[proofplan] The trace operator is constructed as the unique **strictly-continuous extension** of the classical restriction map $u \mapsto u|_{\partial\Omega}$ from $C^\infty(\overline\Omega) \cap BV(\Omega)$ to all of $BV(\Omega)$. The relevant topology on $BV(\Omega)$ is the *strict topology*: $u_j \to u$ strictly if $u_j \to u$ in $L^1(\Omega)$ and $|Du_j|(\Omega) \to |Du|(\Omega)$. In this topology, smooth functions are dense (theorem 3131), but the topology is genuinely weaker than BV-norm convergence — when $u$ has a non-trivial singular part, no smooth sequence converges to $u$ in BV norm, so one cannot apply the smooth-class inequality to differences $u_j - u_l$ to conclude $L^1(\partial\Omega)$-Cauchyness directly. The key analytical input is an $L^1$-boundary estimate of the form $\int_{\partial\Omega} |u| \, d\mathcal{H}^{n-1} \le C(\|u\|_{L^1(\Omega)} + |Du|(\Omega))$, valid for smooth functions, obtained by flattening the boundary in a finite Lipschitz atlas and applying the fundamental theorem of calculus along vertical lines in each chart, with the change of variables performed via the area formula for bi-Lipschitz maps. To extend the smooth-class estimate to all of $BV(\Omega)$ we use a *specific* strict-approximating sequence: chart-by-chart, pull $u$ back to the half-space, vertically translate by a small height $h_j > 0$ so the boundary value at $\{x_n = 0\}$ becomes the interior slice $w_k(\cdot, h_j)$, then mollify at a smaller scale $\varepsilon_j < h_j / 2$ to smoothen. The Cauchyness of the resulting boundary slices is established directly by the Fubini decomposition of the BV measure $D_{x_n} w_k$ along the normal direction: the difference $\|w_k(\cdot, h_j) - w_k(\cdot, h_l)\|_{L^1}$ is bounded by $|D_{x_n} w_k|(\{h_l < x_n < h_j\})$, which vanishes as $h_j, h_l \to 0$ by countable additivity of the finite measure $|D_{x_n} w_k|$. This produces a strictly-continuous extension $T: BV(\Omega) \to L^1(\partial\Omega; \mathcal{H}^{n-1})$. The smooth-class bound persists in the strict-continuous limit, yielding the BV-norm operator estimate $\|Tu\|_{L^1(\partial\Omega)} \le C_\Omega \|u\|_{BV(\Omega)}$ as a *corollary* of strict continuity (since BV-norm convergence implies strict convergence). Linearity, agreement on continuous functions, and uniqueness within the class of strictly-continuous extensions are then verified separately; uniqueness rests on the standard fact that two strictly-continuous extensions agreeing on a strictly-dense subset coincide. [/proofplan] [step:Reduce the boundary estimate to a half-space model via a finite Lipschitz atlas] Since $\partial\Omega$ is Lipschitz, there exists a finite open cover $\{V_k\}_{k=1}^N$ of $\partial\Omega$ such that for each $k$ there is a bi-Lipschitz map $\gamma_k: V_k \to W_k \subseteq \mathbb{R}^n$ that flattens $\partial\Omega \cap V_k$ to a portion of the hyperplane $\{x_n = 0\}$ inside the upper half-space $\mathbb{R}^n_+ = \{x_n > 0\}$. Choose an additional open set $V_0$ with $\overline{V_0} \subset \Omega$ such that $\{V_k\}_{k=0}^N$ covers $\overline{\Omega}$, and let $\{\psi_k\}_{k=0}^N$ be a smooth partition of unity subordinate to this cover. For $u \in C^\infty(\overline{\Omega}) \cap BV(\Omega)$ and $k \ge 1$, write $u_k := u\psi_k$; the support of $u_k$ meets $\partial\Omega$ only inside $V_k$, and $\int_{\partial\Omega} |u| \, d\mathcal{H}^{n-1} \le \sum_{k=1}^N \int_{\partial\Omega} |u_k| \, d\mathcal{H}^{n-1}$ since $u = \sum_{k=0}^N u_k$ and $\psi_0$ has support compactly contained in $\Omega$, hence vanishes on $\partial\Omega$. [guided] The estimate $\int_{\partial\Omega} |u| \, d\mathcal{H}^{n-1} \le C(\|u\|_{L^1} + |Du|(\Omega))$ is a global statement on $\Omega$, but the boundary $\partial\Omega$ is curved Lipschitz geometry. The standard reduction is to a finite atlas of charts in which $\partial\Omega$ is locally a graph; under each chart the estimate becomes a half-space inequality. The Lipschitz hypothesis on $\partial\Omega$ is used