we reduce the measure µ on the form (1.5). Also, the (n − )-dimensional Hausdor measure that can be locally represented in the from δ(x − g(x , . . . , x d ))dx . . . dx d and, after the change of variables z = x − g(x , . . . , x d ), x j = z j , j = , . . . , d, it gets the form (1.5). Moreover, if we augment (1.1) with initial conditions involving an uniformly singular measure, then, at least in the case of rst order scalar equations, the solution will contain the uniformly singular measure as well (since the solution is given along characteristics).
To continue, we assume that all the principal symbols A j , j = , . . . , n, can be represented in the form For instance, the matrix A j can be of the form ξ ξ whereÃ j (ξ ) = ξ andĀ j (ξ ) = ξ . The operator determined by such a matrix A is actually a hyperbolic operator. The latter restriction essentially means that we can separate variablesx andx while for the dual variables ξ , this means that we rst move in the direction ofξ determined by the matrixĀ j , and then in the directioñ ξ determined by the matrixÃ j .

Theorem 2.
Let µ be a solution to (1.1) of the form (1.5). Then, for |µus|-almost everyx ∈ R k and |µ s |-almost everyx ∈ R d−k there existsξ ∈P such that: If for all j = , . . . , n, the manifoldsP j would be the same, sayP, and we have the same set of dominating multi-indices I = I j , j = , . . . , n, then we could rewrite (1.9) in the form for appropriate matrices Aα, α ∈ I . If we do not have the uniformly singular part andP is the sphere in R d , then (1.10) is actually the statement of the main result from [3]. In their elegant proof, the main tool was the concept of tangent measures in the sense of [7]. We will pursue this approach here as well but with slightly more re ned arguments which take into account properties of the measure (splitting on singular and uniformly singular part) as well as properties of the operator A itself (principal symbols do not have to be de ned on a sphere). We will dedicate the last section to the proof of the theorem. In the next section, we shall prove it in the case of rst order constant coe cient operators and the scalar measure which captures all the elements of the general situation. The proof is based on the blow up method [8] (which naturally leads us to the tangent measures) and appropriate usage of Fourier multiplier operators (as in deriving appropriate defect functionals [2,5]).
Let us recall that the Fourier multiplier operator T ψ with the symbol ψ is de ned via the Fourier and inverse Fourier transform where the Fourier and the inverse Fourier transforms are given by For properties of the Fourier multiplier operators one can consult [4].
As for the tangent measure, we shall use the property of any locally nite measure [7, Lemma 2.4] and to recall the following theorem (see also [7,Theorem 2.5]) representing a special case of the tangent measures.
The paper is organized as follows. In the next section, we shall prove the theorem in the case µ = µus(x)dx which contains arguments concerning uniformly singular measures which are new with respect to the ones from [3]. In the last section, we prove the result in the full generality.

Proof of Theorem 2 in the case of the hyperbolic constant coe cients operator
Here, we shall prove Theorem 2 when the scalar nite Radon measure µ ∈ M(R d ) of the form µ(x) = µus(x)dx satis es the equation where aα are constants and σ is also nite scalar Radon measure. The proof is essentially the same for the general operator of the form given in (1.1), but it is a bit less technical for (2.11). The proof in full generality is given in the next section. Before we start, letf (x ) be the Radon-Nykodim derivative of µus with respect to |µus|: We x a convolution kernel ρ : R → R which is a smooth, compactly supported function of total mass one and convolve (2.11) by Then, we take an arbitrary φ ∈ C c (R k ) and φ ∈ C c (R d−k ) and test the convolved equation on the product of such functions. We get (below, we denote µ ε us = µus * ρ ε and φ(ỹ), µus(ỹ) (2.12) We now xx ∈ R k and take φ(x −x ε ) instead of φ in (2.12). We get (below, w = (w,w)): We now introduce in the rst integral above the change of variablesx =x + εw and multiply the entire expression by ε m . We get (we denotex = (x , . . . , x k )) andw = (w , . . . , w k )): (2.14) We consider separately terms involving the measure µus: We rewrite (2.15) in the form (for α and β given in the uniform singularity de nition) (2.16) Now, according to the assumptions for the uniformly singular measures (see De nition 1) and the fact that ρ is compactly supported: β(ε)) and, obviously we get after dividing (2.16) by |µus|(B(x , α(ε)) and letting ε → in (2.16) (for |µus|-a.e.x ∈ R d ): is the Fourier multiplier operator with the symbol ψ(ξ ) |ξ | m , we nd after taking the Plancherel theorem into account: From here, we conclude after letting ε → in (2.14) with φ given by (2.18) (2.20) From here, due to arbitrariness of ψ, we conclude that (1.10) holds (without the ∪ sign since we do not have the singular part, but only the uniformly singular part of the measure).

