TRANSPORT EQUATIONS WITH PARTIALLY BV VELOCITIES

Nicolas Lerner

Université de Rennes 1

September 22, 2004

Abstract. We prove the uniqueness of weak solutions for the Cauchy problem for a class of transport equations whose velocities are partially with bounded variation. Our result deals with the initial value problem ∂_{t}u + Xu = f, u_{|_{t=0}} = g, where X is the vector field

a_{1}(x_{1}) · ∂_{x_{1}} + a_{2}(x_{1}, x_{2}) · ∂_{x_{2}}, a_{1} ∈ BV (ℝ_{x_{1}}^{N_{1}}), a_{2} ∈ L_{x_{1}}^{1}(BV (ℝ_{x_{2}}^{N_{2}})),

with a boundedness condition on the divergence of each vector field a_{1}, a_{2}. This model was studied in the paper [LL] with a W^{1,1} regularity assumption replacing our BV hypothesis. This settles partly a question raised in the paper [Am]. We examine the details of the argument of [Am] and we combine some consequences of the Alberti rank-one structure theorem for BV vector fields with a regularization procedure. Our regularization kernel is not restricted to be a convolution and is introduced as an unknown function. Our method amounts to commute a pseudo-differential operator with a BV function.

Table of contents

1. Introduction. Historical account. The renormalization property. Sketch of the paper.

2. Statement of the results. Theorem 2.1: renormalization property for partially BV vector fields. Theorem 2.2: local uniqueness result. Theorem 2.3: global uniqueness result for transport equations.

3. Following Ambrosio’s argument with some modifications. Theorem 3.1: the renormalization property for BV vector fields. Remarks 3.2: removability of closed sets with H^{N−1} measure zero. Theorem 3.3: Alberti’s rank-one theorem.

4. Sum of Leibnizian vector fields. Theorem 4.1: renormalization property for partially BV vector fields. Lemma 4.2: another commutation lemma. Lemma 4.3: on the disintegration of the measure ∂a_{2}/∂x_{2}.

5. Appendix. Some technical lemmas. Index of notations.

2000 Mathematics Subject Classification 35F05, 34A12, 26A45. Key words and phrases. Vector fields, Transport equation, Weak solutions, BV.

Typeset by AMS-TEX

1. Introduction

In this article, we want to study some transport equations whose velocities are partially with bounded variation. More precisely, we intend to prove the uniqueness of weak solutions u of

∂_{t}u + Xu = 0, u_{|_{t=0}} = 0,

for vector fields X of the following type,

(1.1) \{ \begin{matrix}X = a_{1}(x_{1}) · ∂_{x_{1}} + a_{2}(x_{1}, x_{2}) · ∂_{x_{2}}, a_{1} ∈ BV (ℝ_{x_{1}}^{N_{1}}), a_{2} ∈ L_{x_{1}}^{1}(BV (ℝ_{x_{2}}^{N_{2}})), \\ div_{1} a_{1} ∈ L^{∞}(ℝ^{N_{1}}), div_{2} a_{2} ∈ L^{∞}(ℝ^{N_{1}+N_{2}}). \end{matrix}

Note that the BV vector field a_{1} depends only on the x_{1}-variables, but that the vector field a_{2} is only L^{1} with respect to the x_{1}-variables (and BV with respect to x_{2}). Note also that our condition on the divergence is stronger that div X ∈ L^{∞} since we want to control both divergences of the vector fields a_{1}, a_{2}. This type of question is tackled in a recent paper by C. Le Bris and P.L.Lions ([LL]), in which they examine vector fields of type (1.1), where the BV regularity is replaced by a W^{1,1} assumption.

A short historical account of the problem. Let us recall briefly a part of the recent history of this problem. In 1989, R.DiPerna and P.L.Lions proved in [DL] that the W^{1,1} regularity of a vector field, (along with a condition of boundedness on the divergence and a global condition) is enough to ensure the uniqueness of weak solutions. In 1998, P.L.Lions introduced in [Li] the so-called piecewise W^{1,1} class and extended the results of [DL] for this type of regularity. In 2001, F.Bouchut studied in [Bo] some cases of BV regularity corresponding essentially to W^{1,1} singularities occurring on hyperplanes. The paper [CL2] introduced the invariantly defined class conormal BV, for which the authors prove the uniqueness of weak solutions; moreover, their definition is simplified by the remark that closed sets whose (N −1)-Hausdorff measure is zero are unimportant for locally bounded vector fields. Finally in 2003, L.Ambrosio fully proved in [Am] the conjecture formulated in [DL] that BV vector fields (with bounded divergence) do have a flow. The main new ingredient brought forward by the article [Am] is a deep structure result on BV -vector-valued functions due to G.Alberti ([Al]). Although the full strength of the Alberti theorem is not needed as noted in the remark 3.7 of [Am] and also below in our Remark 3.4, it is nevertheless a very helpful tool for our investigation. It should also be noted that the classical counterexample of M.Aizenman ([Ai]), the recent counterexamples of N.Depauw ([De]) and of F.Colombini, T.Luo, J.Rauch ([CLR]) indicate that the BV regularity is close to optimality for the uniqueness property.

The renormalization property. Following the method introduced in [DL], the main tool for the proof of all these uniqueness results for vector fields X is a commutation lemma devised to ensure that a (bounded) solution u of the equation Xu = 0 should be also such that X(u^{2}) = 0 and more generally should satisfy the renormalization property

(1.2)  X(β(u)) = β′(u)Xu

for any C^{1} function β (to get uniqueness, it is enough to prove the Leibniz formula X(u^{2}) = 2uX(u)). The property (1.2) could fail even if both sides of the equality make sense, as shown by the counterexample constructed in [De]. In that paper, N.Depauw shows that there exists a bounded measurable vector field a ∈ L^{∞}(ℝ_{t} × ℝ_{x}^{2}; ℝ^{2}) with null divergence and a bounded measurable function u ∈ L^{∞}(ℝ_{t} × ℝ_{x}^{2}), supported in {t ≥ 0} such that

∂_{t}u + ∂_{x} · (au) = 0, u^{2} = 1_{ℝ_{+}}(t).

For that particular vector field X_{D} = ∂_{t} +a(t, x)·∂_{x} and that function u, we have X_{D}(u^{2}) = δ(t) and 2uX_{D}(u) = 0, violating (1.2), in spite of the fact that X_{D}(u^{2}) and 2uX_{D}(u) are both meaningful. A somewhat equivalent approach to checking the property (1.2) is the fact that to get uniqueness for a vector field X, one should be able to prove that it behaves like an ordinary vector field with respect to the Leibniz formula, namely, assuming for instance that X is in L_{loc}^{1} with L_{loc}^{1} divergence, and u, v are L_{loc}^{∞} functions such that X(u), X(v) are in L_{loc}^{1}, we have to check

X(uv) = uX(v) + vX(u).

Checking (1.2) can be reduced to a commutation problem. In fact, assuming that X is a L_{loc}^{1} vector field with null divergence and u is a L_{loc}^{∞} function such that Xu = 0, checking X(u^{2}) = 0 amounts to examine the bracket of duality (φ is a test function in C_{c}^{1})

⟨Xu^{2}, φ⟩ = − ∫ u^{2}(x)(Xφ)(x)dx

since the divergence of X is zero. Now assuming that R_{ϵ}u is C^{1}, bounded and converging pointwise a.e. to u when ϵ goes to zero, we get

⟨Xu^{2}, φ⟩ = − lim_{ϵ→0} ∫ u(x) (R_{ϵ}u)(x) (Xφ)(x)dx = ⟨X(uR_{ϵ}u), φ⟩.

Now since R_{ϵ}u is C^{1}, one can use Leibniz formula X(uR_{ϵ}u) = (Xu)(R_{ϵ}u) + uX(R_{ϵ}u) and since Xu = 0, we get

⟨Xu^{2}, φ⟩ = lim_{ϵ→0}⟨uX(R_{ϵ}u), φ⟩ = lim_{ϵ→0} ∫ φ(x) u(x) (XR_{ϵ}u)(x) dx.

Using again that Xu = 0, we obtain

⟨Xu^{2}, φ⟩ = lim_{ϵ→0} ∫ (φu)(x) ([X, R_{ϵ}]u)(x) dx.

Since the function φu is bounded with compact support, to obtain Xu^{2} = 0 is thus reduced to proving that the commutator [X, R_{ϵ}]u goes to zero in L_{loc}^{1}.

