2.1. Whitney cubes

Our goal is to extend a function from a closed set $X \subset \mathbb {R}^N$ to $\mathbb {R}^N$. In the context of truncation, we leave the function $u \colon \mathbb {R}^N \to \mathbb {R}^N$ untouched on some set $X$ and modify it on $X^C$. We often refer to $X$ as the ‘good set’ and to its complement as the ‘bad set’. Further, we assume that $X$ is closed and, therefore, $X^C$ is open.

Following Stein's book [Reference Stein33, Chapter VI], we can cover $X^C$ by open, dyadic cubes $(Q_i^{\ast })_{i \in \mathbb {N}}$ with the following properties:

1. (i*) $X^C = \bigcup _{i \in \mathbb {N}} \bar {Q}_i^{\ast }$;

2. (ii*) $Q_i^{\ast } \cap Q_j^{\ast } = \emptyset$ if $i \neq j$;

3. (iii*) ${1}/{4} \operatorname {dist}(Q_i^{\ast },\,X) \leq l(Q_i^{\ast }) \leq 4\operatorname {dist}(Q_i^{\ast },\,X)$ where $l(Q_i^{\ast })$ denotes the sidelength of $Q_i^{\ast }$;

4. (iv*) if $\bar {Q}_i^{\ast } \cap \bar {Q}_j^{\ast } \neq \emptyset$, then ${1}/{4} l(Q_i^{\ast }) \leq l(Q_j^{\ast }) \leq 4 l(Q_i^{\ast })$;

5. (v*) for each $i \in \mathbb {N}$, the number of cubes $Q_j^{\ast }$ such that $\bar {Q}_i^{\ast } \cap \bar {Q}_j^{\ast } \neq \emptyset$ is bounded by a dimensional constant $C(N)$.

Furthermore, for each cube $Q_i^{\ast }$, we denote by $c_i$ the centre of the cube $Q_i^{\ast }$. We may also find a projection point $z_i \in X$, such that $\operatorname {dist}(Q_i^{\ast },\,z_i) \leq 4 \operatorname {dist}(Q_i^{\ast },\,X)$. Moreover, we define the measure $\mu _i$ as

\[ \mu_i = \mathcal{L}^N\left(\frac{1}{2} Q_i^{{\ast}}\right)^{{-}1} \left( \mathcal{L}^N \llcorner \left(\frac{1}{2}Q_i^{{\ast}}\right)\right), \]

where ${1}/{2} Q_i^{\ast }$ is the open cube with axis parallel faces, centre $c_i$ and sidelength ${1}/{2}l(Q_i^{\ast })$.

In addition, we also consider slightly blown-up open cubes $Q_i=(1+\epsilon ) Q_i^{\ast }$ with the same centre $c_i$ but sidelength $(1+\varepsilon )l(Q_i^{\ast })$. If $\epsilon$ is sufficiently small (e.g. $\varepsilon < {1}/{32}$), then these cubes have the following properties:

1. (i*) $X^C = \bigcup _{i \in \mathbb {N}} Q_i$;

2. (ii*) for all $i \in \mathbb {N}$, the number of cubes $Q_j$ with $Q_j \cap Q_i \neq \emptyset$ is bounded by $C(N)$;

3. (iii*) ${1}/{5} \operatorname {dist}(Q_i,\,X) \leq l(Q_i) \leq 5\operatorname {dist}(Q_i,\,X)$, where $l(Q_i)$ is the sidelength of $Q_i$ ($l(Q_i) =(1+\epsilon )l(Q_i^{\ast })$;

4. (iv*) if $Q_i \cap Q_j \neq \emptyset$, then ${1}/{4} l(Q_i) \leq l(Q_j) \leq 4 l(Q_i)$.

We now take $\varphi \in C_c^{\infty }((-\varepsilon /2,\,1+\varepsilon /2)^N)$ with $\varphi \equiv 1$ on $[0,\,1]^N$. By translation and scaling we get $\varphi _j^{\ast } \in C_c^{\infty }(Q_j)$ with $\varphi ^{\ast }_j \equiv 1$ on $Q_j^{\ast }$.

Using the properties of the cubes $Q_j$ we can show (again, cf. [Reference Stein33]), that

(2.1)\begin{equation} \varphi_j = \frac{\varphi_j^{{\ast}}}{\sum_{i \in \mathbb{N}} \varphi_i^{{\ast}}} \end{equation}

defines a partition of unity on $X^C$, i.e.

\[ \sum_{j \in \mathbb{N}} \varphi_j(y) = \chi_{X^C}(y) := \left\{ \begin{array}{@{}ll} 1 & \text{on } X^C, \\ 0 & \text{on } X. \end{array} \right. \]

Moreover, $\varphi _j \in C_c^{\infty }(Q_j)$ and they satisfy the bound

(2.2)\begin{equation} \Vert D^k \varphi_j \Vert_{L^{\infty}} \leq C(N,k) l(Q_j)^{{-}k}. \end{equation}

Before we continue with Whitney's extension theorem, we again point out that all the dimensional constants $C(N)$ and $C(N,\,k)$ in above construction do not depend on the regularity of the set $X$; all we need is the set $X$ to be closed.

2.2. Whitney extension and the related truncation theorem

Given a closed set $X \subset \mathbb {R}^N$ and a Lipschitz continuous function $u \colon X \to \mathbb {R}$ with Lipschitz constant $L>0$, we define

(2.3)\begin{equation} \tilde{u}(y) = \left\{ \begin{array}{@{}ll} \sum_{i \in \mathbb{N}} \varphi_i(y) u(z_i) & y \in X^{C}, \\ u(y) & y \in X. \end{array} \right. \end{equation}

A detailed proof can be found in [Reference Stein33, Chapter VI]. We give a brief heuristic idea, why lemma 2.1 works, as a similar argument is used in § 3.

Observe that $\tilde {u} \in C^{\infty }(X^C)$ and that locally, only finitely many terms are nonzero. Therefore, we can compute its derivative for $y \in X^C$:

\[ D \tilde{u}(y) = \sum_{i \in \mathbb{N}} D \varphi_i(y) u(x_i) = \sum_{i,j \in \mathbb{N}} \varphi_j D\varphi_i(y) (u(x_i)-u(x_j)). \]

Note that we explicitly used in the second step that $(\varphi _i)_{i \in \mathbb {N}}$ and $(\varphi _j)_{j \in \mathbb {N}}$ are partitions of unity. Applying the bound for the derivative of $\varphi _i$ and the Lipschitz bound for $u(x_i)-u(x_j)$ yields an $L^{\infty }$-bound for the derivative of $\tilde {u}$.

Together with the observation, that $\tilde {u}$ is continuous, this leads to Lipschitz continuity of $\tilde {u}$.

This approach can then be applied to Lipschitz truncation as follows: First, we find a ‘good set’ $X$, on which $u$ is Lipschitz continuous. For the truncation we then take $\tilde {u} \equiv u$ on the good set and redefine is on the bad set as in (2.3). An important part in showing Lipschitz continuity for $\tilde {u}$ is therefore the following lemma 2.3 that connects Lipschitz continuity to the (centred) Hardy–Littlewood maximal function $\mathscr {M} u$ that is defined via

\[ \mathscr{M} u(x) := \sup_{r>0} {\unicode{x2A0F}}_{B_r(x)} \vert u(z) \vert \,{\textrm d}z, \]

where ${\unicode{x2A0F}}$ denotes the average integral. The associated operator then has the following properties.

Lemma 2.2 The maximal function

The maximal operator is sublinear, i.e.

\[ \mathscr{M}(u+v)(y) \leq \mathscr{M} u(y) + \mathscr{M} v(y) \]

for almost every $y\in \mathbb {R}^N$. Moreover, it is bounded as a map from $L^p(\mathbb {R}^N) \to L^p(\mathbb {R}^N)$, whenever $1< p\leq \infty$. For $p=1$, $\mathscr {M}$ is not bounded from $L^1(\mathbb {R}^N)$ to $L^1(\mathbb {R}^N)$, but from $L^1(\mathbb {R}^N)$ to $L^{1,\infty }(\mathbb {R}^N)$, i.e.

\[ \mathcal{L}^N( \{ \vert \mathscr{M} u \vert \geq \lambda \}) \leq C \lambda^{{-}1} \Vert u \Vert_{L^1} \]

for every $\lambda >0$ and $u \in L^1(\mathbb {R}^N)$.