Proof of Theorem 2; general case
In this section, we consider equations (1.2) under the assumption µ(x) = µus(x)µs(x). We have omitted the terms g(x)dx and µus(x)dx appearing in (1.5) since for the purely Lebesgue part there is nothing to prove and the term µus(x)dx is handled as the µ(x) = µus(x)µs(x) by replacing µs(x) by dx. We shall rewrite in the form: whereĨ j andĪ are set of indexes corresponding to the principal symbol of the operator A.
We shall prove Theorem 2 by following the steps from the previous section together with the approach taken in [3] and we refer the reader there for clari cations.
We start by xing j in (1.2) and the convolution kernel ρ : R → R which is smooth, compactly supported with total mass one. We then denote We then apply a test function φ ∈ C ∞ c (R k ) on (3.22) to get (3.23) Now, we x z = (z,z) ∈ R d and take in (3.23) instead of φ. At the same time, for the variablex, we x z = (z,z) ∈ R d we introduce the changē x = (z k+ + ε β k+ w k+ , . . . , z d + ε β d w d ) =z + εβw in (3.23) and apply the test function φ (w). Multiplying the obtained expression by ε and taking into account (1.8), we conclude (3.24) Next, we introduce the change of variablesx = (z + ε β j w , . . . , z k + ε β j k w k ) in the rst term on the left-hand side of (3.24). We get where α(ε) is given in the de nition of the uniform singularity and let ε → . Taking into account the uniform singularity assumptions as in (2.16) and Theorem 3, we get where (along a subsequence; see Theorem 3) We now take where Tm is the Fourier multiplier operator with the symbol m. After inserting this in (3.26) and applying the Plancherel theorem with respect tow, we obtain: If we apply here the Plancherel theorem with respect tow, we get m r= α∈Ĩ j , β∈Ī From here, we conclude that iff (z)f (z) = (f (z)f (z)), . . . ,fm(z)fm(z)) does not satisfy conditions of Theorem 2, we conclude that suppνr = { } which in turn implies that νr is the Lebesgue measure. By repeating the procedure from [3] where we replace the sphere by the manifoldP, we conclude that the convergence from Theorem 3 used to get (3.26) is not only weak but also strong which contradicts the fact that µs is singular. Let us brie y recall the arguments from [3]. First, since ψ and φ in (3.29) are arbitrary, we know that it holds for every (ξ ,ξ ) ∈P j × R d−k , j = , . . . , n: m r= α∈Ĩ j , β∈Ī (− ) |α|+|β| a jr αβ (z)( π iξ ) α (iξ ) βf r(z)fr(z)ν(ξ ) = . (3.30) Due to linearity, with the expense of the small right-hand side, the same holds when we replace ν by ν − ν ε for ν ε given by (3.27). In the matrix notation and after collecting theξ -terms with the coe cients a αβ this means A(ξ )f(z)(ν(ξ ) −ν ε (ξ )) = low order terms.
This implies that ν ε strongly converges toward ν which is impossible since ν is the Lebesgue measure and ν ε are not. Thus, we conclude that for someξ it holds A(ξ )f(z) = which we wanted to prove.