· If the vector field X is W_{loc}^{1,1}, one can take the regularization operator R_{ϵ} as any convolution by a C_{c}^{1} function ρ(x/ϵ)ϵ^{−N} (with integral 1).

· If the vector field has some singularities on affine submanifolds, for instance on {x_{1} = 0}, this translation invariance property leaves open the choice of a convolution operator, but with a structure respecting the geometry such as

ρ(\frac{x_{1}}{ϵ_{1}} , \frac{x_{2}}{ϵ_{2}} )ϵ_{1}^{−N_{1}}ϵ_{2}^{−N_{2}}, 0 < ϵ_{1} ≪ ϵ_{2}.

· If a simple jump for X occurs on a curved hypersurface, no convolution operator will do the job, which is quite natural after all, since no translation invariance is preserved. In fact one has to look for a regularizing kernel of a more general form

(1.3)  (R_{ϵ}u)(x) = ∫ ρ(x, ϵ^{−1}(x − y))ϵ^{−N}u(y)dy.

More pedantically , one could say that we intend to commute a pseudo-differential operator of order 1 with a BV function. Our approach in the present paper will be to take the kernel ρ in (1.3) as an unknown function, and we shall see that the commutation property of [X, R_{ϵ}] can essentially be expressed as some first-order PDE on that kernel ρ.

Note also that going from the property (1.2) to the uniqueness is now rather standard a fact, since, taking for instance β(u) = u^{2}, we produce non-negative solutions, whose uniqueness is easy to establish (see e.g. Lemma 3.1 in [CL2], Lemma 2.2 in [LL]).

A sketch of our paper. Our goal here is in fact twofold. First of all, we wish to revisit the Ambrosio’s argument of [Am] by following our approach of commuting our vector field with a regularizing operator of type (1.3), checking which constraints occur on the unknown kernel ρ (this is done in our section 3). However, our method will follow closely the arguments of [Am] and we shall try to be as explicit as possible in our construction. In particular, if X is our vector field, the canonical decomposition of its derivative can be written as

DX = DX^{ac} + DX^{s}, |DX^{ac}| ≪ m, DX^{s} ⊥ m (m is the Lebesgue measure on ℝ^{N}).

Using the polar decomposition of the singular part, we get DX^{s} = M|DX^{s}|, where M(x) is a N × N matrix. An “ideal” kernel ρ(x, z) to be used in (1.3) should satisfy

(1.4)  \frac{∂ρ}{∂z}(x, z)M(x)z = 0

which is a PDE in the variable z, which should be satisfied |DX^{s}|-a.e in the variable x. Also the support in the z-variable should be compact. These notions have to be clarified, at least for questions of regularity, and it is done in details in section 3. However it is interesting to note that if M is an antisymmetric matrix, one can choose ρ as a convolution kernel (i.e. independent of the variable x) depending only on |z|^{2}, and the equation (1.4) becomes tzMz = 0 which is satisfied since M is antisymmetric: so we recover also the remark made in [CP] that we could also generalize. Anyhow, to get compactly supported solutions (in z) for the equation (1.4) requires some spectral condition on the matrix M, and at least spectrum M ⊂ iℝ (naturally that condition is satisfied by an antisymmetric matrix). At any rate, the spectral structure of the matrix M(x) is playing a key role and we shall use some consequences of Alberti’s rank-one theorem [Al]. More details are given in our Remark 3.4 below.

Our second aim is to use that constructive approach to tackle vector fields of type (1.1) and to obtain a generalization to BV regularity of the results of [LL]. This gives also a partial answer to the remark 3.8(3) of [Am]. In our section 4, we concentrate our attention on the proof of the renormalization property for vector fields of type (1.1). We shall use a regularizing kernel in (1.3) of type ρ(x_{1}, x_{2}, z_{2}), which means in particular that we regularize only in the x_{2}-variable but in a way depending on the point (x_{1}, x_{2}). We have to deal with another commutation problem between the vector field a_{1}(x_{1})∂_{x_{1}} in (1.1) and our regularization operator. Moreover, we follow the construction in section 3 with parameters x_{1} and we use the disintegration of the measure ∂a_{2}/∂x_{2}. Also in our Remark 5.5, we give some invariance properties of the matrix M under C^{1,1} diffeomorphism.

2. Statement of the results

We concentrate our attention on the so-called renormalization property for the vector field

(2.1) X = \underbrace{a_{1}(x_{1})∂_{x_{1}}}_{X_{1}} + \underbrace{a_{2}(x_{1}, x_{2})∂_{x_{2}}}_{X_{2}}, a_{1} ∈ BV_{loc}(ℝ^{N_{1}}), a_{2} ∈ L_{loc}^{1}(ℝ^{N_{1}}; BV_{loc}(ℝ^{N_{2}}))

satisfying also

(2.2)  div X_{1} ∈ L_{loc}^{1}(ℝ^{N_{1}}), div X_{2} ∈ L_{loc}^{1}(ℝ^{N_{1}+N_{2}}).

Theorem 2.1. Let N, N_{1}, N_{2} be non-negative integers such that N = N_{1} + N_{2}. Let X be a vector field on ℝ^{N} satisfying (2.1-2) and let w be a L_{loc}^{∞}(ℝ^{N}) function such that Xw ∈ L_{loc}^{1}(ℝ^{N}). Then, with α ∈ C^{1}(ℝ; ℝ),

(2.3)  X(α(w)) = α′(w)Xw.

The proof of this theorem is given in section 4. Theorem 2.1 above along with Lemma 3.1 in [CL1] imply readily the following local uniqueness result.

Theorem 2.2. Let X be a vector field satisfying the assumptions of Theorem 2.1 such that X ∈ L_{loc}^{∞}, div X ∈ L_{loc}^{∞}. Let Ω be an open subset of ℝ^{N} and S be a C^{1} hypersurface of Ω such that X is transverse to S. Let c and w be L_{loc}^{∞}(Ω) functions such that Xw = cw on Ω, and supp w ⊂ S_{+} (here S_{+} is the “half-space” above S). Then w = 0 on a neighborhood of S.

Theorem 2.1 is also the key step to get the uniqueness of bounded solutions for transport equations of type ∂_{t} + X. Let T > 0 be given and X be a vector field as above. Let c be a L_{loc}^{1}(ℝ^{N+1}) function and w be a L_{loc}^{∞}(ℝ^{N+1}) function. Let us recall that the following equation

(2.4)  \{ \begin{matrix}∂_{t}w + Xw = cw & on (0, T) × ℝ^{N}, \\ w(0, x) = 0 & on ℝ^{N}, \end{matrix}

holds weakly means that

(2.4)′  ∀φ ∈ C_{c}^{∞}([0, T) × ℝ^{N}), ∫_{0}^{T} ∫_{ℝ^{N}} w(∂_{t}φ + Xφ + φ div X + cφ)dxdt = 0.

The following theorem is a consequence of Theorem 2.1 above and of Lemma 3.3 in [CL1].

Theorem 2.3. Let X be a vector field satisfying the assumptions of Theorem 2.1 such that

(2.5)  \frac{a_{1}(x_{1})}{1 + |x_{1}|} ∈ L^{1}(ℝ^{N_{1}}), \frac{a_{2}(x_{1}, x_{2})}{1 + |x_{2}|} ∈ L^{1}(ℝ^{N}),
(2.6)  div X_{1} ∈ L^{∞}(ℝ^{N_{1}}), div X_{2} ∈ L^{∞}(ℝ^{N}).

Let T be a positive number. Let c(t, x) and w(t, x) be L^{∞}((0, T) × ℝ^{N}) functions such that (2.4) holds weakly (i.e. (2.4)′). Then w = 0 on (0, T) × ℝ^{N}.

We refer the reader to the paper [LL] for the statements of similar uniqueness theorems that we are able to generalize by replacing in (H1) and (H4) of [LL] the W^{1,1} regularity by the BV regularity.

In our Remark 5.6, we point out that an invariant formulation can be found to express an assumption such as (2.1).

3. Following Ambrosio’s argument with some modifications

In this section, we follow closely Ambrosio’s argument in [Am], based on the Alberti rank-one theorem (Theorem 2.3 in [Am]). We give a few modifications and we take the regularizing kernel as an unknown function.