Key to the truncation statement then are the following two observations. The first one, proven in [Reference Acerbi and Fusco1, Reference Liu26] proves that a function $u$ is Lipschitz continuous on sublevel sets of its maximal function.

Lemma 2.3 Let $u \in W^{1,p}(\mathbb {R}^N;\mathbb {R}^d)$ be continuous. There exists a dimensional constant $C$, such that for all $\lambda >0$ and all $x,\,y \in X_{\lambda }= \{ \mathscr {M} (Du) \leq \lambda \}$ we have

(2.4)\begin{equation} \vert u(x) - u(y) \vert \leq C \lambda \vert x- y \vert. \end{equation}

The second ingredient is an estimate on the measure of the ‘bad set’, the complement of $X_{\lambda }$, cf. [Reference Zhang35].

Lemma 2.4 Let $v \in L^p(\mathbb {R}^N;\mathbb {R}^d)$, $1 \leq p< \infty$. There is a dimensional constant $C$, such that for all $\lambda >0$ we have

(2.5)\begin{equation} \mathcal{L}^N(\{ \mathscr{M} v > \lambda \}) \leq C \lambda^{{-}p} \int_{\{ \vert v \vert \geq \lambda/2\}} \vert v \vert^p \,{\textrm d}x. \end{equation}

These results are combined to get a Lipschitz truncation in the fashion we already mentioned: We take $u \colon \mathbb {R}^N \to \mathbb {R}^d$, set $\tilde {u}=u$ on the ‘good set’ $X_{\lambda }$ and then extend this restriction via Lipschitz extension (2.3) to the full space.

Defining the extension as in (2.3) only bears the problem that $u(z_i)$ might not be well-defined for an arbitrarily chosen $z_i$. There are several workarounds for this problem, e.g. only taking Lebesgue points, considering $u \in C^1(\mathbb {R}^N)$ first and then using a density argument or using an averaged value of the function instead.

In the following, we pursue the third approach. Therefore, we define:

(2.6)\begin{equation} T_{\operatorname{Lip}} u(y) = \left\{ \begin{array}{ll} \sum_{i \in \mathbb{N}} \varphi_i(y) \int u(x_i) \,{\textrm d}\mu_i(x_i) & y \in X_{\lambda}^C,\\ u(y) & y \in X_{\lambda}. \end{array} \right. \end{equation}

A suitable adaptation of lemma 2.3 and the estimate (2.5) then yield the following:

2.3. Solenoidal truncation

In this section we shortly describe previous approaches to solenoidal Lipschitz truncation and compare it to the approach in the present work.

2.3.1. Potential truncation.

First of all, in [Reference Breit, Diening and Schwarzacher9, Section 4] (see also [Reference Behn, Gmeineder and Schiffer5, Reference Gallenmüller21] for related discussions), the truncation is obtained by writing a divergence-free function $u \in W^{1,p}(\mathbb {R}^3;\mathbb {R}^3)$ as

\[ u= \operatorname{curl} v \]

for a function $v \in \dot {W}^{2,p}(\mathbb {R}^3;\mathbb {R}^3)$, truncate $v$ to obtain some $\tilde {v} \in W^{2,\infty }(\mathbb {R}^3;\mathbb {R}^3)$ and set $\tilde {u}= \operatorname {curl} \tilde v$.

This approach is particularly well-suited to evolutionary problems (i.e. $u \colon [0,\,\infty ) \times \mathbb {R}^N \to \mathbb {R}^N)$ and can (together with parabolic Lipschitz truncation) also tackle the usually quite low regularity of the time derivative $\partial _t u$.

Seeking for a sharper result in space and not in space-time (which severely complicates the matter), this form of truncation can, however, only achieve property (T2a’) and not (T2b’). Even if $u \in W^{1,\infty }(\mathbb {R}^3;\mathbb {R}^3)$, its potential $v$ is not in $W^{2,\infty }(\mathbb {R}^3;\mathbb {R}^3)$ in general (Ornstein's non-inequality [Reference de Leeuw and Mirkil13, Reference Ornstein30]) and $\tilde {v} \neq v$ regardless of the truncation parameter $L$.

Let us further note that in dimension $N=2$, (T2b’) is satisfied; the reason being that the map $u \mapsto v$ is bounded from $W^{1,\infty }(\mathbb {R}^2;\mathbb {R}^2)$ to $W^{2,\infty }(\mathbb {R}^2)$. Hence, the truncation of [Reference Breit, Diening and Schwarzacher9] (together with the correct spaces), actually yields property (T2b’) both in space and space-time. Moreover, the truncation constructed in [Reference Breit, Diening and Schwarzacher9] in the stationary setting coincides with the truncation constructed in this work in space dimension two.

2.3.2. Truncation via local corrections.

In contrast, the work [Reference Breit, Diening and Fuchs7] is directly suited towards stationary problems in more involved spaces, e.g. Orlicz spaces. In particular, the truncation outlined in [Reference Breit, Diening and Fuchs7] is able to obtain (T2b’). Instead of writing $u=\operatorname {curl} v$, we work with $u$ directly, find $\tilde {u}$ as in (2.3) and then add corrector terms to restore solenoidality. As the ‘good set’ one takes the set where the maximal function of $u$ and of $Du$ is smallFootnote ¹ .

In more detail, we first add a global corrector term $\Pi u$, such $\tilde {u} + \Pi u =: \tilde {T}u$ is still a $W^{1,\infty }$-truncation of $u$. In the second step, one adds local corrector terms $\mathrm {Cor}_i \in W^{1,\infty }_0(Q_i;\mathbb {R}^3)$, that satisfy the divergence-equation

(2.7)\begin{equation} \operatorname{div} \mathrm{Cor}_i = \operatorname{div} (\varphi_i(\tilde{u}+\Pi u)). \end{equation}

Finally, one obtains the divergence-free truncation via

\[ T_{\operatorname{div}} u = \tilde{T} u - \sum_{i \in \mathbb{N}} \mathrm{Cor}_i. \]

As one only modifies the function on the bad set $\{ \mathscr {M} u > \lambda \} \cup \{\mathscr {M}(Du) >\lambda \}$, one is able to obtain (T2b’). One crucial observation (cf. [Reference Breit, Diening and Fuchs7, Lemma 2.11]) to get $T_{\operatorname {div}} u \in W^{1,\infty }$ is the following:

First note that the observation that $\mathrm {Cor}_i$ belongs to $W^{1,\infty }$ is not trivial; the divergence equation (2.7) and its solution operator (the Bogovskiĭ operator [Reference Bogovskiĭ6]) is not bounded from $L^{\infty }$ to $W^{1,\infty }$. Instead, one uses that $\operatorname {div} (\varphi _i(\tilde {u}+\Pi u))$ has a very specific form. It is a much smoother function, hence the solution to the divergence equation is in $W^{1,\infty }$. A uniform $W^{1,\infty }$-bound for the solution is achieved via the following argument: The partition of unity is obtained by formula (2.1) for a cover consisting of $Q_i^{\ast }$ and associated functions $\varphi _i^{\ast }$. But actually, as all cubes $Q_i^{\ast }$ are dyadic, there are only finitely many configurations how the cover can locally look like. In particular, up to scaling and translation, we only need to solve

\[ \operatorname{div} \mathrm{Cor}_i = \operatorname{div} \bigl(\varphi_i \cdot (\tilde{u}+\Pi u)\bigr) \quad \text{in } Q_i^{{\ast}} \text{ with zero boundary values} \]

a finite amount of times as, up to scaling and translation, there are only finitely many functions the right-hand-side can realize. As we then only argue about finitely many different corrector terms, the uniform $W^{1,\infty }$-bound is an easy consequence.

2.3.3. Truncations via local corrections II.

In this work, we study a modification of the second approach by explicitly constructing the corrector terms using that our truncation has a very particular form. This approach only relies on

1. (a) some purely geometric estimates;

2. (b) the existence of a cover with sets $Q_i$ (that are not necessarily cubes);

3. (c) a partition of unity $\varphi _i$ that satisfies estimate (2.2).

In particular, the assumptions on the cover over cubes and the partition of unity are less restrictive than in [Reference Breit, Diening and Fuchs7], which, for instance, might be useful for geometries on manifolds.