here in a specific way: it provides a finite open cover $\{V_k\}_{k=1}^N$ of $\partial\Omega$ together with bi-Lipschitz maps $\gamma_k: V_k \to W_k \subseteq \mathbb{R}^n$ such that \begin{align*} \gamma_k(V_k \cap \Omega) &= W_k \cap \mathbb{R}^n_+, \\ \gamma_k(V_k \cap \partial\Omega) &= W_k \cap \{x_n = 0\}. \end{align*} This is the unpacking of "$\partial\Omega$ is Lipschitz" as locally the graph of a Lipschitz function. The Lipschitz constants of $\gamma_k$ and $\gamma_k^{-1}$ are uniformly bounded in $k$ since the cover is finite, and $\mathcal{H}^{n-1}(\partial\Omega) < \infty$ for the same reason. We complete the cover of $\overline{\Omega}$ by an interior set $V_0$ with $\overline{V_0} \subset \Omega$, so that $\{V_k\}_{k=0}^N$ is a finite open cover of the compact set $\overline{\Omega}$. Existence of a smooth partition of unity $\{\psi_k\}_{k=0}^N$ subordinate to this cover is standard. For $u \in C^\infty(\overline{\Omega}) \cap BV(\Omega)$, we decompose $u = \sum_{k=0}^N u\psi_k$. Setting $u_k := u\psi_k$, each $u_k$ is smooth with $\operatorname{supp}(u_k) \subseteq V_k \cap \overline{\Omega}$. For $k = 0$, $\operatorname{supp}(u_0) \subseteq \overline{V_0} \subset \Omega$, so $u_0 \equiv 0$ on $\partial\Omega$. For $k \ge 1$, $u_k$ is supported in $V_k \cap \overline{\Omega}$, and on $\partial\Omega$, \begin{align*} |u| = \left|\sum_{k=1}^N u_k\right| \le \sum_{k=1}^N |u_k|. \end{align*} Integrating over $\partial\Omega$ with respect to $\mathcal{H}^{n-1}$ produces the localised bound. The point of the localisation is that it suffices to prove the boundary estimate for each $u_k$ separately; summing $N$ such estimates gives the global bound. [/guided] [/step] [step:Prove the boundary $L^1$ estimate on the half-space model via FTC and Tonelli] In the half-space model, the boundary estimate for smooth $v \in C^\infty(\overline{\mathbb{R}^n_+}) \cap BV(\mathbb{R}^n_+)$ with compact support reads \begin{align*} \int_{\{x_n = 0\}} |v| \, d\mathcal{H}^{n-1} \le \int_{\mathbb{R}^n_+} |\partial_{x_n} v| \, d\mathcal{L}^n. \end{align*} This is obtained by applying the fundamental theorem of calculus along vertical lines: for $\mathcal{L}^{n-1}$-a.e. $x' \in \mathbb{R}^{n-1}$ and any $x_n > 0$, \begin{align*} v(x', 0) = -\int_0^{x_n} \partial_{x_n} v(x', t) \, d\mathcal{L}^1(t) + v(x', x_n), \end{align*} which by compact support and letting $x_n \to \infty$ along the support yields $|v(x', 0)| \le \int_0^\infty |\partial_{x_n} v(x', t)| \, d\mathcal{L}^1(t)$. Integrating in $x' \in \mathbb{R}^{n-1}$ via Tonelli's theorem (valid since the integrand $|\partial_{x_n} v|$ is nonnegative and measurable) gives the displayed inequality. [guided] We need to control $\int_{\partial\Omega} |u| \, d\mathcal{H}^{n-1}$ by an interior quantity. The cleanest model is the upper half-space $\mathbb{R}^n_+ = \{x_n > 0\}$ with smooth $v$ vanishing for $x_n$ large. The fundamental theorem of calculus, applied to the smooth function $v$ along the vertical line $t \mapsto v(x', t)$ for fixed $x' \in \mathbb{R}^{n-1}$, gives \begin{align*} v(x', 0) - v(x', T) = -\int_0^T \partial_{x_n} v(x', t) \, d\mathcal{L}^1(t) \end{align*} for any $T > 0$. Since $\operatorname{supp}(v) \cap \overline{\mathbb{R}^n_+}$ is compact, we may pick $T > 0$ large so that $v(x', T) = 0$, giving \begin{align*} v(x', 0) = -\int_0^T \partial_{x_n} v(x', t) \, d\mathcal{L}^1(t), \end{align*} hence $|v(x', 0)| \le \int_0^\infty |\partial_{x_n} v(x', t)| \, d\mathcal{L}^1(t)$. Integrating in $x'$, since $|\partial_{x_n} v|$ is nonnegative and measurable on $\mathbb{R}^n_+$, Tonelli's theorem allows iterated integration without an a priori integrability check on the inner integral: \begin{align*} \int_{\mathbb{R}^{n-1}} |v(x', 0)| \, d\mathcal{H}^{n-1}(x') &\le \int_{\mathbb{R}^{n-1}} \int_0^\infty |\partial_{x_n} v(x', t)| \, d\mathcal{L}^1(t) \, d\mathcal{H}^{n-1}(x') \\ &= \int_{\mathbb{R}^n_+} |\partial_{x_n} v| \, d\mathcal{L}^n, \end{align*} where on $\{x_n = 0\}$ we identified $\mathcal{H}^{n-1}$ with the Lebesgue measure $\mathcal{L}^{n-1}$ on $\mathbb{R}^{n-1}$ via the canonical projection. The right-hand side is finite because $v \in BV(\mathbb{R}^n_+)$ implies $\partial_{x_n} v \in L^1$. This is the half-space inequality. [/guided] [/step] [step:Pull the half-space estimate back to the chart via the area formula] For each $k \ge 1$, define the bi-Lipschitz constant \begin{align*} L_k := \max\big(\operatorname{Lip}(\gamma_k),\, \operatorname{Lip}(\gamma_k^{-1})\big), \end{align*} so that both $\gamma_k$ and $\gamma_k^{-1}$ are $L_k$-Lipschitz. Set $v_k := u_k \circ \gamma_k^{-1}: W_k \cap \overline{\mathbb{R}^n_+} \to \mathbb{R}$. Since $u_k$ is smooth on $\overline{\Omega}$ and $\gamma_k^{-1}$ is bi-Lipschitz, $v_k$ is Lipschitz on $W_k \cap \overline{\mathbb{R}^n_+}$ with compact support, hence in $W^{1,1}$ and a fortiori in $BV$. Apply the previous step's half-space inequality to $v_k$: \begin{align*} \int_{W_k \cap \{x_n = 0\}} |v_k| \, d\mathcal{H}^{n-1} \le \int_{W_k \cap \mathbb{R}^n_+} |\partial_{x_n} v_k| \, d\mathcal{L}^n. \end{align*} The change of variables back to $\Omega$ uses the **area formula for bi-Lipschitz maps**: for a Lipschitz map $\Phi: A \to B$ between subsets of $\mathbb{R}^m$ and any non-negative Borel function $f$, \begin{align*} \int_A f(\Phi(x)) J\Phi(x) \, d\mathcal{H}^m(x) = \int_B f(y) \, \mathcal{H}^0(\Phi^{-1}(y)) \, d\mathcal{H}^m(y), \end{align*} where $J\Phi$ is the Jacobian of $\Phi$, which exists $\mathcal{H}^m$-a.e. by Rademacher's theorem and satisfies $J\Phi(x) \le \operatorname{Lip}(\Phi)^m$. With $L_k$ as defined above, both maps $\gamma_k$ and $\gamma_k^{-1}$ have Lipschitz constant $\le L_k$, giving the explicit Jacobian bounds \begin{align*} J\gamma_k(x) \le L_k^n, \qquad J\gamma_k^{-1}(y) \le L_k^n \qquad \text{(volume Jacobians)}, \end{align*} and on the boundary slice $\{y_n = 0\}$ (an $(n-1)$-dimensional set), \begin{align*} J\gamma_k^{-1}|_{\{y_n=0\}}(y) \le L_k^{n-1}. \end{align*} Applied to $\gamma_k^{-1}$ on $W_k \cap \{x_n = 0\}$ (a subset of an $(n-1)$-plane mapped onto $V_k \cap \partial\Omega$, with $\mathcal{H}^0$-fibre equal to $1$ since $\gamma_k$ is bijective): \begin{align*} \int_{V_k \cap \partial\Omega} |u_k| \, d\mathcal{H}^{n-1} = \int_{W_k \cap \{x_n=0\}} |v_k(y)| \, J\gamma_k^{-1}|_{\{y_n=0\}}(y) \, d\mathcal{H}^{n-1}(y) \le L_k^{n-1} \int_{W_k \cap \{x_n=0\}} |v_k| \, d\mathcal{H}^{n-1}. \end{align*} The chain rule for Lipschitz maps gives $|\partial_{x_n} v_k(y)| \le L_k |\nabla u_k(\gamma_k^{-1}(y))|$ at points where both are differentiable (Rademacher); the factor $L_k$ here is the bound $|\partial_{y_n} \gamma_k^{-1}| \le \operatorname{Lip}(\gamma_k^{-1}) \le L_k$. The area formula for $\gamma_k$ acting on the $n$-dimensional volume integrand $|\nabla u_k(\gamma_k^{-1}(y))|$ yields \begin{align*} \int_{W_k \cap \mathbb{R}^n_+} |\partial_{x_n} v_k| \, d\mathcal{L}^n \le L_k \int_{W_k \cap \mathbb{R}^n_+} |\nabla u_k(\gamma_k^{-1}(y))| \, d\mathcal{L}^n(y) = L_k \int_{V_k \cap \Omega} |\nabla u_k(x)| \, J\gamma_k(x) \, d\mathcal{L}^n(x) \le L_k \cdot L_k^n \int_{V_k \cap \Omega} |\nabla u_k| \, d\mathcal{L}^n, \end{align*} the second equality from the area formula applied to $\gamma_k$ (one-to-one onto $W_k \cap \mathbb{R}^n_+$), the final inequality from $J\gamma_k \le L_k^n$. Chaining the three bounds gives an explicit total constant \begin{align*} \underbrace{L_k^{n-1}}_{\text{(boundary Jacobian)}} \cdot \underbrace{L_k}_{\text{(chain rule)}} \cdot \underbrace{L_k^n}_{\text{(volume Jacobian)}} = L_k^{2n}. \end{align*} Setting \begin{align*} C_k := L_k^{2n}, \end{align*} we obtain \begin{align*} \int_{V_k \cap \partial\Omega} |u_k| \, d\mathcal{H}^{n-1} \le C_k \int_{V_k \cap \Omega} |\nabla u_k| \, d\mathcal{L}^n \le C_k \, |Du_k|(\Omega), \end{align*} where the second step uses that for smooth $u_k \in C^\infty(\overline{\Omega})$ the total variation $|Du_k|(\Omega) = \int_\Omega |\nabla u_k| \, d\mathcal{L}^n$. [guided] The half-space inequality of Step 2 lives on $\mathbb{R}^n_+$; we need its image under $\gamma_k$ on $\Omega$. The transfer is done by the **area formula for Lipschitz maps**, which is the rigorous tool for changing variables when the map is only Lipschitz (not $C^1$). The area formula states that for a Lipschitz $\Phi$ between Euclidean subsets of dimension $m$, \begin{align*} \int_A f \circ \Phi \cdot J\Phi \, d\mathcal{H}^m = \int_B f \cdot \#\Phi^{-1} \, d\mathcal{H}^m, \end{align*} where $J\Phi$ is the (a.e.