Theorem 3.1. Let Ω be an open subset of ℝ^{N}, X be a real BV_{loc} vector field on Ω such that div X ∈ L_{loc}^{1} and w ∈ L_{loc}^{∞} such that Xw ∈ L_{loc}^{1}. Then, with α ∈ C^{1}(ℝ; ℝ),

(3.1)  X(α(w)) = α′(w)Xw.

Remarks 3.2. Note that proving the uniqueness of weak solutions requires only to get X(w^{2}) = 0 whenever Xw = 0 for w ∈ L_{loc}^{∞} (see e.g. lemma 3.1 in [CL1]). Also to obtain (3.1) on the open set Ω, it is enough to prove it on an open subset Ω_{0} such that

H^{N−1}(Ω\Ω_{0}) = 0,

provided that the vector field X is also locally bounded as well as its divergence (here H^{N−1} stands for the N − 1 Hausdorff measure). For a L_{loc}^{1} vector field X = ∑ a_{j}∂_{j} with an L_{loc}^{1} divergence and a L_{loc}^{∞} function w, the distribution Xw is defined as

(3.2)  Xw = ∑_{1≤j≤n} ∂_{j}(a_{j}w) − w div X.

Proof of Theorem 3.1. Step 1: Preliminaries. Proving (3.1) amounts to checking that for any test function φ ∈ C_{c}^{1}(Ω),

∫ α(w)(Xφ + φ div X)dm + ∫ α′(w)(Xw)φdm = 0,

where dm is the Lebesgue measure on ℝ^{N}. Let χ be a C_{c}^{1}(Ω) function identically equal to 1 on the support of φ. Then for x ∈ supp φ, α(w(x)) = α(χ(x)w(x)), so that we need only to check

(3.3)  − ∫ α(χw)(Xφ + φ div X)dm = ∫ α′(χw)(X(χw))φdm.

Note that, from the assumptions of Theorem 3.1,

v = χw ∈ L_{comp}^{∞}, Xv = wXχ + χXw ∈ L_{comp}^{1},

and we can use a mollifier with the properties of Lemma 5.2 of our appendix to write

− ∫ α(v)(Xφ + φ div X)dm = − lim_{ϵ→0} ∫ α(R_{ϵ}v)(Xφ + φ div X)dm

= lim_{ϵ→0} {∫ α′(R_{ϵ}v)(X(R_{ϵ}v) − R_{ϵ}(Xv))φdm + ∫ α′(R_{ϵ}v)R_{ϵ}(Xv)φdm} .

Lemma 5.2 shows that R_{ϵ}(Xv) converges to Xv in L^{1} and since R_{ϵ}v is bounded independently of ϵ, and converges almost everywhere toward v, proving (3.3) amounts to proving

lim_{ϵ→0} ∫ φα′(R_{ϵ}v)[X, R_{ϵ}]vdm = 0.

Using Lemma 5.3 in the appendix, we get

∫ φα′(R_{ϵ}v)[X, R_{ϵ}]vdm = ∫ φα′(R_{ϵ}v)T_{ϵ,ρ}vdm + ∫ \overbrace{α′(R_{ϵ}v)}^{bounded} \overbrace{φN_{ϵ}v}^{{→0 \atop in L^{1}}} dm

so that we need only to prove

(3.4)  lim_{ϵ→0} ∫ φα′(R_{ϵ}v)T_{ϵ,ρ}vdm = 0,

that is, using (5.3)

lim_{ϵ→0} ∫∫ ∂_{2}ρ(x, z)ϵ^{−1}(X(x) − X(x − ϵz))(v(x − ϵz) − v(x))φ(x)α′((R_{ϵ}v)(x))dxdz = 0.

Now if (v_{k})_{k∈ℕ} is a sequence of C_{c}^{1} functions converging almost everywhere to the L_{comp}^{∞} function v so that ∥v_{k}∥_{L^{∞}} ≤ ∥v∥_{L^{∞}}, we set

ω(ϵ, k) = ∫∫ ∂_{2}ρ(x, z)ϵ^{−1}(X(x) − X(x − ϵz))(v_{k}(x − ϵz) − v_{k}(x))φ(x)α′((R_{ϵ}v)(x))dxdz

= ∫∫∫_{0}^{1} ∂_{2}ρ(x, z)DX(x − ϵθz)z(v_{k}(x − ϵz) − v_{k}(x))φ(x)α′((R_{ϵ}v)(x))dxdzdθ, (3.5)

which makes sense as a bracket of duality since the distribution derivative DX is of order ≤ 1. We have to prove

(3.6)  lim_{ϵ→0}( lim_{k→∞} ω(ϵ, k)) = 0.

Step 2: Getting rid of the absolutely continuous part. So far our discussion required only that X and div X should belong to L_{loc}^{1}(Ω). In fact our assumption on X in Theorem 3.1 makes DX a Radon measure. We consider now the canonical decomposition of that measure DX

DX = DX^{a} + DX^{s}, |DX^{a}| ≪ m, |DX^{s}| ⊥ m,

where m is the Lebesgue measure on ℝ^{N}. We note that defining

ω_{0}(ϵ, k) = ∫∫∫_{0}^{1} ∂_{2}ρ(x, z)DX^{a}(x − ϵθz)z(v_{k}(x − ϵz) − v_{k}(x))φ(x)α′((R_{ϵ}v)(x))dxdzdθ,

we get with

(3.7)  C_{0} = sup_{|s|≤∥v∥_{L^{∞}}} |α′(s)|,

using that DX^{a} ∈ L^{1},

lim sup_{k→∞} |ω_{0}(ϵ, k)| ≤ C_{0} ∫∫∫_{0}^{1} |DX^{a}(x − ϵθz)||(τ_{ϵz}v − v)(x)||φ(x)|dx|z|ρ_{0}(z)dzdθ

and using Lemma 5.1, we get (3.6) for ω_{0}.

Step 3: Handling the singular part. We are left with the bracket of duality (that we write as an integral),

ω_{1}(ϵ, k) =

∫∫∫_{0}^{1}∂_{2}ρ(x+ϵθz, z)DX^{s}(x)z(τ_{ϵz−ϵθz}v_{k}−τ_{−ϵθz}v_{k})(x)τ_{−ϵθz}φ(x)α′((τ_{−ϵθz}R_{ϵ}v)(x))dxdzdθ,

so that, using the polar decomposition of the measure DX^{s} = M|DX^{s}| with the notation μ = |DX^{s}|, we get

ω_{1}(ϵ, k) = ∫∫∫_{0}^{1} ∂_{2}ρ(x + ϵθz, z)M(x)z(τ_{ϵz−ϵθz}v_{k} − τ_{−ϵθz}v_{k})(x)(τ_{−ϵθz}φ)(x)

α′((τ_{−ϵθz}R_{ϵ}v)(x))dμ(x)dzdθ.

Using now that sup_{k,x} |v_{k}(x)| ≤ ∥v∥_{L^{∞}} and that the measure μ is positive, we obtain

|ω_{1}(ϵ, k)| ≤ C_{0}2 ∥v∥_{L^{∞}} ∫∫∫_{0}^{1} |∂_{2}ρ(x + ϵθz, z)M(x)z||φ(x + ϵθz)|dμ(x)1_{supp ρ_{0}}(z)dzdθ.

Since ∂_{2}ρ and φ are continuous functions, |M(x)| ≤ 1, μ-a.e., the dominated convergence theorem for the measure dμdzdθ gives

(3.8)  lim sup_{ϵ→0}(sup_{k} |ω_{1}(ϵ, k)|) ≤ C_{0}2 ∥v∥_{L^{∞}} ∫∫ |∂_{2}ρ(x, z)M(x)z||φ(x)|dμ(x)dz.

We reach now the main point of the proof which amounts to choosing properly the mollifier ρ so that the z-integral of the dot product Mz · ∂_{2}ρ in (3.8) is small. Let us consider the function

(3.9)  ρ(x, z) = F_{0}(U(x)z)| det U(x)|

where, using the notation 𝕄_{N}(ℝ) for the N × N real matrices, and C_{b}^{1} for the C^{1} functions bounded as well as their first derivatives,

(3.10)  U ∈ C_{b}^{1}(ℝ^{N}; 𝕄_{N}(ℝ)), ^{t}U(x)U(x) ≥ Id,

and F_{0} ∈ C_{c}^{1}(ℝ^{N}; ℝ_{+}), ∫ F_{0}(ζ)dζ = 1. Note that the function ρ given by (3.9) satisfies the assumptions of Lemma 5.2; to check (5.1) we assume supp F_{0} ⊂ B(0, r_{0}) and we get, since |z| ≤ |U(x)z|,

sup_{x} ρ(x, z) ≤ sup |F_{0}|1_{B(0,r_{0})}(z) ∥det U∥_{L^{∞}} .