Comparing the present ansatz to [Reference Breit, Diening and Fuchs7], we also add correctors such that $\tilde {u}$ first given by (2.3) is modified to be divergence-free. Instead of solving the divergence-equation via the Bogovskiĭ operator, we give explicit formulas for the corrector terms that rely on certain cancellations and Stokes’ theorem. In space dimension three, those corrections are defined on pairs and tuples of cubes, i.e.

\[ T_{\operatorname{div}} u = \tilde{u} + \sum_{i,j \in \mathbb{N}} \mathrm{Cor}_{i,j} + \sum_{i,j,k \in \mathbb{N}} \mathrm{Cor}_{i,j,k}. \]

In general dimension, we need more corrector terms, which is connected to the nice algebraic structure provided for the divergence operator.

2.4. Truncation for differential forms

The construction done for the divergence (in three space dimensions) fits into a more general framework for closed differential forms - we introduce the exact notation in § 3. Recall that a divergence-free function $u \colon \mathbb {R}^N \to \mathbb {R}^N$ can be identified with a closed differential form (meaning the exterior derivative vanishes) $\hat {u} \colon \mathbb {R}^N \to \Lambda ^{N-1}$; in particular the divergence operator might be identified with the operator of exterior differentiation.

The approach of constructing the corrector terms heavily builds on this algebraic structure and, correspondingly, on Stokes’ theorem. This connection becomes easily visible when dealing with $\operatorname {curl}$-free functions, these are $v \colon \mathbb {R}^N \to \mathbb {R}^N$ obeys

\[ (\operatorname{curl} v)_{ij} := \partial_i v_j - \partial_j v_i =0. \]

These can also be put into the framework of closed forms, i.e. $\hat {v} \colon \mathbb {R}^N \to \Lambda ^1$. The results then reads as follows.

The proof of this result can be achieved quite directly by writing $v= \nabla V$ for some $V \in \dot {W}^{2,p}(\mathbb {R}^N,\,\mathbb {R}^N)$, truncating $V$ to obtain some $\bar {V}$ and then considering the derivative of $\bar {V}$. The existence of such a $V$ is ensured by the differential constraint $\operatorname {curl} v=0$.

In particular, on the ’bad set’ $X^C$, we have (recall that $z_i$ was a projection point of $Q_i$ onto $X$)

\[ \bar{V}(y) = \sum_{i \in \mathbb{N}} \varphi_i(y) \bigl(v(z_i)+Dv(z_i)\cdot(y-z_i)\bigr). \]

Now, one can calculate the derivative of $\bar {V}$ and then replace $Du$ by $v$ to obtain

(2.8)\begin{equation} \bar{v}(y) = \sum_{i \in \mathbb{N}} \varphi_i(y) v(z_i) + \sum_{i,j \in \mathbb{N}} \varphi_j(y) D\varphi_i(y) \int_{[z_i,z_j]} (y-\xi)^T \cdot Dv(z) \cdot \frac{z_i-z_j}{\vert z_i -z_j \vert}~{\textrm d}\xi; \end{equation}

if $z_i=z_j$, we define the integral in (2.8) to be zero.

Observe that (2.8) features the previously discussed structure. That is, the first sum in (2.8) coincides with the unconstrained extension and the second is a correction to satisfy the constraint $\operatorname {curl} \bar {v}=0$.

The general case of differential forms ($\operatorname {curl}$-free functions are closed $1$-forms) requires multiple steps of corrections. This is carried out in § 3. On the other hand, such a construction involving corrections does not seem to be limited to differential forms, but probably can be achieved for a wider class of differential operators (cf. § 4.2).

3.1. Definition of the truncation in three space dimensions

We shortly outline the three-dimensional case. To this end, let $X \subset \mathbb {R}^3$ be a closed set and $(Q_i)_{i \in \mathbb {N}}$ be a Whitney cover of $X^C$ featuring a partition of unity $\varphi _i$ and measures $\mu _i$ (cf. § 2.1). For simplicity, we write

\[ \,{\textrm d}\mu_{i,j} := \,{\textrm d}\mu_i(x_i) \,{\textrm d}\mu_j(x_j), \quad \,{\textrm d}\mu_{i,j,k}= \,{\textrm d}\mu_i(x_i) \,{\textrm d}\mu_j(x_j) \,{\textrm d}\mu_k(x_k). \]

For points $x_{i_1},\,\ldots,\,x_{i_k} \in \mathbb {R}^3$ denote by

\[ \operatorname{Sim}(i_1,\ldots,i_k) = \operatorname{Sim}(x_{i_1},\ldots,x_{i_k)} := \mathrm{conv} (x_{i_1},...,x_{i_k}) \]

the convex hull of those points. If $k=2$, we denote this by the direct line $[x_{i_1},\,x_{i_2}]:=\operatorname {Sim}(i_1,\,i_2)$ between these points. If $k=2,\,3,\,4$, we call this object a non-degenerate simplex if its dimension is $(k-1)$, i.e. $\mathcal {H}^{k-1}(\operatorname {Sim}(i_1,\,\ldots,\,i_k))>0$.

Definition 3.1 Given $u \in W^{1,p}(\mathbb {R}^3;\mathbb {R}^3)$, tuples $(a,\,b,\,c) \in \{(1,\,2,\,3),\,(2,\,3,\,1), (3,\,1,\,2)\}$ and $X \subset \mathbb {R}^3$, we define the truncation operator $T$ as follows:

(3.1)\begin{equation} (T u (y)) _a= \left\{ \begin{array} {@{}ll} \displaystyle\sum\limits_{i \in \mathbb{N}} \varphi_i(y) \int u_a(x_i) \,{\textrm d}\mu_i(x_i) + (S u(y))_a + (Ru(y))_a & y \in X^C, \\ u (y) & y \in X, \end{array} \right. \end{equation}

where the correction term $S$ is defined as

(3.2)\begin{align} & (S u)_a:={-}\frac{1}{2} \sum_{i,j \in \mathbb{N}} \Bigl(\varphi_j \partial_b \varphi_i (A_{ab}(i,j) - A_{ba}(i,j))\Bigr) + \Bigl( \varphi_j \partial_c \varphi_i (A_{ac}(i,j) - A_{ca}(i,j))\Bigr), \end{align}

(3.3)\begin{align} & A_{\alpha \beta}(i,j) :=\int {\unicode{x2A0F}}_{[x_i,x_j]} D u_\beta(z) \cdot (x_i-x_j) (y-z)_{\alpha} \,{\textrm d}\mathcal{H}^1(z) \,{\textrm d}\mu_{i,j}, \end{align}

if $i \neq j$ and $A_{\alpha \beta }(i,\,i) =0$ for all $i \in \mathbb {N}$.

The corrector $R$ is defined via

(3.4)\begin{align} & (R u)_a :={-}\sum_{i,j,k \in \mathbb{N}} \varphi_k \partial_b \varphi_j \partial_c \varphi_i B(i,j,k), \end{align}

(3.5)\begin{align} & B(i,j,k) := \int {\unicode{x2A0F}}_{\operatorname{Sim}(i,j,k)} \frac{1}{2} (x_i-x_j) \times (x_j-x_k) \cdot \Bigl(\partial_1 u (y-z)_1 + \partial_2 u (y-z)_2 \end{align}

(3.6)\begin{align} & \quad + \partial_3 u (y-z)_3 \Bigr) \,{\textrm d}\mathcal{H}^{2}(z) \,{\textrm d}\mu_{i,j,k}, \end{align}

if $i$, $j$ and $k$ are pairwise disjoint and $B(i,\,i,\,k)=B(i,\,j,\,i)=B(i,\,j,\,j)=0$.

We may prove the following statement, which is part of the higher dimensional treatment in § 3.3 ff.

Theorem 3.2 If $u \in W^{1,p}(\mathbb {R}^3;\mathbb {R}^3)$ with $\operatorname {div} u=0$ and $\lambda >0$, set

\[ X = X_{\lambda} = \{ \mathscr{M} u \leq \lambda \} \cup \{ \mathscr{M} (D u) \leq \lambda \}. \]

Then the truncated $T u$ defined in definition 3.1 satisfies all the assertions of theorem 1.1 for a.e. $L>0$ with $L={\lambda }/{2}$.