-defined, by Rademacher's theorem) Jacobian and $\#\Phi^{-1}(y) = \mathcal{H}^0(\Phi^{-1}(y))$ counts pre-images. For our $\gamma_k$ which are bi-Lipschitz bijections, $\#\gamma_k^{-1} \equiv 1$, and we have the two-sided Jacobian bound \begin{align*} L_k^{-m} \le J\gamma_k(x) \le L_k^m, \qquad L_k^{-m} \le J\gamma_k^{-1}(y) \le L_k^m, \end{align*} where $L_k = \max(\operatorname{Lip}(\gamma_k), \operatorname{Lip}(\gamma_k^{-1}))$, applied with $m = n$ for the bulk and $m = n-1$ for the boundary slice. Apply the area formula to transfer the boundary integral. With $v_k := u_k \circ \gamma_k^{-1}$, the surface integral on $V_k \cap \partial\Omega$ pulls back to one on $W_k \cap \{y_n = 0\}$: \begin{align*} \int_{V_k \cap \partial\Omega} |u_k(x)| \, d\mathcal{H}^{n-1}(x) = \int_{W_k \cap \{y_n = 0\}} |v_k(y)| \, J\gamma_k^{-1}|_{\{y_n=0\}}(y) \, d\mathcal{H}^{n-1}(y). \end{align*} Bounding $J\gamma_k^{-1}|_{\{y_n=0\}} \le L_k^{n-1}$ pointwise: \begin{align*} \int_{V_k \cap \partial\Omega} |u_k| \, d\mathcal{H}^{n-1} \le L_k^{n-1} \int_{W_k \cap \{y_n = 0\}} |v_k| \, d\mathcal{H}^{n-1}. \end{align*} The half-space inequality from Step 2 then gives \begin{align*} \int_{W_k \cap \{y_n=0\}} |v_k| \, d\mathcal{H}^{n-1} \le \int_{W_k \cap \mathbb{R}^n_+} |\partial_{y_n} v_k| \, d\mathcal{L}^n. \end{align*} For the bulk integral, the chain rule for Lipschitz functions (valid a.e. by Rademacher) yields $|\partial_{y_n} v_k(y)| = |\partial_{y_n}(u_k \circ \gamma_k^{-1})(y)| \le L_k |\nabla u_k(\gamma_k^{-1}(y))|$ since $|\partial_{y_n} \gamma_k^{-1}| \le L_k$. Pulling back via the area formula on the $n$-dimensional bulk: \begin{align*} \int_{W_k \cap \mathbb{R}^n_+} |\partial_{y_n} v_k(y)| \, d\mathcal{L}^n(y) &\le L_k \int_{W_k \cap \mathbb{R}^n_+} |\nabla u_k(\gamma_k^{-1}(y))| \, d\mathcal{L}^n(y) \\ &= L_k \int_{V_k \cap \Omega} |\nabla u_k(x)| \, J\gamma_k(x) \, d\mathcal{L}^n(x) \\ &\le L_k \cdot L_k^n \int_{V_k \cap \Omega} |\nabla u_k| \, d\mathcal{L}^n. \end{align*} Combining the three bounds with $C_k := L_k^{2n}$ (a finite constant depending only on the Lipschitz constants of the chart) and using $|Du_k|(\Omega) = \int_\Omega |\nabla u_k| \, d\mathcal{L}^n$ for smooth $u_k$: \begin{align*} \int_{V_k \cap \partial\Omega} |u_k| \, d\mathcal{H}^{n-1} \le C_k |Du_k|(\Omega). \end{align*} The constant $C_k$ depends only on $n$ and $L_k = \max(\operatorname{Lip}(\gamma_k), \operatorname{Lip}(\gamma_k^{-1}))$. [/guided] [/step] [step:Globalise via the partition of unity to obtain the smooth-class boundary estimate] Summing over $k = 1, \dots, N$ and using $\nabla u_k = \psi_k \nabla u + u \nabla \psi_k$ for smooth $u$, \begin{align*} |Du_k|(\Omega) = \int_\Omega |\nabla u_k| \, d\mathcal{L}^n \le \|\psi_k\|_\infty \int_\Omega |\nabla u| \, d\mathcal{L}^n + \|\nabla \psi_k\|_\infty \int_\Omega |u| \, d\mathcal{L}^n. \end{align*} Set \begin{align*} C_\Omega := N \cdot \max_{1 \le k \le N} C_k \cdot \max_{1 \le k \le N} \max(\|\psi_k\|_\infty, \|\nabla \psi_k\|_\infty), \end{align*} a finite constant depending only on $\Omega$ (through the chart constants $C_k$, the cover size $N$, and the partition-of-unity bounds). Combining the previous step's per-chart estimate with the product-rule bound on $|Du_k|(\Omega)$ and summing gives, for every $u \in C^\infty(\overline{\Omega}) \cap BV(\Omega)$, \begin{align*} \int_{\partial\Omega} |u| \, d\mathcal{H}^{n-1} \le C_\Omega \big(\|u\|_{L^1(\Omega)} + |Du|(\Omega)\big) = C_\Omega \|u\|_{BV(\Omega)}. \end{align*} [guided] The