Similar estimates are true for sup_{x} |d_{x,z}ρ(x, z)|. We get

∫∫ |∂_{2}ρ(x, z)M(x)z||φ(x)|dμ(x)dz

= ∫∫ |F′_{0}(U(x)z)U(x)M(x)z|| det U(x)||φ(x)|dμ(x)dz

≤ ∫ ∥U(x)M(x)U(x)^{−1}∥|φ(x)|dμ(x)∫ |F′_{0}(ζ)||ζ|dζ,

so that we obtain from (3.8) that we need only to prove

(3.11)  0 = inf_{U satisfying (3.10)} ∫ ∥U(x)M(x)U(x)^{−1}∥|φ(x)|dμ(x).

It is possible to simplify further that condition by getting rid of the continuity properties of U, U′ required in (3.10). First of all, (3.11) is a consequence of

(3.12) 0 = inf_{V ∈C_{c}^{1}(ℝ^{N};𝕄_{N} (ℝ))} ∫ ∥(Id +^{t}V (x)V (x))M(x)(Id +^{t}V (x)V (x))^{−1}∥|φ(x)|dμ(x),

since the matrix U(x) = Id +^{t}V (x)V (x) satisfies (3.10) for V ∈ C_{c}^{1}(ℝ^{N}; 𝕄_{N}(ℝ)). We claim now that it is enough to obtain

(3.13)  0 = inf_{V ∈L^{∞}(|φ|dμ)} ∫ ∥(Id +^{t}V (x)V (x))M(x)(Id +^{t}V (x)V (x))^{−1}∥|φ(x)|dμ(x).

To prove that claim, we consider a matrix V ∈ L^{∞}(|φ|dμ) ; since |φ|dμ is a finite Radon measure on ℝ^{N}, can find a sequence (V_{l}) ∈ C_{c}^{0}(ℝ^{N}; 𝕄_{N}(ℝ)) converging to V in L^{1}(|φ|dμ) with

sup_{x} ∥V_{l}(x)∥ ≤ ∥V ∥_{L^{∞}(|φ|dμ)} .

Regularizing by a standard mollifier the matrices V_{l}, we may suppose that they are in C_{c}^{1}(ℝ^{N}; 𝕄_{N}(ℝ)) and, extracting a subsequence, we may also assume that they converge pointwise |φ|dμ-a.e. to V. We note that |φ|dμ-a.e.,

∥(Id +^{t}V_{l}(x)V_{l}(x))M(x)(Id +^{t}V_{l}(x)V_{l}(x))^{−1}∥ ≤ ∥ Id +^{t}V_{l}(x)V_{l}(x)∥ ≤ 1 + ∥V ∥_{L^{∞}(|φ|dμ)}^{2} , so that the Lebesgue dominated convergence theorem for the measure with finite mass |φ|dμ gives

(3.14) lim_{l→+∞} ∫ ∥(Id +^{t}V_{l}(x)V_{l}(x))M(x)(Id +^{t}V_{l}(x)V_{l}(x))^{−1}∥|φ(x)|dμ(x)

= ∫ ∥(Id +^{t}V (x)V (x))M(x)(Id +^{t}V (x)V (x))^{−1}∥|φ(x)|dμ(x).

Thus the infimum in the rhs of (3.13), a priori smaller than the rhs of (3.12) is actually the same, proving our claim. We are then reduced to proving (3.13). The key argument relies on

Theorem 3.3 (Alberti’s rank one theorem[Al]). Let Ω be an open subset of ℝ^{N}, a ∈ BV (Ω, ℝ^{N′}) and let Da = M|Da| be the polar decomposition of its distribution derivative. Then M(x) has rank one, i.e.

M(x) = ξ(x) ⊗ η(x), |D^{s}a| almost everywhere.

The product ξ ⊗ η is the linear map defined by ⟨ξ, T ⟩η and if a is a vector field on Ω (Ω ⊂ ℝ^{N}, N′ = N), the divergence of a is ⟨ξ, η⟩|Da| so that the absolute continuity of the divergence with respect to the Lebesgue measure amounts to the orthogonality of the unit vectors ξ, η, |D^{s}a| almost everywhere. We apply this theorem to the matrix-valued measure DX^{s} = M|DX^{s}|, μ = |DX^{s}|, and we use the notation M = ξ ⊗ η, μ-almost everywhere. We choose now the L^{∞}(μ) matrix V (x) = γ^{1/2}M(x), where γ ≥ 0, and we note that, from Lemma 5.4,

(3.15)  ∥(Id +^{t}V (x)V (x))M(x)(Id +^{t}V (x)V (x))^{−1}∥ ≤ (1 + γ)^{−1}.

Since γ is an arbitrary positive number and |φ|dμ is finite, we obtain (3.13), completing the proof of Theorem 3.1. □

Remark 3.4. Looking at our proof, it seems quite obvious that the full strength of Alberti’s theorem is not needed. For instance, in the paper [CL2], the key argument could be modified to rely on the fact that the matrix M is triangular with zeros on the diagonal. It is also pointed out in remark 3.7 of [Am] that a recent still unpublished proof of Alberti is using only the absolute continuity of the divergence with respect to the Lebesgue measure, that is Tr M = 0. As far as our proof is concerned, from the analysis in (3.8) , for fixed x, we need to have a compactly supported (in z) solution ρ(x, z) of the equation

(3.16)  \frac{∂ρ}{∂z}(x, z)M(x)z = 0.

The previous equation is simply given by a vector field in the z-variables, with coefficients depending linearly on z (and with parameters x),

∑_{1≤j≤N}( ∑_{1≤k≤N} M_{kj}(x)z_{j})\frac{∂ρ}{∂z_{j}} (x, z) = 0.

For this vector field, to have a compactly supported solution require the spectral condition,

(3.17)  spectrum(M(x)) ⊂ iℝ.

However, the equation (3.16) need not to be satisfied exactly, and (3.17) may certainly be relaxed.

Remark 3.5. If we follow our remark 5.5 below, we see that, although the matrix M(x) = (\frac{∂a_{j}}{∂x_{k}})_{1≤j,k≤N} does not carry any geometric meaning, its class modulo L_{loc}^{1} has actually some invariance properties, so that it is not hopeless to expect that a spectral condition of type (3.17) could be meaningful, even for a vector field ∑_{j} a_{j}∂_{j} more singular than BV.

4. Sum of Leibnizian vector fields

In this section, we prove that the renormalization property holds for the vector field

(4.1) X = \underbrace{a_{1}(x_{1})∂_{x_{1}}}_{X_{1}} + \underbrace{a_{2}(x_{1}, x_{2})∂_{x_{2}}}_{X_{2}}, a_{1} ∈ BV_{loc}(ℝ^{N_{1}}), a_{2} ∈ L_{loc}^{1}(ℝ^{N_{1}}; BV_{loc}(ℝ^{N_{2}}))

provided that

(4.2)  div X_{1} ∈ L_{loc}^{1}(ℝ^{N_{1}}), div X_{2} ∈ L_{loc}^{1}(ℝ^{N_{1}+N_{2}}).

In Remark 5.6 below, we point out that an invariant formulation of our statement can be given, using a codimension N_{1}-foliation of the reference open set.

Theorem 4.1. Let N, N_{1}, N_{2} be non-negative integers such that N = N_{1} + N_{2}. Let X be a vector field on ℝ^{N} satisfying (4.1-2) and w ∈ L_{loc}^{∞}(ℝ^{N}) such that Xw ∈ L_{loc}^{1}(ℝ^{N}). Then, with α ∈ C^{1}(ℝ; ℝ),

(4.3)  X(α(w)) = α′(w)Xw.

Proof. To simplify our argument, we shall only prove that, if w ∈ L^{∞}(ℝ^{N}) satisfies Xw = 0, then X(w^{2}) = 0. Also, we shall assume that both divergences in (4.2) are vanishing identically. Let us consider ρ ∈ C^{1}(ℝ_{x_{1}}^{N_{1}} × ℝ_{x_{2}}^{N_{2}} × ℝ_{z_{2}}^{N_{2}}; ℝ_{+}) such that

(4.4)  ∫_{ℝ^{N_{2}}} ρ(x_{1}, x_{2}, z_{2})dz_{2} = 1,
(4.5)  sup_{x_{1},x_{2}}(|ρ(x_{1}, x_{2}, z_{2})| + |d_{x_{1},x_{2},z_{2}}ρ(x_{1}, x_{2}, z_{2})|) = ρ_{0}(z_{2}) ∈ L_{comp}^{∞}.