This is of course just a special case of the statement for general dimension below, cf. theorem 3.5. In 3D, however, every computation can also be done in the local coordinates. The proof then roughly proceeds as follows:

1. St.1: Argue that $T_0 u \in W^{1,p}(\mathbb {R}^3;\mathbb {R}^3)$ and that $Su,\,Ru \in W^{1,p}_0(X^C;\mathbb {R}^3)$;

2. St.2: Compute that in $X^C$ we have $\operatorname {div} Tu=0$ (which we can do via pointwise computation).

3. St.3: Show $L^{\infty }$-bounds on $D (Tu)$.

Furthermore, we mention that the treatment of $\operatorname {curl}$-free functions, i.e. closed $1$-forms (lemma 2.8), is also done in a coordinate-wise fashion, cf. formula (2.8). Extending above to arbitrary space dimension, one may get a truncation result for closed two-forms. The goal of the following subsections is to generalize it to arbitrary dimension.

3.2. Intermezzo: differential forms and Stokes’ theorem

We shortly recall some basic notation and properties of differential forms. Taking $\mathbb {R}^N$ as the physical space, $(\mathbb {R}^N)^{\ast }$ denotes its dual and

\[ \Lambda^r = \underbrace{(\mathbb{R}^N)^{{\ast}} \wedge \ldots \wedge (\mathbb{R}^N)^{{\ast}}}_{r \text{ copies}} \]

denotes the $r$-fold wedge product of the dual. Likewise, $V^r$ denotes the $r$-fold wedge product of $\mathbb {R}^N$ with itself. Observe that $\Lambda ^0=\mathbb {R}$, $\Lambda ^1= (\mathbb {R}^N)^{\ast }$ and that $\Lambda ^{N}$ is isomorphic to $\mathbb {R}$. Moreover, $\Lambda ^r$ is the dual of $V^r$ and we write $\omega [\nu ]$ to indicate the dual pairing of $\omega \in \Lambda ^r$ with $\nu \in V^r$.

We call a (smooth) map $u \colon \mathbb {R}^N \to \Lambda ^r$ an $r$-form. Moreover, there is the exterior derivative $d$, that maps $r$-forms into $(r+1)$-forms with the following properties:

1. (i) $d \circ d=0$;

2. (ii) If $\alpha \in C^{\infty }(\mathbb {R}^N;\Lambda ^r)$ and $\beta \in C^{\infty }(\mathbb {R}^N;\Lambda ^s)$, then

   \[ d (\alpha \wedge \beta) = d \alpha \wedge \beta + ({-}1)^r \alpha \wedge d \beta \]

3. (iii) $d \colon C^{\infty }(\mathbb {R}^N;\Lambda ^0) \to C^{\infty }(\mathbb {R}^N;\Lambda ^1)$ is the gradient.

Usually, one may identify a divergence-free function $u \colon \mathbb {R}^N \to \mathbb {R}^N$ with a function $\bar {u} \in \mathbb {R}^N \to \Lambda ^1$ obeying $d^{\ast } u=0$ through seeing the divergence as the adjoint to the gradient. Instead of this, we use a different identification with $(N-1)$-forms. In particular, for the standard basis $e_i$ of $\mathbb {R}^N$ denote by $dx_i$ the map in $(\mathbb {R}^N)^{\ast }$ given by

\[ \sum_{i =1}^N u_i e_i \mapsto u_i. \]

Then the vectors

\[ \omega_i := ({-}1)^i dx_1 \wedge \ldots dx_{i-1} \wedge dx_{i+1} \wedge \ldots \wedge dx_N \]

form a basis of $\Lambda ^{N-1}$ and we may define an isomorphism $\Psi \colon \mathbb {R}^N \to \Lambda ^{N-1}$ that is defined on the standard basis as $e_i \mapsto \omega _i$. Then we have

\[ \operatorname{div} u =0 \quad \Longleftrightarrow \quad d (\Psi(u)) =0 \]

and therefore we routinely identify $u$ with the $(N-1)$-form $\Psi (u)$.

One then can formulate Stokes’ theorem that, for a $(r+1)$-dimensional manifold $M$ links the integral of $d u$ on $M$ to the boundary integral of $u$. Of special interest in the present setting is the case where $M$ is a simplex; to this end recall that for an index $I=(i_1,\,\ldots,\,i_k)$ and $x_{i_1},\,\ldots,\,x_{i_k} \in \mathbb {R}^N$ we denote by

\[ \operatorname{Sim}(x_I) = \operatorname{Sim}(x_{i_1},\ldots,x_{i_k}) := \mathrm{conv}(x_{i_1},...,x_{i_{k}}). \]

If the dimension of $\operatorname {Sim}(x_I)$ is $(k-1)$, we call the simplex non-degenerate. In that case, we define

\[ \nu(x_I) = \tfrac{1}{(k-1)!}(x_{i_1}-x_{i_2}) \wedge \ldots \wedge (x_{i_{k-1}} - x_{i_k}) \in V^{k-1} \]

to be the generalized normal. For any $(k-2)$-dimensional face opposite to $x_{i_l}$, i.e. $\operatorname {Sim}^l(x_I) = \operatorname {Sim}(x_{i_1},\,\ldots,\,x_{i_{l-1}},\,x_{i_{l+1}},\,\ldots,\,x_{i_k})$ we can accordingly define a normal vector $\nu ^l(x_I)$ on this face. Further observe that $\vert \nu (x_I) \vert = \mathcal {H}^{k-1}(\operatorname {Sim}(x_I))$.

Stokes’ theorem on simplices now reads as follows:

(3.7)\begin{equation} {\unicode{x2A0F}}_{\operatorname{Sim}(x_I)} d u [\nu(x_I)] \,{\textrm d}\mathcal{H}^{k-1} = \sum_{l=1}^{k} ({-}1)^l {\unicode{x2A0F}}_{\operatorname{Sim}^l(x_I)} u [\nu^l(x_I)] \,{\textrm d}\mathcal{H}^{k-2}. \end{equation}

3.3. Definition of the truncation for divergence-free functions

Similarly to definition 3.1, we are now able to define the truncation through one part, that corresponds to $W^{1,\infty }$-truncation, and another part consisting of a bunch of corrector terms.

From now on, instead of working with $u \colon \mathbb {R}^N \to \mathbb {R}^N$, through the identification discussed in the previous subsection we instead consider a non-renamed $(N-1)$-form $u \colon \mathbb {R}^N \to \Lambda ^{N-1}$. The condition of solenoidality then coincides with closedness of the form, i.e. that the exterior derivative vanishes.

To define correct analogues of $A_{\alpha \beta }$ and $B$ as in definition 3.1, for $u \colon \mathbb {R}^N \to \Lambda ^{N-1}$ we define the objects

\[ Du [\nu] \quad \text{and} \quad Du [\nu] \cdot (y-z) \]

as follows: We first understand $v=Du$ as an element of $\Lambda ^1 \otimes \Lambda ^{N-1}$ and consider the pairing of the first component of this tensor with some $\nu \in V^r$ as

\begin{align*} (v_1 \otimes v_2) [\nu]& \mapsto (v_1[\nu] \otimes v_2 ) \in V^{r-1} \otimes \Lambda^{N-1}, \\ ((v_1 \otimes v_2) [\nu])(y-z) & \mapsto (v_1[\nu] \wedge (y-z)) \otimes v_2) \in V^{r} \otimes \Lambda^{N-1}. \end{align*}

and then map $V^s \otimes \Lambda ^{N-1}$, $s=r-1,\,r$ to $\Lambda ^{N-1-s}$ through a duality pairing, i.e. finally

\begin{align*} & (v_1 \otimes v_2) [\nu] := v_2\left(v_1(\nu)\right) \in \Lambda^{N-r}, \\ & (v_1 \otimes v_2) [\nu] (y-z):= v_2\left(v_1(\nu) \wedge (y-z) \right) \in \Lambda^{N-r-1}. \end{align*}

Corresponding to definition 3.1, let $X \subset \mathbb {R}^N$ be a closed set and cover its complement with Whitney cubes $Q_i$. Consider a partition of unity $\varphi _i$ on $Q_i$, measures $\mu _i$ and the product measure $\mu _I$ for $I=(i_1,\,\ldots,\,i_r)$ and, for simplicity, write

\[ \mathrm{d}\mu_I := \mathrm{d} \mu_{i_1}(x_{i_1}) \ldots \mathrm{d} \mu_{i_r}(x_{i_r}). \]