previous step gives the boundary inequality for each localised piece $u_k$ in terms of $|Du_k|(\Omega)$. To combine, we expand $\nabla u_k = \nabla(u\psi_k) = \psi_k \nabla u + u \nabla \psi_k$ — the product rule for smooth functions. Taking absolute values: \begin{align*} |\nabla u_k| \le \psi_k |\nabla u| + |u| \, |\nabla \psi_k| \le \|\psi_k\|_\infty |\nabla u| + \|\nabla \psi_k\|_\infty |u|. \end{align*} Integrating over $\Omega$, \begin{align*} |Du_k|(\Omega) \le \|\psi_k\|_\infty |Du|(\Omega) + \|\nabla \psi_k\|_\infty \|u\|_{L^1(\Omega)}. \end{align*} Substituting into the per-chart bound from Step 3, \begin{align*} \int_{V_k \cap \partial\Omega} |u_k| \, d\mathcal{H}^{n-1} \le C_k\big(\|\psi_k\|_\infty |Du|(\Omega) + \|\nabla \psi_k\|_\infty \|u\|_{L^1(\Omega)}\big). \end{align*} Summing over $k = 1, \dots, N$ and using $|u| = |\sum_{k=0}^N u\psi_k| \le \sum_{k=0}^N |u_k| = \sum_{k=1}^N |u_k|$ on $\partial\Omega$ (since $u_0$ vanishes there): \begin{align*} \int_{\partial\Omega} |u| \, d\mathcal{H}^{n-1} &\le \sum_{k=1}^N \int_{V_k \cap \partial\Omega} |u_k| \, d\mathcal{H}^{n-1} \\ &\le \sum_{k=1}^N C_k\big(\|\psi_k\|_\infty |Du|(\Omega) + \|\nabla \psi_k\|_\infty \|u\|_{L^1(\Omega)}\big) \\ &\le C_\Omega \big(\|u\|_{L^1(\Omega)} + |Du|(\Omega)\big), \end{align*} where the final inequality uses \begin{align*} C_\Omega := N \cdot \max_{1 \le k \le N} C_k \cdot \max_{1 \le k \le N} \max(\|\psi_k\|_\infty, \|\nabla \psi_k\|_\infty). \end{align*} The factor $N$ accounts for the sum over $k$, the factor $\max_k C_k$ uniformly bounds the per-chart constants, and the partition-of-unity factor $\max_k \max(\|\psi_k\|_\infty, \|\nabla \psi_k\|_\infty)$ controls both terms in the product-rule expansion. The constant depends only on the geometry of $\Omega$ (the cover and the partition of unity), not on $u$. The right-hand side is exactly $C_\Omega \|u\|_{BV(\Omega)}$, since $\|u\|_{BV(\Omega)} = \|u\|_{L^1(\Omega)} + |Du|(\Omega)$. This completes the inequality on the smooth subclass. The next step extends it to all of $BV(\Omega)$ by approximation. [/guided] [/step] [step:Construct shifted-and-mollified smooth approximants whose boundary slices are $L^1(\partial\Omega)$-Cauchy via Fubini] For arbitrary $u \in BV(\Omega)$, by [BV Smooth Approximation](/theorems/3131) one has $C^\infty(\Omega) \cap BV(\Omega)$ approximants converging in $L^1$ and strictly in total variation. For the boundary trace we need approximants in $C^\infty(\overline\Omega) \cap BV(\Omega)$, and crucially we need a sequence whose *boundary values themselves* are Cauchy in $L^1(\partial\Omega; \mathcal{H}^{n-1})$. The smooth-class inequality of Step 4 applied to differences $u_j - u_l$ would yield this from BV-norm Cauchyness, but strict convergence of 3131 does *not* imply BV-norm Cauchyness (oscillating gradients aligned with $Du$ can preserve total variation while keeping $|D(u_j - u_l)|(\Omega)$ bounded below). We instead establish $L^1(\partial\Omega)$-Cauchyness of boundary slices directly, by applying the half-space inequality of Step 2 to the difference of two *normal-shifted* slices and using the BV property along the normal direction via Fubini. We carry out the construction chart by chart, using the smooth partition of unity $\{\psi_k\}_{k=0}^N$ from Step 1 to localise. Fix $k \ge 1$. Define the localised piece \begin{align*} \tilde u_k := \psi_k u \in BV(\Omega), \end{align*} which has $\operatorname{supp}(\tilde u_k) \subseteq \operatorname{supp}(\psi_k) \subset V_k$. Because $\psi_k \in C_c^\infty(V_k)$ has compact support strictly inside $V_k$, the product $\psi_k u$ has no jump along $\partial V_k$: outside $\operatorname{supp}(\psi_k)$ both $\psi_k$ and $\tilde u_k$ vanish, and on this open set $\tilde u_k \equiv 0$ extends across $\partial V_k$ smoothly. The product rule for $BV$ functions with smooth multipliers gives $\tilde u_k \in BV(\Omega)$ with \begin{align*} |D\tilde u_k|(\Omega) \le \|\psi_k\|_\infty \, |Du|(\Omega) + \|\nabla \psi_k\|_\infty \, \|u\|_{L^1(\Omega)}. \end{align*} Pull back via $\gamma_k$: set \begin{align*} w_k := \tilde u_k \circ \gamma_k^{-1} \quad \text{on } W_k \cap \mathbb{R}^n_+, \end{align*} extended by zero outside $W_k \cap \mathbb{R}^n_+$ (the extension is genuinely zero, not a jump-extension, since $\tilde u_k$ vanishes near $\partial V_k$). Bi-Lipschitz maps preserve $BV$, so $w_k \in BV(\mathbb{R}^n_+)$ with compact support inside $W_k \cap \mathbb{R}^n_+$, and \begin{align*} |Dw_k|(\mathbb{R}^n_+) \le L_k^{n-1} \cdot L_k \cdot |D\tilde u_k|(V_k \cap \Omega) \le C_k \, |D\tilde u_k|(\Omega), \end{align*} with $C_k = L_k^{2n}$ as in Step 3. For $h > 0$ small, define the *normal-shift-then-mollify* approximation \begin{align*} w_k^{(h, \varepsilon)}(x', x_n) := (\eta_\varepsilon * w_k)(x', x_n + h), \end{align*} where $\eta_\varepsilon$ is a standard mollifier on $\mathbb{R}^n$ and $0 < \varepsilon < h/2$. The shift moves the evaluation point a distance $h$ into the half-space, so $w_k^{(h, \varepsilon)}$ is well-defined and smooth on a neighbourhood of $\overline{\mathbb{R}^n_+}$. Since $w_k$ has compact support inside $W_k \cap \mathbb{R}^n_+$, so does $w_k^{(h, \varepsilon)}$ for $h, \varepsilon$ small. As $\varepsilon \to 0$ with $h$ fixed, $w_k^{(h, 0)}(x', x_n) := w_k(x', x_n + h)$ is the unmollified shift, an element of $BV$ on $\mathbb{R}^n_+$, and the slice $w_k^{(h, 0)}(\cdot, 0) = w_k(\cdot, h)$ is an $L^1$ function on $\{x_n = 0\}$ for $\mathcal{L}^1$-a.e. $h > 0$ (by Fubini applied to $|w_k| \in L^1(W_k \cap \mathbb{R}^n_+)$). Pick a sequence $h_j \downarrow 0$ avoiding the (Lebesgue null) exceptional set, and for each $j$ pick $\varepsilon_j < h_j / 2$ so small that \begin{align*} \|w_k^{(h_j, \varepsilon_j)} - w_k^{(h_j, 0)}\|_{L^1(\mathbb{R}^n_+)} + \|w_k^{(h_j, \varepsilon_j)}(\cdot, 0) - w_k(\cdot, h_j)\|_{L^1(\{x_n = 0\})} < \frac{1}{j N}. \end{align*} The first quantity is small for small $\varepsilon_j$ by $L^1$-continuity of mollification. The second quantity is small because, for the smooth function $\eta_\varepsilon * w_k$, restriction to the hyperplane $\{x_n = 0\}$ is a continuous operation, and by Fubini $\int_{\{x_n = 0\}} |(\eta_\varepsilon * w_k)(\cdot, h) - w_k(\cdot, h)| \, d\mathcal{H}^{n-1} \to 0$ as $\varepsilon \to 0$ for $\mathcal{L}^1$-a.e. fixed $h$ (since $\eta_\varepsilon * w_k \to w_k$ in $L^1(W_k \cap \mathbb{R}^n_+)$ and Fubini converts the $L^1$-norm into iterated integration). Push forward each chart-piece by $\gamma_k$ and reassemble: \begin{align*} u_j := \sum_{k=1}^N \big(w_k^{(h_j, \varepsilon_j)} \circ \gamma_k\big) + (\eta_{\varepsilon_j} * (\psi_0 u))_{\text{ext}}, \end{align*} where the last term denotes a standard interior mollification of the interior piece $\psi_0 u$ (with $\operatorname{supp}(\psi_0) \subset \subset \Omega$, this can be done with a fixed-scale mollifier supported well inside $\Omega$, contributing nothing on $\partial\Omega$). Each $u_j \in C^\infty(\overline\Omega)$ since each pushed-forward smooth piece $w_k^{(h_j, \varepsilon_j)} \circ \gamma_k$ extends smoothly across $\partial\Omega \cap V_k$ (via the bi-Lipschitz chart and the fact that $w_k^{(h_j, \varepsilon_j)}$ is smooth on a neighbourhood of $\overline{\mathbb{R}^n_+}$ with compact support inside $W_k$). By construction the sum reproduces $u$ on $\overline\Omega$ in the limit, since $\sum_{k=0}^N \psi_k \equiv 1$ and the construction is in fact built directly from $\tilde u_k = \psi_k u$ (rather than reattaching $\psi_k$ after pull-back). Choosing $\varepsilon_j, h_j$ small enough, $u_j \to u$ in $L^1(\Omega)$ and $|Du_j|(\Omega) \to |Du|(\Omega)$, by the convergence analysis at the end of this step. The boundary slices are now Cauchy in $L^1(\partial\Omega; \mathcal{H}^{n-1})$. To see this, on $V_k \cap \partial\Omega$ pull back via $\gamma_k$ to obtain the slice on $W_k \cap \{x_n = 0\}$. The chart-$k$ contribution to $u_j|_{\partial\Omega}$ is $(w_k^{(h_j,\varepsilon_j)} \circ \gamma_k)|_{\partial\Omega}$ (the $k=0$ term contributes nothing on $\partial\Omega$). For $j > l$, \begin{align*} \|u_j|_{\partial\Omega} - u_l|_{\partial\Omega}\|_{L^1(V_k \cap \partial\Omega)} &\le L_k^{n-1}\,\|w_k^{(h_j, \varepsilon_j)}(\cdot, 0) - w_k^{(h_l, \varepsilon_l)}(\cdot, 0)\|_{L^1(\{x_n=0\})} \\ &\le L_k^{n-1}\Big(\|w_k(\cdot, h_j) - w_k(\cdot, h_l)\|_{L^1(\{x_n=0\})} + \tfrac{2}{lN}\Big), \end{align*} using the boundary Jacobian bound $L_k^{n-1}$ from Step 3 and the choice of $\varepsilon_j, \varepsilon_l$. The remaining slice difference is controlled by Fubini applied to the BV property of $w_k$ along the normal direction: \begin{align*} \|w_k(\cdot, h_j) - w_k(\cdot, h_l)\|_{L^1(\{x_n = 0\})} = \int_{\mathbb{R}^{n-1}} |w_k(x', h_j) - w_k(x', h_l)| \, d\mathcal{H}^{n-1}(x'), \end{align*} and for the $BV$ function $w_k$, the FTC bound along vertical lines (Step 2's argument applied to $w_k$ rather than a smooth $v$, which is licit because $w_k$ has $L^1$ vertical traces for a.e.\ $x_n > 0$ and $\partial_{x_n} w_k$ is a finite measure) gives \begin{align*} \int_{\mathbb{R}^{n-1}} |w_k(x', h_j) - w_k(x', h_l)| \, d\mathcal{H}^{n-1}(x') &\le \int_{\mathbb{R}^{n-1}} |D_{x_n} w_k|(\{x'\} \times [h_l, h_j]) \, d\mathcal{H}^{n-1}(x') \\ &= |D_{x_n} w_k|(W_k \cap \{h_l < x_n < h_j\}), \end{align*} the last equality by the disintegration / Fubini decomposition of the vector measure $D_{x_n} w_k$ along the normal direction (a standard property of $BV$ functions: $|D_{x_n} w_k|$ is a finite Borel measure on the half-space, and slicing by $x'$ gives the one-dimensional total variation on each line, with $|D_{x_n} w_k|(W_k) = \int_{\mathbb{R}^{n-1}} |D_{x_n}^{(x')} w_k(x', \cdot)|(\mathbb{R}_+) \, d\mathcal{H}^{n-1}(x')$ for the slice variation $|D_{x_n}^{(x')} w_k(x', \cdot)|$ on the vertical line through $x'$). Since $|D_{x_n} w_k|$ is finite on $W_k \cap \mathbb{R}^n_+$, the strip $\{h_l < x_n < h_j\}$ shrinks to $\emptyset$ as $h_l, h_j \to 0$ along $h_j > h_l$, so $|D_{x_n} w_k|(W_k \cap \{h_l < x_n < h_j\}) \to 0$. Summing over $k$ and using compactness of $\partial\Omega$: \begin{align*} \|u_j|_{\partial\Omega} - u_l|_{\partial\Omega}\|_{L^1(\partial\Omega)} \le \big(\sum_{k=1}^N L_k^{n-1}\big)\big(\,\sup_k |D_{x_n} w_k|(W_k \cap \{h_l < x_n < h_j\}) + \tfrac{2}{l}\,\big) \to 0 \end{align*} as $j, l \to \infty$. Hence $(u_j|_{\partial\Omega})$ is Cauchy in $L^1(\partial\Omega; \mathcal{H}^{n-1})$. [guided] *Why the previous "BV-norm Cauchyness" route fails.* The natural temptation is to find smooth $u_j \to u$ with $\|u_j - u_l\|_{BV} \to 0$, then apply the smooth-class boundary inequality of Step 4 to $u_j - u_l$ to read off $L^1(\partial\Omega)$-Cauchyness. But theorem 3131 produces only *strict* convergence: $u_j \to u$ in $L^1$ and $|Du_j|(\Omega) \to |Du|(\Omega)$. Strict convergence does *not* upgrade to BV-norm Cauchyness; for $u$ with non-trivial singular part $D^s u$, no smooth sequence converges to $u$ in $BV$-norm at all (the $W^{1,1}$ closure of smooth functions does not contain $u$ in this case). So the route through "BV-Cauchy approximants" is genuinely closed off. *The correct route: control the boundary slices directly.* Instead of bounding $\|u_j - u_l\|_{BV}$, we bound $\|u_j|_{\partial\Omega} - u_l|_{\partial\Omega}\|_{L^1(\partial\Omega)}$ directly, by exploiting the structure of how the approximants are built — specifically, that their boundary values are *horizontal slices* of $u$ at heights $h_j \downarrow 0$, up to a small mollification error. *The construction.