We define also for ϵ > 0 the operator R_{ϵ} by

(4.6)  (R_{ϵ}u)(x_{1}, x_{2}) = ∫_{ℝ^{N_{2}}} ρ(x_{1}, x_{2}, ϵ^{−1}(x_{2} − y_{2}))ϵ^{−N_{2}}u(x_{1}, y_{2})dy_{2}.

We need now to commute R_{ϵ} with the vector field X_{1}.

Lemma 4.2. Let X_{1}, R_{ϵ}, w be as above. Then, X_{1}R_{ϵ}w and R_{ϵ}X_{1}w belong to L_{loc}^{1} and lim_{ϵ→0_{+}}[X_{1}, R_{ϵ}]w = 0 in L_{loc}^{1}.

Proof of the lemma. We have, since ρ is C^{1},

X_{1}R_{ϵ}w = \frac{∂}{∂x_{1}} · ∫ a_{1}(x_{1})ρ(x_{1}, x_{2}, ϵ^{−1}(x_{2} − y_{2}))ϵ^{−N_{2}}w(x_{1}, y_{2})dy_{2}

= ∫ a_{1}(x_{1}) · \frac{∂ρ}{∂x_{1}} (x_{1}, x_{2}, ϵ^{−1}(x_{2} − y_{2}))ϵ^{−N_{2}}w(x_{1}, y_{2})dy_{2}

+ ∫ ρ(x_{1}, x_{2}, ϵ^{−1}(x_{2} − y_{2}))ϵ^{−N_{2}} \frac{∂}{∂x_{1}} · (a_{1}(x_{1})w(x_{1}, y_{2}))dy_{2}

= ∫ a_{1}(x_{1}) · \frac{∂ρ}{∂x_{1}} (x_{1}, x_{2}, z_{2})(w(x_{1}, x_{2} − ϵz_{2}) − w(x_{1}, x_{2}))dz_{2}

+ a_{1}(x_{1}) · \underbrace{∫ \frac{∂ρ}{∂x_{1}} (x_{1}, x_{2}, z_{2})dz_{2}}_{=0 from (4.4)} w(x_{1}, x_{2}) + R_{ϵ}X_{1}w,

which entails [X_{1}, R_{ϵ}]w ∈ L_{loc}^{1} and from Lemma 5.1 that lim_{ϵ→0}[X_{1}, R_{ϵ}]w = 0 in L_{loc}^{1}. Moreover, we have, using that ρ is C^{1} and the equation Xw = 0,

R_{ϵ}X_{1}w = −R_{ϵ}X_{2}w = − ∫ ρ(x_{1}, x_{2}, ϵ^{−1}(x_{2} − y_{2}))ϵ^{−N_{2}}(∂_{2} · a_{2}w)(x_{1}, y_{2})dy_{2}

= − ∫ \frac{∂ρ}{∂z_{2}} (x_{1}, x_{2}, ϵ^{−1}(x_{2} − y_{2}))ϵ^{−N_{2}−1} · (a_{2}w)(x_{1}, y_{2})dy_{2}

which belongs to L_{loc}^{1}, completing the proof of the lemma. □

We have, using Theorem 3.1 and Lemma 4.2 (that is the fact that X_{1} is Leibnizian, R_{ϵ}w ∈ L_{loc}^{∞}, X_{1}R_{ϵ}w ∈ L_{loc}^{1})

X((R_{ϵ}w)^{2}) = X_{1}((R_{ϵ}w)^{2}) + X_{2}((R_{ϵ}w)^{2}) = 2(R_{ϵ}w)(X_{1}R_{ϵ}w) + X_{2}((R_{ϵ}w)^{2}).

We note also that the function R_{ϵ} is C^{1} with respect to x_{2}, so that the ordinary Leibniz formula can be used to get X_{2}((R_{ϵ}w)^{2}) = 2(R_{ϵ}w)(X_{2}R_{ϵ}w). We obtain,

X((R_{ϵ}w)^{2})= 2(R_{ϵ}w)([X_{1}, R_{ϵ}]w) + 2(R_{ϵ}w)(R_{ϵ}X_{1}w) + 2(R_{ϵ}w)([X_{2}, R_{ϵ}]w) + 2(R_{ϵ}w)R_{ϵ}X_{2}w
= 2(R_{ϵ}w)([X_{1}, R_{ϵ}]w) + 2(R_{ϵ}w)(R_{ϵ}Xw) + 2(R_{ϵ}w)([X_{2}, R_{ϵ}]w)

which gives, since Xw = 0,

(4.7)  X((R_{ϵ}w)^{2}) = 2(R_{ϵ}w)([X_{1}, R_{ϵ}]w) + 2(R_{ϵ}w)([X_{2}, R_{ϵ}]w).

Since the term [X_{1}, R_{ϵ}]w goes to zero in L_{loc}^{1} from the Lemma 4.2 and the function R_{ϵ}w is locally bounded from Lemma 5.2 we are left with the bracket [X_{2}, R_{ϵ}]w. We recall that, with φ ∈ C_{c}^{1}(ℝ^{N}),

⟨(X_{1} + X_{2})(w^{2}), φ⟩_{D^{′(1)},C_{c}^{1}} = − lim_{ϵ→0} ∫_{ℝ^{N}} (R_{ϵ}w)^{2}(Xφ)dm

= lim_{ϵ→0} ∫_{ℝ^{N}} X((R_{ϵ}w)^{2})φdm = 2 lim_{ϵ→0} ∫_{ℝ^{N}} (R_{ϵ}w)([X_{2}, R_{ϵ}]w)φdm.

We need only to prove that the last term above is 0. We define

ω̃(ϵ) = ∫_{ℝ^{N}} (R_{ϵ}w)([X_{2}, R_{ϵ}]w)φdm =

= ∫∫∫ (R_{ϵ}w)(x_{1}, x_{2})φ(x_{1}, x_{2})\frac{∂ρ}{∂z_{2}} (x_{1}, x_{2}, z_{2})ϵ^{−1}(X_{2}(x_{1}, x_{2}) − X_{2}(x_{1}, x_{2} − ϵz_{2}))

(w(x_{1}, x_{2} − ϵz_{2}) − w(x_{1}, x_{2}))dx_{2}dz_{2}dx_{1}.

Now, we consider a sequence of continuous functions w_{k} bounded by ∥w∥_{L^{∞}} converging a.e. to w and we define

ω̃(ϵ, k) = ∫∫∫ (R_{ϵ}w)(x_{1}, x_{2})φ(x_{1}, x_{2})\frac{∂ρ}{∂z_{2}} (x_{1}, x_{2}, z_{2})ϵ^{−1}(X_{2}(x_{1}, x_{2}) − X_{2}(x_{1}, x_{2} − ϵz_{2}))

(w_{k}(x_{1}, x_{2} − ϵz_{2}) − w_{k}(x_{1}, x_{2}))dx_{1}dx_{2}dz_{2}.

We need only to prove that

(4.8)  lim_{ϵ→0}( lim_{k→∞} ω̃(ϵ, k)) = 0.

We have, using an integral notation for the bracket of duality,

ω̃(ϵ, k) = ∫_{0}^{1}∫∫∫ (R_{ϵ}w)(x_{1}, x_{2})φ(x_{1}, x_{2})\frac{∂ρ}{∂z_{2}} (x_{1}, x_{2}, z_{2})\frac{∂a_{2}}{∂x_{2}} (x_{1}, x_{2} − ϵθz_{2})z_{2}

(w_{k}(x_{1}, x_{2} − ϵz_{2}) − w_{k}(x_{1}, x_{2}))dθdx_{1}dx_{2}dz_{2}

= ∫_{0}^{1}∫∫∫ (R_{ϵ}w)(x_{1}, x_{2} + ϵθz_{2})φ(x_{1}, x_{2} + ϵθz_{2})\frac{∂ρ}{∂z_{2}} (x_{1}, x_{2} + ϵθz_{2}, z_{2})\frac{∂a_{2}}{∂x_{2}} (x_{1}, x_{2})z_{2}

(w_{k}(x_{1}, x_{2} + ϵθz_{2} − ϵz_{2}) − w_{k}(x_{1}, x_{2} + ϵθz_{2}))dθdx_{1}dx_{2}dz_{2}.