Definition 3.3 Formula for divergence-free extension

Let $u \in W^{1,p}(\mathbb {R}^N;\Lambda ^{N-1})$, $X \subset \mathbb {R}^N$ closed. We define

(3.8)\begin{equation} Tu(y) := \begin{cases} T_0 u(y) + \sum_{k=1}^{N-1} S_k u(y) & y \in X^C, \\ u(y) & y \in X, \end{cases} \end{equation}

where

(3.9)\begin{equation} T_0 u(y) = \sum_{i \in \mathbb{N}} \varphi_i(y) \int u(x_i) \,{\textrm d}\mu_i(x_i) \end{equation}

and

(3.10)\begin{align} S_k u(y) & := \tfrac{({-}1)^k(N-k)!}{N!}\sum_{I \in \mathbb{N}^{k+1}} \varphi_{i_{k+1}}(y) \wedge d \varphi_{i_k}(y) \wedge \ldots \wedge d \varphi_{i_1}(y) \wedge A_k(I); \end{align}

(3.11)\begin{align} A_k(I) & := \int {\unicode{x2A0F}}_{\operatorname{Sim}(x_I)} Du[\nu(x_I)](y-z) \,{\textrm d}\mathcal{H}^k(z) \,{\textrm d}\mu(x_I). \end{align}

We define the integral in (3.11) to be zero if the simplex $\operatorname {Sim}(x_I)$ is degenerate.

Observe that the definition of $A_k(I)$ is antisymmetric in the index, i.e. if $\sigma \in S(k+1)$ is a permutation and $\tilde {I}=(i_{\sigma (1)},\,\ldots,\,i_{\sigma (k+1)})$, then

\[ A_k(I) = \mathrm{sgn}(\sigma) A_k(\tilde{I}); \]

and in particular $A_k(I)=0$ if $i_{j_1}=i_{j_2}$ for some $j_1 \neq j_2$. Further, as a minor note, $S_k u$ maps to $\Lambda ^{N-1}$: The wedge product of all $\varphi$'s is an element of $\Lambda ^k$ and, by definition, $A_k(I) \in \Lambda ^{N-k-1}$.

Remark 3.4 Definition 3.1 vs definition 3.3

For clarity, we shortly argue why definition 3.3 is the generalization of the 3D-case, definition 3.1. As mentioned before, we identify $(\mathbb {R}^3)^{\ast } \wedge (\mathbb {R}^3)^{\ast }$ with $\mathbb {R}^3$ in the following way:

\[ e_1 \mapsto dx_2 \wedge dx_3,~e_2 \mapsto dx_3 \wedge dx_1,~e_3 \mapsto dx_1 \wedge dx_2. \]

The terms for $S$, $R$, $A$ and $B$ in definition 3.1 then are nothing else than $S_1$, $S_2$, $A_1$ and $A_2$ in above definition 3.3 written in a coordinate-wise fashion.

The remainder of this section is devoted to prove the following result.

Theorem 3.5 Let $u \in W^{1,p}(\mathbb {R}^N,\,\Lambda ^{N-1})$ with $d u=0$ and $\lambda >0$, set

\[ X = X_{\lambda} = \{\mathscr{M} u \leq \lambda \} \cup \{ \mathscr{M} (Du) \leq \lambda\}. \]

Then the truncated $Tu$ defined in definition 3.3 satisfies all the assertions of theorem 1.1 (with the identification of closed $(N-1)$-forms to divergence-free functions) for a.e. $L>0$ with $L=\lambda /2$.

3.4. Structure of the proof and auxiliary results

In this section, we outline the different steps of the proof. First of all, we remind the reader of the following unconstrained truncation result.

Lemma 3.6 [Reference Zhang35]

Let $X$ be as in theorem 3.5 and $u \in W^{1,p}(\mathbb {R}^N;\Lambda ^{N-1})$. Then $\tilde {u}$ defined as

\[ \tilde{u}(y) := \begin{cases} T_0u(y) & y \in X^C, \\ u(y) & y \in X, \end{cases} \]

satisfies $\tilde {u} \in W^{1,\infty }(\mathbb {R}^N;\Lambda ^{N-1})$ and $\Vert D \tilde {u} \Vert _{W^{1,\infty }} \leq C \lambda$.

Moreover, recall that due to lemma 2.4 we have the following bound on the measure of $X_{\lambda }$.

Lemma 3.7 Let $u \in W^{1,p}(\mathbb {R}^N;\Lambda ^{N-1})$. Then the set $X_{\lambda }$ defined in theorem 3.5 obeys

(3.12)\begin{equation} \mathcal{L}^N(X_{\lambda}) \leq C \lambda^{{-}p} \int_{\{\vert u \vert \geq \lambda/2\} \cup \{ \vert Du \vert \geq \lambda/2 \} } \vert u \vert^p + \vert Du \vert^p \,{\textrm d}x. \end{equation}

It remains to show that the corrector terms $S_1,\,\ldots,\,S_k$ are well-behaved and achieve their goal, i.e. that those terms are bounded in $W^{1,\infty }(\mathbb {R}^N;\Lambda ^{N-1})$ and that together with to $T_0u$ they give a solenoidal vector field. We formulate these properties in two separate lemmas.

The unconstrained truncation $T_0 u$ and all $S_k u$ are smooth functions in the open set $X_{\lambda }^C$. Therefore, to prove that $d T u =0$ globally, it suffices to compute the exterior derivative pointwisely.

Lemma 3.9 On the bad set $X_{\lambda }^C$ we have $Tu \in C^{\infty }(X^C;\Lambda ^{N-1})$ and, moreover, the strong derivative satisfies

\[ d Tu = d T_0 u + \sum_{k=1}^{N-1} d S_k u =0. \]

This lemma is a quite straightforward calculation and depends on certain cancellations and Stokes’ theorem. Before proving lemmas 3.8 and 3.9, we shortly realize that these lead directly to the proof of theorem 3.5.

Proof Proof of theorem 3.5:

Consider $Tu$ as given in (3.8). Lemma 3.6 combined with lemma 3.8 gives that

\[ \sum_{k=1}^{N-1} S_k u \in W^{1,1}_0(X_{\lambda}^C;\Lambda^{N-1}) \]

and therefore $Tu \in W^{1,\infty }(\mathbb {R}^N;\Lambda ^{N-1})$. Furthermore, we obtain the bound $\Vert T u \Vert _{W^{1,\infty }} \leq C \lambda$.

Consequently, to prove that $d (Tu) =0$, it suffices to check that $d (Tu)=0$ pointwise almost everywhere. But we have $d (Tu) (y)=0$ for almost every $y \in X_{\lambda }^{\circ }$, as $Tu =u$ in this (open) set. On the other hand, lemma 3.9 shows that $d (Tu) =0$ for every $y \in (X_{\lambda }^C)$. As for almost every $\lambda >0$ we have $\mathcal {L}^N(\partial X_{\lambda }) =0$, we are finished in showing that $d(Tu)=0$.

It remains to check the bound for the set where $u \neq T u$, i.e. (T2). But $\{ u \neq Tu\} \subset X_{\lambda }^C$ and the measure of the latter is bounded in lemma 3.7.

3.5. $L^{\infty }$-bounds

This section is devoted to the proof of lemma 3.8. Before proving the lemma, shortly recall that $Q_i$ is a cover of the bad set, every point is only contained in finitely many cubes $Q_i$ and that $\varphi _i \in C^{\infty }(Q_i;[0,\,1])$ is a partition of unity.

In order to tackle the terms appearing in the definition of $A_k$ in the appropriate fashion, let us formulate the following lemma.