From our assumption on the L^{1}(ℝ^{N}) function a_{2}, that we shall make globally for simplicity, we know that for m_{N_{1}}-almost all x_{1} in ℝ^{N_{1}}, the function ℝ^{N_{2}} ∋ x_{2} ↦ a_{2}(x_{1}, x_{2}) ∈ ℝ^{N_{2}} is in BV (ℝ^{N_{2}}) with an L^{1} divergence and

(4.9)  ∫ [∥a_{2}(x_{1}, ·)∥_{BV (ℝ^{N_{2}} )} + ∥div a_{2}(x_{1}, ·)∥_{L^{1}(ℝ^{N_{2}} )}]dx_{1} < ∞.

As a consequence, from the canonical decomposition of D_{2}a_{2}(x_{1}, ·), the polar decomposition of D_{2}a_{2}(x_{1}, ·)^{s}, and the Theorem 3.3 along with Lemma 5.4, we get that for m_{N_{1}}-almost all x_{1} in ℝ^{N_{1}},

(4.10)  D_{2}a_{2}(x_{1}, ·) = (D_{2}a_{2}(x_{1}, ·))^{ac} + (D_{2}a_{2}(x_{1}, ·))^{s} ,
(4.11)  (D_{2}a_{2}(x_{1}, ·))^{s} = M_{x_{1}}(x_{2})μ_{x_{1}}(x_{2}), μ_{x_{1}} = | (D_{2}a_{2}(x_{1}, ·))^{s} |

and

(4.12) ∥(Id +γ^{t}M_{x_{1}}(x_{2})M_{x_{1}}(x_{2}))M_{x_{1}}(x_{2})(Id +γ^{t}M_{x_{1}}(x_{2})M_{x_{1}}(x_{2}))^{−1}∥ ≤ (1 + γ)^{−1},

with

(4.13)  ∫_{ℝ^{N_{1}}} [∥D_{2}a_{2}(x_{1}, ·)^{ac}∥_{L^{1}(ℝ^{N_{2})}} + ∥D_{2}a_{2}(x_{1}, ·)^{s}∥_{M(ℝ^{N_{2}} )}]dx_{1} < ∞.

In particular, setting k_{x_{1}}(x_{2}) = (D_{2}a_{2}(x_{1}, ·))^{ac} we get that ∫_{ℝ^{N_{1}}} ∥k_{x_{1}}∥_{L^{1}(ℝ^{N_{2}} )} dx_{1} < ∞ and thus the function (x_{1}, x_{2}) ↦ k_{x_{1}}(x_{2}) belongs to L^{1}(ℝ^{N}). Let us recall the standard (see e.g. theorem 2.28 in [AFP])

Lemma 4.3 (disintegration of the measure ∂a_{2}/∂x_{2}). Let N, N_{1}, N_{2} as above and a_{2} ∈ L^{1}(ℝ_{x_{1}}^{N_{1}}; BV (ℝ_{x_{2}}^{N_{2}})). We denote by π_{1} the projection ℝ^{N_{1}} × ℝ^{N_{2}} → ℝ^{N_{1}}, by ν the measure ∂a_{2}/∂x_{2} and we set λ = π_{1}_{∗}(|ν|). Our assumption implies that |ν|(ℝ^{N}) < ∞. The disintegration theorem gives

ν = λ ⊗ ν_{x_{1}}

where for λ-almost all x_{1} ∈ ℝ^{N_{1}}, the 𝕄_{N_{2}}-valued measure ν_{x_{1}} is such that

|ν_{x_{1}}|(ℝ^{N_{2}}) = 1.

It means that for F (x_{1}, x_{2}) ∈ L^{1}(ℝ^{N}, dν), we have, for λ-almost all x_{1} ∈ ℝ^{N_{1}}, F (x_{1}, ·) ∈ L^{1}(ℝ^{N_{2}}, d|ν_{x_{1}}|), x_{1} ↦ ∫_{ℝ^{N_{2}}} F (x_{1}, x_{2})dν_{x_{1}}(x_{2}) belongs to L^{1}(ℝ^{N_{1}}, dλ) and

∫_{ℝ^{N_{1}} ×ℝ^{N_{2}}} F (x_{1}, x_{2})dν(x_{1}, x_{2}) = ∫_{ℝ^{N_{1}}} (∫_{ℝ^{N_{2}}} F (x_{1}, x_{2})dν_{x_{1}}(x_{2})) dλ(x_{1}).

We note also that the measure λ is absolutely continuous with respect to the Lebesgue measure: let A be a Borelian subset of ℝ^{N_{1}} of Lebesgue measure 0. We have from (4.9)

λ(A) = |ν|(A × ℝ^{N_{2}}) ≤ ∫_{A} \overbrace{∥|D_{2}a_{2}|(x_{1}, ·)∥_{M(ℝ^{N_{2}} )}}^{∈L^{1}(ℝ^{N_{1}} )} dx_{1} = 0.

We thus obtain that, with h ∈ L^{1}(ℝ^{N_{1}}), ν = λ ⊗ ν_{x_{1}} = hm_{N_{1}} ⊗ ν_{x_{1}}, and thus we have the disintegration formulas

(4.14)  \frac{∂a_{2}}{∂x_{2}} = ν = m_{N_{1}} ⊗ h(x_{1})ν_{x_{1}} = m_{N_{1}} ⊗ M_{x_{1}}μ_{x_{1}} + L^{1}(ℝ^{N}).

In fact for F ∈ C_{c}^{0}(ℝ^{N}), we have with the notations of (4.10–13)

∫_{ℝ^{N_{1}}} (∫_{ℝ^{N_{2}}} F (x_{1}, x_{2})M_{x_{1}}(x_{2})dμ_{x_{1}}(x_{2})) dx_{1} + ∫_{ℝ^{N_{1}}} (∫_{ℝ^{N_{2}}} F (x_{1}, x_{2})k_{x_{1}}(x_{2})dx_{2}) dx_{1}

= ∫_{ℝ^{N_{1}} ×ℝ^{N_{2}}} F (x_{1}, x_{2})dν(x_{1}, x_{2})

= ∫_{ℝ^{N_{1}}} (∫_{ℝ^{N_{2}}} F (x_{1}, x_{2})dν_{x_{1}}(x_{2})) h(x_{1})dx_{1}.

The term belonging to L^{1}(ℝ^{N}) in (4.14) can be given the same treatment as DX^{a} in section 3 and the same method along with (4.14) gives

(4.15) lim sup_{ϵ→0} ω̃(ϵ)

≤ 2 ∥w∥_{L^{∞}}^{2} ∫∫ (∫ |φ(x_{1}, x_{2})| |\frac{∂ρ}{∂z_{2}} (x_{1}, x_{2}, z_{2})M_{x_{1}}(x_{2})z_{2}| dμ_{x_{1}}(x_{2})) dz_{2}dx_{1}.

We inspect then the arguments of section 3, between (3.9) and (3.15). We consider a function

(4.16)  ρ(x_{1}, x_{2}, z_{2}) = F_{0}(U(x_{1}, x_{2})z_{2})| det U(x_{1}, x_{2})|

where U ∈ C_{b}^{1}(ℝ^{N}; 𝕄_{N_{2}}(ℝ)) is such that ^{t}U(x)U(x) ≥ Id, and F_{0} ∈ C_{c}^{1}(ℝ^{N_{2}}; ℝ_{+}) satisfies ∫_{ℝ^{N_{2}}} F_{0}(ζ)dζ = 1. We would like to choose

U(x) = Id +γ^{t}M_{x_{1}}(x_{2})M_{x_{1}}(x_{2}),

but, as in section 3, it is not directly possible because of the lack of regularity of that function. The matrices M_{x_{1}}(x_{2}) have norm ≤ 1. For all x_{1}, we can find a sequence (V_{x_{1},l}(x_{2}))_{l∈ℕ} of functions in C_{b}^{1}(ℝ^{N_{2}}; unit ball of 𝕄_{N_{2}}(ℝ)) converging pointwise to the bounded matrix M_{x_{1}}(x_{2}). We can also regularize these matrices with respect to x_{1} by a standard mollifier, which will be enough since the integral in the variables x_{1}, z_{2} takes place on a fixed compact set. We can conclude as in section 3 by using (4.12). The proof of Theorem 4.1 is complete. □

5. Appendix

Lemma 5.1. Let a ∈ L^{1}(ℝ^{N}) and v ∈ L^{1}(ℝ^{N}) ∩ L^{∞}(ℝ^{N}). Then, we have

lim_{t→0} ∫ |a(x)||v(x + t) − v(x)|dx = 0.

Proof. We have for λ > 0,

∫ |a(x)||v(x − t) − v(x)|dx ≤ λ ∫_{|a|≤λ} |v(x − t) − v(x)|dx + 2 ∥v∥_{L^{∞}} ∫_{|a|≥λ} |a(x)|dx

≤ λ ∥τ_{t}v − v∥_{L^{1}} + 2 ∥v∥_{L^{∞}} ∫ 1_{|a|≥λ}|a(x)|dx,

so that, since v ∈ L^{1}, lim sup_{t→0} ∫ |a(x)||v(x − t) − v(x)|dx ≤ 2 ∥v∥_{L^{∞}} ∫ 1_{|a|≥λ}|a(x)|dx, which gives the result by taking the limit of the rhs when λ goes to infinity. □

Remark. Assuming v ∈ L^{∞} is enough, as proven in the lemma 5.1 of [CL2].

Lemma 5.2. Let ρ ∈ C^{1}(ℝ^{N} × ℝ^{N}; ℝ_{+}) such that ∫_{ℝ^{N}} ρ(x, z)dz = 1 and

(5.1)  ρ_{0}(z) = sup_{x} |ρ(x, z)| + sup_{x} |d_{x,z}ρ(x, z)| ∈ L_{comp}^{∞}.

For ϵ > 0, we consider the operator R_{ϵ} with kernel ρ(x, ϵ^{−1}(x − y))ϵ^{−N}, defined for u ∈ L_{loc}^{1} by

(5.2)  (R_{ϵ}u)(x) = ∫ ρ(x, ϵ^{−1}(x − y))ϵ^{−N}u(y)dy.

Let 1 ≤ p < +∞ and u ∈ L^{p}; then lim_{ϵ→0} R_{ϵ}u = u in L^{p}. If u ∈ L^{∞}, ∥R_{ϵ}u∥_{L^{∞}} ≤ ∥u∥_{L^{∞}} . If u belongs to L_{loc}^{1}, the function R_{ϵ}u belongs to C^{1}(ℝ^{N}) and for almost all x ∈ ℝ^{N}, lim_{ϵ→0}(R_{ϵ}u)(x) = u(x).

Proof. This lemma is classical for a convolution. We check here that this more general regularizing kernel does not introduce any new difficulty. Let us first assume that u ∈ L^{p} with 1 ≤ p < +∞. We have

(R_{ϵ}u)(x) − u(x) = ∫ ρ(x, z)(u(x − ϵz) − u(x))dz

so that, defining α = ∥ρ_{0}∥_{L^{1}}, we get from Jensen’s inequality

∥R_{ϵ}u − u∥_{L^{p}}^{p} ≤ ∫ (∫ ρ_{0}(z)|(τ_{ϵz}u − u)(x)|dz)^{p} dx ≤ α^{p−1} ∫ ρ_{0}(z) ∥τ_{ϵz}u − u∥_{L^{p}}^{p} dz.

Since lim_{ϵ→0} ∥τ_{ϵz}u − u∥_{L^{p}} = 0 and ∥τ_{ϵz}u − u∥_{L^{p}} ≤ 2 ∥u∥_{L^{p}}, Lebesgue’s dominated convergence theorem gives the result. Let us assume now that u ∈ L^{∞} ; the estimate on ∥R_{ϵ}u∥_{L^{∞}} is trivial. Moreover, we have for u ∈ L_{loc}^{1}

|(R_{ϵ}u)(x) − u(x)| ≤ ∫_{z∈supp ρ_{0}} |u(x − ϵz) − u(x)|dz ∥ρ_{0}∥_{L^{∞}}

and since supp ρ_{0} is compact, the Lebesgue differentiation theorem gives that this quantity is going to 0 with ϵ for almost all x. Assuming u ∈ L_{loc}^{1}, the function

y ↦ ρ(x, ϵ^{−1}(x − y))ϵ^{−N}u(y)

belongs to L^{1} for all x since

∫ ϵ^{−N}ρ_{0}(ϵ^{−1}(x − y))|u(y)|dy ≤ ϵ^{−N} ∥ρ_{0}∥_{L^{∞}} ∫_{x−ϵ supp ρ_{0}} |u(y)|dy < ∞.

Moreover the function x ↦ ρ(x, ϵ^{−1}(x − y))ϵ^{−N}u(y) is continuously differentiable and for K compact

∫ sup_{x∈K} |∂_{1}ρ(x, ϵ^{−1}(x − y))ϵ^{−N} + ϵ^{−1}∂_{2}ρ(x, ϵ^{−1}(x − y))ϵ^{−N} | |u(y)|dy

≤ 2 ∥ρ_{0}∥_{L^{∞}} ϵ^{−N−1} ∫_{K−ϵ supp ρ_{0}} |u(y)|dy < ∞.

We obtain that the function R_{ϵ}u belongs to C^{1}(ℝ^{N}), completing the proof of the lemma. □

Lemma 5.3. Let Ω be an open subset of ℝ^{N} and let X be a L_{loc}^{1} vector field on Ω such that div X ∈ L_{loc}^{1}(Ω). Let v be in L_{comp}^{∞}(Ω) and R_{ϵ} be given by Lemma 5.1. Then we have

(XR_{ϵ}v − R_{ϵ}Xv)(x) = (T_{ϵ,ρ}v)(x) + (N_{ϵ}v)(x)

with with lim_{ϵ→0} N_{ϵ}v = 0 in L_{loc}^{1} and

(5.3)  (T_{ϵ,ρ}v)(x) = ∫ ∂_{2}ρ(x, z)ϵ^{−1}(X(x) − X(x − ϵz))(v(x − ϵz) − v(x))dz.

Proof. To avoid confusion between the vector field X (for each x ∈ Ω, X(x) is a vector tangent to Ω at x) and the operator X acting on functions, we shall denote by 𝕏 the operator: we use the notation (𝕏w)(x) = dw(x)X(x) (a scalar quantity as the product of the 1 × N covector dw with the N × 1 vector X) and we write

𝕏R_{ϵ}v = ∫ ∂_{1}ρ(x, ϵ^{−1}(x−y))ϵ^{−N}v(y)dyX(x)+∫ ∂_{2}ρ(x, ϵ^{−1}(x−y))ϵ^{−1−N}v(y)dyX(x).

We note that ∂_{2}ρ(x, ϵ^{−1}(x−y))ϵ^{−1} = −\frac{∂}{∂y} (ρ(x, ϵ^{−1}(x−y))) which gives, using the identity ∫ ρ(x, z)dz ≡ 1,

𝕏R_{ϵ}v = \overbrace{∫ ∂_{1}ρ(x, z)X(x)(v(x − ϵz) − v(x))dz}^{:=N_{ϵ,1}v(x)} + \overbrace{∫ ∂_{1}ρ(x, z)dz}^{=0} X(x)v(x)

+ ∫ \frac{∂}{∂y} (ρ(x, ϵ^{−1}(x − y))ϵ^{−N})(X(y) − X(x))v(y)dy

− ∫ \frac{∂}{∂y} (ρ(x, ϵ^{−1}(x − y))ϵ^{−N})X(y)v(y)dy,

so that

𝕏R_{ϵ}v = N_{ϵ,1}v(x) + ∫ ∂_{2}ρ(x, z)ϵ^{−1}(X(x) − X(x − ϵz))v(x − ϵz)dz

+ ∫ ρ(x, ϵ^{−1}(x − y))ϵ^{−N} \frac{∂}{∂y} · (X(y)v(y))dy,

where the last term is in fact a bracket of duality. Since we have from (3.2)

\frac{∂}{∂y} · (X(y)v(y)) = (𝕏v)(y) + (div X)(y)v(y),

we obtain

𝕏R_{ϵ}v = N_{ϵ,1}v + \overbrace{∫ ∂_{2}ρ(x, z)ϵ^{−1}(X(x) − X(x − ϵz))(v(x − ϵz) − v(x))dz}^{=(T_{ϵ,ρ}v)(x)}