Lemma 3.10 Let $1\leq k \leq N-1$, $Q_1=[-1/2,\,1/2]^N$, $Q_2,\,\ldots Q_{k+1}$ cubes of sidelength $1/4 \leq l(Q_r) \leq 4$ with $Q_r \subset B_{10}(0)$, $2 \leq r \leq k+1$. Suppose that $v \in L^1(B_{10}(0))$. Then we can estimate

(3.13)\begin{equation} {\unicode{x2A0F}}_{Q_1 \times \ldots \times Q_{k+1}} \int_{\operatorname{Sim}(x_1,\ldots, x_{k+1})} \vert v(z) \vert \,{\textrm d}\mathcal{H}^k(z) \,{\textrm d}x_{k+1} \ldots \,{\textrm d}x_1 \leq C \int_{B_{10}(0)} \vert v(z) \vert ~\mathrm{d}\mathcal{L}^N(z). \end{equation}

Observe that this inequality scales as follows: If $Q_1=[-1/2 \rho,\,1/2 \rho ]$ and $v \in L^1(B_{10\rho }(0))$, then the left-hand-side scales with a factor of $\rho ^k$ and the right-hand-side scales with $\rho ^N$.

It is possible to prove the lemma in various different ways (e.g. by transformation rule or coarea formula), below we use Young's convolution inequality.

Proof Proof of lemma 3.10:

We extend $v$ by zero outside of $B_{10}(0)$. First of all, we rewrite the integral in the simplex by transformation rule

\begin{align*} & ={\unicode{x2A0F}}_{Q_1 \times \ldots \times Q_{k+1}} \int_{\operatorname{Sim}(x_1,\ldots, x_{k+1})} \vert v(z) \vert \,{\textrm d}\mathcal{H}^k(z) \,{\textrm d}x_{k+1} \ldots \,{\textrm d}x_1 \\ & = {\unicode{x2A0F}}_{Q_1 \times \ldots \times Q_{k+1}} \int_{\mathbb{D}_k} \vert v \vert (t_1x_1+\ldots\\ & \quad +t_{k+1}x_{k+1}) \mathcal{H}^k(\operatorname{Sim}(x_1,\ldots,x_{k+1})) \,{\textrm d}t \,{\textrm d}x_{k+1} \ldots \,{\textrm d}x_1, \end{align*}

where $\mathbb {D}_k= \{t \in [0,\,1]^{k+1} \colon t_1+\ldots t_{k+1}=1\}$. As $\mathcal {H}^k(\operatorname {Sim}(x_1,\,\ldots,\,x_{k+1}))$ is uniformly bounded, we can absorb it into a constant. We proceed by estimating the integral over $\mathbb {D}_k^1 = \{t \in \mathbb {D}_k \colon t_1 \geq {1}/({k+1})\}$ instead of $\mathbb {D}_k$, the estimate over the full integral follows from symmetry arguments. We now use Fubini's theorem and write

\begin{align*} & {\unicode{x2A0F}}_{Q_1 \times \ldots \times Q_{k+1}} \int_{\mathbb{D}_k^1} \vert v \vert (t_1x_1+\ldots+t_{k+1}x_{k+1}) \,{\textrm d}t \,{\textrm d}x_{k+1} \ldots \,{\textrm d}x_1 \\ & \quad=\int_{\mathbb{D}_k^1} \int_{Q_1} \int_{Q_2 \times \ldots \times Q_{k+1}} \vert v \vert (t_1 x_1 + t_2x_2 + \ldots t _{k+1} x_{k+1}) \,{\textrm d}x_{k+1} \ldots \,{\textrm d}x_{2} \,{\textrm d}x_1 \,{\textrm d}t \\ & \quad\leq \int_{\mathbb{D}_k^1} \int_{\mathbb{R}^N} \int_{(\mathbb{R}^N)^{k}} \vert v \vert(t_1 x_1+\tilde{x}_2+\ldots \tilde{x}_{k+1}) \prod_{i=2}^{k+1} t_i^{{-}N} 1_{t_iQ_i}(\tilde{x}_i) ~\mathrm{d}\tilde{x}_{k+1} \ldots \mathrm{d}\tilde{x}_{2} \,{\textrm d}x_1 \,{\textrm d}t, \end{align*}

where in this context (and this context only) $t_iQ_i$ is the cube with sidelength $t_il(Q_i)$ and centre $t_ic_i$. Applying Young's convolution inequality yields that

\begin{align*} & {\unicode{x2A0F}}_{Q_1 \times \ldots \times Q_{k+1}} \int_{\mathbb{D}_k^1} \vert v \vert (t_1x_1+\ldots+t_{k+1}x_{k+1}) \,{\textrm d}t \,{\textrm d}x_{k+1} \ldots \,{\textrm d}x_1 \\ & \quad\leq C \int_{\mathbb{D}_k^1} \Vert v(t_1 \cdot) \Vert_{L^1(\mathbb{R}^N)} \leq C \Vert v({\cdot}) \Vert_{L^1(\mathbb{R}^N)}, \end{align*}

where in the last inequality we use that $t_1 \geq {1}/{k+1}$ in $\mathbb {D}_k^1$.

We are now ready to prove the Lipschitz bounds, i.e. lemma 3.8.

Proof Proof of lemma 3.8:

For (i) observe that all $\varphi _i$'s are smooth and therefore any summand is smooth. Hence, as locally the sum is finite, we conclude $Tu \in C^{\infty }(X_{\lambda }^C;\Lambda ^{N-1})$. In particular, only summing over a finite index set (e.g. $I \in \{1,\,\ldots,\,M\}^k$), these sums are contained in $C_c^{\infty }(X_{\lambda };\Lambda ^{N-1})$.

Due to this observation, for (ii), it suffices to prove that the sum converges absolutely in $W^{1,1}(\mathbb {R}^N;\mathbb {R}^N)$. Fix the first index of $I$, i.e. fix $i_1 \in \mathbb {N}$. Then only finitely many terms are nonzero. Moreover, $Q_{i_r} \cap Q_{i_1} \neq \emptyset$ and therefore the sidelength of the cubes are comparable. Therefore,

(3.14)\begin{equation} \vert \varphi_{i_k} \wedge d \varphi_{i_{k-1}} \wedge \ldots \wedge d \varphi_{i_1} \vert \leq C l(Q_{i_1})^{-(k-1)}. \end{equation}

On the other hand, we apply lemma 3.10 to $v = D u[ \nu ] \cdot (y-z)$. Observe that due to the definition of $Q_i$, $20 Q_i \cap X_{\lambda } \neq \emptyset$ and therefore, due to the definition of $X_{\lambda }$ as a sublevel set of the maximal function of $Du$ and the estimate $\vert y- z \vert \leq Cl(Q_i)$,

\[ {\unicode{x2A0F}}_{20Q_i} \vert v \vert \,\mathrm{d} x \leq C \lambda l(Q_i). \]

Therefore,

(3.15)\begin{equation} \vert A_k(I) \vert \leq C \lambda l(Q_{i_1})^k \end{equation}

which leads to

\[ \Vert \varphi_{i_{k+1}} \wedge d \varphi_{i_k} \wedge \ldots \wedge d \varphi_{i_1} \wedge A_k(I) \Vert_{L^{\infty}} \leq C l(Q_{i_1}). \]

The same bounds including the derivative lead to

(3.16)\begin{equation} \Vert D( \varphi_{i_k} \wedge d \varphi_{i_{k-1}} \wedge \ldots \wedge d \varphi_{i_1}) \Vert_{L^{\infty}} \leq C l(Q_{i_1})^{{-}k}, \quad \Vert D_y A_k(I) \Vert_{L^{\infty}} \leq C l(Q_{i_1})^{k-1}, \end{equation}

and we conclude

\[ \left \Vert \varphi_{i_{k+1}}(y) \wedge d \varphi_{i_k}(y) \wedge \ldots \wedge d \varphi_{i_1}(y) \wedge A_k(I) \right \Vert_{W^{1,\infty}} \leq C, \]

and, as this function is supported on $Q_{i_1}$, we obtain

\[ \left \Vert \varphi_{i_{k+1}}(y) \wedge d \varphi_{i_k}(y) \wedge \ldots \wedge d \varphi_{i_1}(y) \wedge A_k(I) \right \Vert_{W^{1,1}} \leq C l(Q_{i_1})^{{-}N}, \]

Again, for only finitely many indices $I$ with this fixed $i_1$, the summand is nonzero, hence for any $j \in \mathbb {N}$

\[ \sum_{I \in \mathbb{N}^k \colon i_1=j} \Vert \varphi_{i_{k+1}}(y) \wedge d \varphi_{i_k}(y) \wedge \ldots \wedge d \varphi_{i_1}(y) \wedge A_k(I) \Vert_{W^{1,1}} \leq Cl(Q_{j})^{{-}N}. \]

Summing over all $j$ and realizing that the $Q_j$'s cover $X_{\lambda }^C$ only a finite number of times, we finally obtain

\[ \sum_{I \in \mathbb{N}^k} \Vert \varphi_{i_{k+1}}(y) \wedge d \varphi_{i_k}(y) \wedge \ldots \wedge d \varphi_{i_1}(y) \wedge A_k(I) \Vert_{W^{1,1}} \!\leq\! \sum_{j \in \mathbb{N}} Cl(Q_{j})^{{-}N} \!\leq\! C \mathcal{L}^N(X_{\lambda}^C). \]

Therefore, the sum converges absolutely in $W^{1,1}_0(X_\lambda ^C)$, i.e. each $S_k u \in W^{1,1}_0(X_\lambda ^C)$.