+ ∫ ∂_{2}ρ(x, z)ϵ^{−1}(X(x) − X(x − ϵz))dzv(x)

+ ∫ ρ(x, ϵ^{−1}(x − y))ϵ^{−N}((𝕏v)(y) + (div X)(y)v(y))dy,

which implies

𝕏R_{ϵ}v = N_{ϵ,1}v + T_{ϵ,ρ}v + ∫ ∂_{2}ρ(x, z)ϵ^{−1}(X(x) − X(x − ϵz))dzv(x) + R_{ϵ}𝕏v + ℝ_{ϵ}v div X. Since z ↦ ρ(x, z) is compactly supported and C^{1}, the covector ∫ ∂_{2}ρ(x, z)dz = 0. Moreover, we have

− ∫ ∂_{2}ρ(x, z)ϵ^{−1}X(x − ϵz)dzv(x) + (ℝ_{ϵ}v div X)(x) =

− v(x) ∫ ρ(x, z)(div X)(x − ϵz) + ∫ ρ(x, z)v(x − ϵz)(div X)(x − ϵz)dz = ∫ ρ(x, z)(v(x − ϵz) − v(x))(div X)(x − ϵz)dz := N_{ϵ,2}v(x) and consequently

𝕏R_{ϵ}v = N_{ϵ,1}v + T_{ϵ,ρ}v + N_{ϵ,2}v + R_{ϵ}𝕏v.

We have for K compact subset of ℝ^{N} and ρ_{0} ∈ L^{1}(dz) given in (5.1)

∫_{K} |N_{ϵ,1}(x)|dx ≤ ∫ \overbrace{∫ |1_{K}(x)X(x)||(τ_{ϵz}v − v)(x)|dx}^{goes to 0 with ϵ from lemma 5.1, bounded above by 2∥1_{K} X∥_{L^{1}} ∥v∥_{L^{∞}}} ρ_{0}(z)dz,

so that Lebesgue’s dominated convergence theorem gives lim_{ϵ→0} ∥N_{ϵ,1}v∥_{L^{1}(K)} = 0. We have for K compact subset of ℝ^{N}

∫_{K} |N_{ϵ,2}v(x)|dx ≤ ∫∫ 1_{K−ϵ supp ρ_{0}}(x)|(div X)(x)||v(x) − v(x + ϵz)|ρ_{0}(z)dzdx

which goes to zero with ϵ from Lemma 5.1. The proof of Lemma 5.3 is complete. □

Lemma 5.4. Let E be a real Euclidean finite dimensional vector space. Let M be an endomorphism of E such that M = ξ ⊗ η with ξ, η orthogonal unit vectors. Then for all γ ≥ 0,

(5.4)  ∥(Id +γ^{t}MM)M(Id +γ^{t}MM)^{−1}∥ ≤ (1 + γ)^{−1}.

Proof. Since for T ∈ E, we have MT = ⟨ξ, T ⟩η we get M^{2}T = ⟨ξ, T ⟩⟨ξ, η⟩η = 0 and

⟨M^{t}MMT_{1}, T_{2}⟩ = ⟨ξ,^{t}MMT_{1}⟩⟨η, T_{2}⟩ = ⟨η, η⟩⟨ξ, T_{1}⟩⟨η, T_{2}⟩ = ⟨MT_{1}, T_{2}⟩,

which means M = M^{t}MM and implies (1 + γ)M = M + γM^{t}MM = M(Id +γ^{t}MM) and

(5.5)  (Id +γ^{t}MM)M(Id +γ^{t}MM)^{−1} = M(Id +γ^{t}MM)^{−1} = (1 + γ)^{−1}M

implying (5.4). □

Remark 5.5. It worth noticing that for a BV_{loc} vector field X = ∑_{1≤j≤N} a_{j}(x)∂_{x_{j}}, the matrix

(5.6)  M(x) = (\frac{∂a_{j}}{∂x_{k}} )_{1≤j,k≤N}

has some invariance properties, under C^{1} diffeomorphism, at least modulo L_{loc}^{1} matrices. In fact if x = κ(y), y = ν(x) is such a diffeomorphism, the vector field X in the y-chart is

X = ∑_{1≤j,k≤N} a_{j}(κ(y))\frac{∂y_{k}}{∂x_{j}} ∂_{y_{k}} = ∑_{1≤k≤N} b_{k}(y)∂_{y_{k}}, b_{k} = ∑_{1≤j≤N} a_{j} \frac{∂y_{k}}{∂x_{j}} ,
so that \frac{∂b_{k}}{∂y_{l}} = ∑_{1≤j≤N} \frac{∂a_{j}}{∂x_{m}} \frac{∂x_{m}}{∂y_{l}} \frac{∂y_{k}}{∂x_{j}} + L_{loc}^{1},

which means

(5.7)  N (y) = (\frac{∂b_{k}}{∂y_{l}} )_{1≤k,l≤N} = ν′(x)M(x)κ′(y) + L_{loc}^{1}

and since ν′(κ(y))κ′(y) = Id, we find that matrices M(x) and N (y) are equivalent, modulo a L_{loc}^{1} matrix.

Remark 5.6. Let us point out here that an invariant formulation of our statement of Theorem 2.1 can be given, using a codimension N_{1}-foliation of the reference open set. Let Ω be an open subset of ℝ^{N} equipped with a codimension N_{1} foliation (in our coordinates the leaves are x_{1} = cte). A vector field T is tangent to the foliation means, in our coordinates, that T = β(x_{1}, x_{2})∂_{x_{2}} since T (x_{1}) should be identically 0. Let us call T the vector fields tangent to the foliation. We introduce a vector field X such that,

(5.8)  ∀T ∈ T , [X, T] ∈ T .

In our coordinates, it means if X = α_{1}(x_{1}, x_{2})∂_{x_{1}} + α_{2}(x_{1}, x_{2})∂_{x_{2}} and T = β(x_{1}, x_{2})∂_{x_{2}} is any tangent vector field

[X, β(x_{1}, x_{2})∂_{x_{2}}] = α_{1}∂_{x_{1}}(β)∂_{x_{2}} − β∂_{x_{2}}(α_{1})∂_{x_{1}} + α_{2}∂_{x_{2}}(β)∂_{x_{2}} − β∂_{x_{2}}(α_{2})∂_{x_{2}}

is tangent to the foliation. So to ask for this commutator to be tangent is the requirement β∂_{x_{2}}(α_{1}) = 0 for all β, which means ∂_{x_{2}}(α_{1}) = 0, so in our coordinates

X = a_{1}(x_{1})∂_{x_{1}} + a_{2}(x_{1}, x_{2})∂_{x_{2}}.

If the open set Ω is equipped with a Riemannian structure, X can be decomposed in the sum of a tangential part to the foliation (a_{2}(x_{1}, x_{2})∂_{x_{2}}) and a normal part (a_{1}(x_{1})∂_{x_{1}}) ; to get the divergence property, we shall assume that the divergence of both parts is zero. The geometric hypothesis (5.8) allows us to produce an invariant result.

Index of notations. · Let X = ∑_{1≤j≤N} a_{j} \frac{∂}{∂x_{j}} be a vector field on an open set of ℝ^{N}. DX stands for the matrix (\frac{∂a_{j}}{∂x_{k}})_{1≤j,k≤N}. When X is a BV_{loc} vector field, DX is a matrix of Radon measures and we can write the canonical decomposition

DX = DX^{a} + DX^{s}, |DX^{a}| ≪ m, μ = |DX^{s}| ⊥ m

where m is the Lebesgue measure on ℝ^{N}. The polar decomposition of the matrix DX^{s} is

DX^{s} = Mμ.

· For y ∈ ℝ^{N} and u ∈ D′(ℝ^{N}), ⟨τ_{y}u, φ⟩_{D′,D} = ⟨u, τ_{−y}φ⟩_{D′,D}, (τ_{−y}φ)(x) = φ(x + y). · C_{b}^{1} stands for the C^{1} functions bounded as well as their first derivatives. · For ξ, η ∈ ℝ^{N}, the product ξ ⊗ η is the linear map ℝ^{N} ∋ T ↦ ⟨ξ, T ⟩η ∈ ℝ^{N}.

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Université de Rennes 1, Irmar, Campus de Beaulieu, 35042 Rennes cedex, France

   E-mail address: nicolas.lerner@univ-rennes1.fr
   Web-page: http://perso.univ-rennes1.fr/nicolas.lerner/