The $W^{1,\infty }$ bound (iii) now follows from the previously made estimates.

3.6. Calculations of the pointwise divergence/exterior derivative

In this section we show that for each $y \in X_{\lambda }^C$ we have $d Tu (y) =0$. This is pure calculation combined with Stokes’ theorem. We formulate it in the following lemma. For further reference denote by

(3.17)\begin{align} R_k u & := \frac{({-}1)^k(N-k)!}{N!} \sum_{I \in \mathbb{N}^{k+1}} \varphi_{i_{k+1}} \wedge d \varphi_{i_k(y)} \wedge \ldots \wedge d \varphi_{i_1} \wedge B_k(I) , \end{align}

(3.18)\begin{align} B_k(I) & := \int {\unicode{x2A0F}}_{\operatorname{Sim}(x_I)} Du[\nu(x_I)] \,{\textrm d}\mathcal{H}^k(z) \,{\textrm d}\mu_I(x_I). \end{align}

Observe that lemma 3.9 instantly follows by summing over parts (i)–(ii).

Proof. We start with the exterior derivative of $T_0 u$, i.e. (i). As the only term depending on the variable $y$ is $\varphi _i$, we obtain for the derivative with respect to $y$

\[ dT_0 u(y) = \sum_{i \in \mathbb{N}} d \varphi_i(y) \wedge \int u(x_i) \,{\textrm d}\mu_i(x_i). \]

Using that $\varphi _i$ is a partition of unity, i.e. $\sum _{j \in \mathbb {N}} \varphi _j=1$ and $\sum _{i \in \mathbb {N}} d \varphi _i =0$ in the interior of $X^C$, we obtain:

\[ dT_0 u (y) = \sum_{i,j \in \mathbb{N}} \varphi_j \wedge d \varphi_i \wedge \left(\int u(x_i) \,{\textrm d}\mu_i(x_i) - \int u(x_j) \,{\textrm d}\mu_j(x_j) \right). \]

Application of the fundamental theorem of calculus (i.e. one-dimensional Stokes’ theorem) component-wise yields

\[ dT_0 u(y) ={-} R_1 u. \]

A similar trick leads to the second statement, (ii). Observe that, in contrast to statement (i), $A_k(I)$ does depend on $y$, hence

(3.19)\begin{equation} \begin{aligned} dS_ku & = \frac{({-}1)^k(N-k)!}{N!} \sum_{I \in \mathbb{N}^{k+1}} d \varphi_{i_{k+1}}(y) \wedge \ldots \wedge d \varphi_{i_1}(y) \wedge A_k(I) \\ & \quad+ \frac{({-}1)^k(N-k)!}{N!} \sum_{I \in \mathbb{N}^{k+1}} \varphi_{i_{k+1}}(y) \wedge \varphi_{i_k}(y) \ldots \wedge d\varphi_{i_1}(y) \wedge d A_k(I) \end{aligned} \end{equation}

Now a computation (it, for instance, suffices to check this coordinate-wise and while assuming that $Du[\nu (x_I)] = dx_1 \wedge \ldots \wedge dx_{N-k}$) reveals that

\[ d A_k(I) = (N-k) B_k(I). \]

Hence, the second sum in (3.19) equals $R_k u$. For the first term, we observe as $\varphi _{i_r}$ is a partition of unity on $X^C$

(3.20)\begin{equation} \begin{aligned} & \frac{({-}1)^k(N-k)!}{N!} \sum_{I \in \mathbb{N}^{k+1}} d \varphi_{i_{k+1}}(y) \wedge \ldots \wedge d \varphi_{i_1}(y) \wedge A_k(I) \\ & \quad= \frac{({-}1)^k(N-k)!}{N!} \sum_{\tilde{I} \in \mathbb{N}^{k+2}} \varphi_{i_{k+2}}(y) d \varphi_{i_{k+1}}(y) \wedge \ldots \wedge d \varphi_{i_1}(y) \wedge \bigl(A_k(\tilde{I}_{k+2})\\ & \quad - \sum_{r=1}^{k+1} A_k(\tilde{I}_r)\bigr), \end{aligned} \end{equation}

where $\tilde {I} =(i_1,\,\ldots,\,i_{k+2}) \in \mathbb {N}^{k+2}$ and $\tilde {I}_r=(i_1,\,\ldots,\,i_{r-1},\,i_{r+1},\,\ldots,\,i_{k+2})$. Recall that

\[ A_k(\tilde{I}_r) = \int {\unicode{x2A0F}}_{\operatorname{Sim}(x_{\tilde{I}_r})} Du[\nu(x_{\tilde{I}_r})] (y-z) \,{\textrm d}\mathcal{H}^k(z) \,{\textrm d}\mu_{\tilde{I}_r}(x_{\tilde{I}_r}). \]

Thus, the terms in (3.20) form a boundary integral of a $(k+1)$-dimensional simplex. After a suitable identification of spaces (i.e. identifying a wedge product space $\Lambda ^{N-1}$ with suitable (skew-symmetric) elements in $\Lambda ^k \otimes \Lambda ^{N-1-k}$), we can apply Stokes’ theorem by writing

\begin{align*} & \int {\unicode{x2A0F}}_{\operatorname{Sim}(x_{\tilde{I}_r})} Du[\nu(x_{\tilde{I}_r})] (y-z) \,{\textrm d}\mathcal{H}^k(z) \,{\textrm d}\mu_{\tilde{I}_r}(x_{\tilde{I}_r})\\ & \quad = \int {\unicode{x2A0F}}_{\operatorname{Sim}(x_{\tilde{I}_r})} F(y,z) [\nu(x_{\tilde{I}_r}] \,{\textrm d}\mathcal{H}^k(z) \,{\textrm d}\mu_{\tilde{I}_r} (x_{\tilde{I}_r}) \end{align*}

for suitable $F\colon \mathbb {R}^N \to \Lambda ^k \otimes \Lambda ^{N-k-1}$. Applying Stokes’ theorem componentwise, one obtains

\begin{align*} & \frac{({-}1)^k(N-k)!}{(N-1)!} \sum_{I \in \mathbb{N}^{k+1}} d \varphi_{i_{k+1}}(y) \wedge \ldots \wedge d \varphi_{i_1}(y) \wedge A_k(I) \\ & \quad=\frac{({-}1)^k(N-k)!}{(N-1)!} \sum_{\tilde{I} \in \mathbb{N}^{k+2}} \varphi_{i_{k+2}}(y)\wedge \ldots \wedge d \varphi_{i_1}(y)\\ & \quad\wedge \int {\unicode{x2A0F}} d_zF(y,z)[\nu(x_{\tilde{I}})] \,{\textrm d}\mathcal{H}^{k+1}(z) \,{\textrm d}\mu_{\tilde{I}}(x_{\tilde{I}}). \end{align*}

Again, a for instance coordinate-wise computation (using that $du=0$) yields that the latter integral is exactly $-(N-k)B_{k+1}(\tilde {I})$.

For (iii) the calculation is exactly the same. It suffices to see that $R_N u =0$, i.e. that $B_N(I)=0$. Note that for any index $I$

\[ B_N(I) = \int {\unicode{x2A0F}}_{\operatorname{Sim}(x_I)} Du[\nu(x_I)] \,{\textrm d}\mathcal{H}^N(z) \,{\textrm d}\mu_I(x_I). \]

Considering the definition of $Du[\nu (x_I)]$ for $\nu (x_I) \in V^N \simeq \mathbb {R}$, we obtain $Du[\nu (x_I)] = du (\nu (x_I))$. As $u$ is solenoidal, $du=0$ and hence $B_N(I)=0$. Therefore $R_{N}=0$, finishing the proof.

4.1. Truncation for closed differential forms

The solenoidal truncation is achieved by identifying functions $u \in W^{1,p}(\mathbb {R}^N;\mathbb {R}^N)$ with a non-renamed closed form $u \in W^{1,p}(\mathbb {R}^N;\Lambda ^{N-1})$. Via careful construction of the corrector terms using the structure of differential forms we then obtained a solenoidal trunction. This construction with the same arguments can also be repeated if $u \in W^{1,p}(\mathbb {R}^N;\Lambda ^{N-1})$ is replaced by a lower-order form, i.e. $u \in W^{1,p}(\mathbb {R}^N;\Lambda ^r)$. In particular, we get the following result.

As the proof is very similar to the proof in § 3 we only give a brief sketch. We take the same setup as in § 3.3, i.e. we take the bad set as a sublevel set of the maximal function, cover it with cubes $Q_i$, consider a partition of unity $\varphi _i$ and measures $\mu _i$.

Proof Proof(Sketch):

Parallel to the beginning of § 3.3, for $u \colon \mathbb {R}^N \to \Lambda ^{r}$ and a normal $\nu \in V^s$, $1 \leq s \leq r$, we may give a meaning to

\[ Du[\nu] \quad \text{and} \quad Du[\nu]\cdot (y-z) \]

by understanding $Du$ as an element of $\Lambda ^1 \otimes \Lambda ^{r}$. Then,

\[ Du[\nu] \in \Lambda^{r+1-s} \quad \text{and} \quad Du[\nu] \cdot (y-z) \in \Lambda^{r-s}. \]

Now as in definition 3.3, consider

(4.1)\begin{equation} Tu(y) := \begin{cases} T_0(y) + \sum_{k=1}^r S_k^r u(y) & y \in X^C, \\ u(y) & y \in X, \end{cases} \end{equation}

where again $T_0 u$ is the usual Lipschitz extension

\[ T_0u(y) = \sum_{i \in \mathbb{N}} \varphi_i(y) \int u(x_i) \,{\textrm d}\mu_i(x_i) \]

and

(4.2)\begin{align} S_k^r u(y) & := \frac{({-}1)^k(r+1-k)!}{(r+1)!}\sum_{I \in \mathbb{N}^{k+1}} \varphi_{i_{k+1}}(y) \wedge d \varphi_{i_k}(y) \wedge \ldots \wedge d \varphi_{i_1}(y) \wedge A_k^r(I); \end{align}

(4.3)\begin{align} A_k^r(I) & := \int {\unicode{x2A0F}}_{\operatorname{Sim}(x_I)} Du[\nu(x_I)](y-z) \,{\textrm d}\mathcal{H}^k(z) \,{\textrm d}\mu(x_I). \end{align}

Define the integral in (4.3) to be zero if the simplex $\operatorname {Sim}(x_I)$ is degenerate.

We claim that with $X=X_{\lambda }= \{ M u \leq \lambda \} \cup \{\mathscr {M} (Du) \leq \lambda \}$ and $\lambda =L/2$, $Tu$ obeys the properties of the proposition.

To show this claim, might argue exactly as in § 3: First of all, parallel to lemma 3.6, $T_0$ is a sufficient unconstrained truncation and the measure of the bad set is also already bounded; lemma 3.7 still works as it never really used the structure of the target space $\Lambda ^{N-1}$.

Furthermore, by using the same calculation, we can show the natural counterpart to lemma 3.8 (the Lipschitz bounds). Finally, we may show the analogue of lemma 3.11, i.e. if for $1 \leq k \leq r$

\begin{align*} R_k^r u & := \frac{({-}1)^k(r+1-k)!}{(r+1)!} \sum_{I \in \mathbb{N}^{k+1}} \varphi_{i_{k+1}} \wedge d \varphi_{i_k(y)} \wedge \ldots \wedge d \varphi_{i_1} \wedge B_k^r(I), \\ B_k(I) & := \int {\unicode{x2A0F}}_{\operatorname{Sim}(x_I)} Du[\nu(x_I)] \,{\textrm d}\mathcal{H}^k(z) \,{\textrm d}\mu_I(x_I). \end{align*}

then for any $1 \leq k \leq r-1$ we have

\[ dT_0 ={-}R_1^r u, \quad dS_k^r u= R_k u - R_{k+1}^r u, \quad dS_r^r u = R_{r}^r u. \]

As seen in § 3, these statements then complete the proof, as we can show the Lipschitz bound, the bound on the set $\{ u \neq Tu\}$ and closedness of the differential form $Tu$.

4.2. General differential constraints

Generalizing above question, taking some $u \in W^{1,p}(\mathbb {R}^N;\mathbb {R}^d)$, one may ask to truncate under the first-order differential constraint

\[ \mathscr{A} u =0, \]

where

\[ \mathscr{A} u = \sum_{\alpha=1}^N A_{\alpha} \partial_{\alpha} u, \quad A_{\alpha} \in \operatorname{Lin}(\mathbb{R}^d;\mathbb{R}^m). \]

Unconstrained truncation corresponds to $\mathscr {A} =0$ and the case of differential forms is represented by $\mathscr {A} = d$ and, in particular, closed $(N-1)$ forms by $\mathscr {A}=\operatorname {div}$. While this general case is out of reach with the results of this paper, the technique of constructing corrector terms does not seem to be entirely hopeless. Meanwhile, similar to [Reference Behn, Gmeineder and Schiffer5, Conjecture 6.4], we conjecture that a truncation result á la 1.1 is possible for some $\mathscr {A}$, whenever $\mathscr {A}$ satisfies the complex constant rank condition (i.e. the Fourier symbol has constant rank over all complex Fourier modes).

4.3. Lower regularity

Corresponding to the $W^{1,p}$-case one might ask to truncate divergence-free $L^p$ functions to be in $L^{\infty }$ (cf. [Reference Schiffer32]). While these questions turn out to be quite different, they both fit in a larger common framework. In our setting of divergence-free $W^{1,p}$-truncation, consider $w=Du$. The solenoidal Lipschitz truncation now corresponds to a $L^{\infty }$-truncation of $w$ under the following constraints

\[ \mathrm{tr}(w)=0, \quad \operatorname{curl} w =0, \quad \operatorname{div}^T w=0, \]

i.e. $\operatorname {curl}$ is taken column-wise and $\operatorname {div}$ row-wise. This can be abstractly set as $\mathscr {B} w=0$. Consequently, Lipschitz truncation of a function $u$ under a constraint $\mathscr {A} u=0$ is closely connected to the $L^{\infty }$-truncation of $w$ ($=Du$) under some constraint $\mathscr {B} w=0$. As mentioned before, the latter has been examined in [Reference Behn, Gmeineder and Schiffer5, Reference Schiffer32] in greater detail; in particular (as above) the authors in [Reference Behn, Gmeineder and Schiffer5] have conjectured that a low-regularity truncation is possible whenever $\mathscr {B}$ obeys the complex constant rank condition.

4.4. Solenoidal extensions for $W^{1,\infty }$

As alluded to in § 2.2, extensions and truncations are connected from a technical perspective. Indeed, in the unconstrained setting, truncation is derived by extending the function of a cleverly chosen subset to the full space. As it becomes visible through the formula for the truncation (3.8), the same analogy is not true for constrained truncation and extension. In particular, all the approaches to constrained truncation (cf. [Reference Breit, Diening and Fuchs7, Reference Breit, Diening and Schwarzacher9] etc. and (3.8)) use to a certain extent that the function is already defined on the bad set. Therefore, constructing an extension is actually more challenging than constructing a truncation, and while certain key ideas stay the same, there are a lot of challenges coming from the geometric properties of the underlying domain. We refer to the forthcoming article [Reference Gmeineder and Schiffer23] for a thorough investigation of solenoidal extension, especially for the borderline cases $p=1$ and $p=\infty$ (also cf. [Reference Acosta, Durán and Muschietti3, Reference Kato, Mitrea, Ponce and Taylor24] for $1< p<\infty$).