* The localisation must be done with the *smooth* partition of unity $\{\psi_k\}$, not with characteristic functions $\mathbf{1}_{V_k}$. The reason is BV: multiplying $u$ by $\mathbf{1}_{V_k}$ would introduce a jump along $\partial V_k \cap \Omega$, contributing a singular part to $D(\mathbf{1}_{V_k} u)$ unrelated to $u$ itself; this would corrupt the Fubini-slice estimate downstream. With the smooth cutoff $\psi_k \in C_c^\infty(V_k)$, the product $\tilde u_k := \psi_k u$ has $\operatorname{supp}(\tilde u_k) \subseteq \operatorname{supp}(\psi_k)$ compactly contained inside $V_k$, so extension by zero across $\partial V_k$ is genuine zero-extension (no jump), and $\tilde u_k \in BV(\Omega)$ with the product-rule bound recorded above. Chart by chart, pull $\tilde u_k = \psi_k u$ back to the half-space to get $w_k = \tilde u_k \circ \gamma_k^{-1}$, which is in $BV(\mathbb{R}^n_+)$ with compact support inside $W_k \cap \mathbb{R}^n_+$. Define the $j$-th approximant in the chart by \begin{align*} w_k^{(h_j, \varepsilon_j)}(x', x_n) := (\eta_{\varepsilon_j} * w_k)(x', x_n + h_j), \end{align*} i.e., translate $w_k$ vertically downward by $h_j$ (so its boundary value at $x_n = 0$ is the slice $w_k(\cdot, h_j)$ from the interior), then mollify at scale $\varepsilon_j < h_j / 2$ to smooth it. The mollification scale is dominated by the shift, so the convolution stays inside the half-space and the result is smooth up to and across $\{x_n = 0\}$. Crucially, the boundary slice $w_k(\cdot, h_j)$ is now the slice of $\psi_k u \circ \gamma_k^{-1}$ at height $h_j$, not the slice of $u \circ \gamma_k^{-1}$ at height $h_j$ — but this is what we want, since after assembly the partition of unity sum reproduces $u$ on $\partial\Omega$ in the limit. The boundary value at $\{x_n = 0\}$ equals $(\eta_{\varepsilon_j} * w_k)(\cdot, h_j)$. By choosing $\varepsilon_j$ small (depending on $h_j$), this is close in $L^1(\{x_n = 0\})$ to the unmollified slice $w_k(\cdot, h_j)$ (Fubini converts $L^1$-convergence of mollification on the bulk into $L^1$-convergence of slices for a.e.\ height; we choose $h_j$ to lie outside the null exceptional set). *Why slices are Cauchy.* The key estimate is the one-dimensional FTC for BV functions along vertical lines. For a $BV$ function $w_k$ on $W_k \cap \mathbb{R}^n_+$, the vector measure $D_{x_n} w_k$ disintegrates along the $x'$-direction: for $\mathcal{H}^{n-1}$-a.e.\ $x'$, the function $w_k(x', \cdot)$ is in $BV(\mathbb{R}_+)$ as a function of one variable, with $|D_{x_n}^{(x')} w_k(x', \cdot)|$ being its total-variation measure, and \begin{align*} |D_{x_n} w_k|(A) = \int_{\mathbb{R}^{n-1}} |D_{x_n}^{(x')} w_k(x', \cdot)|(A_{x'}) \, d\mathcal{H}^{n-1}(x'), \end{align*} for any Borel set $A$, where $A_{x'} = \{t : (x', t) \in A\}$. This is the slicing/Fubini decomposition for $BV$ functions (one of the foundational properties of $BV$ — see e.g.\ Ambrosio-Fusco-Pallara). Applied to $A = W_k \cap \{h_l < x_n < h_j\}$: \begin{align*} \int_{\mathbb{R}^{n-1}} |w_k(x', h_j) - w_k(x', h_l)| \, d\mathcal{H}^{n-1}(x') &\le \int_{\mathbb{R}^{n-1}} |D_{x_n}^{(x')} w_k(x', \cdot)|([h_l, h_j]) \, d\mathcal{H}^{n-1}(x') \\ &= |D_{x_n} w_k|(W_k \cap \{h_l < x_n < h_j\}). \end{align*} The first inequality is the $1$D FTC for a $BV$ function on $\mathbb{R}_+$: for the right-continuous representative $f \in BV(\mathbb{R}_+)$, the bound $|f(b) - f(a)| \le |Df|([a,b])$ holds at every pair of points; for an arbitrary representative one obtains the same bound at points of approximate continuity (a full-measure set). We work with the precise (right-continuous, at the half-space normal) representative of $w_k(x', \cdot)$ throughout, which exists for $\mathcal{H}^{n-1}$-a.e.\ $x'$ by the slicing decomposition. The exceptional $\mathcal{L}^1$-null set of "non-Lebesgue" heights $h$ (heights where the slice $w_k(\cdot, h)$ as an $L^1$-function fails to coincide with the slice of the precise representative on a set of positive $\mathcal{H}^{n-1}$-measure of $x'$'s) is what we explicitly avoid when choosing $h_j$. The second equality in the displayed bound is the disintegration identity. The right-hand side tends to zero as $h_j, h_l \to 0$ (from above): $|D_{x_n} w_k|$ is a finite Borel measure on $W_k \cap \mathbb{R}^n_+$, the strip $\{h_l < x_n < h_j\}$ has Lebesgue measure $|h_j - h_l|$, and as both $h$'s tend to zero the strip is contained in the slab $\{0 < x_n < h_j\}$ which shrinks to $\partial(W_k \cap \mathbb{R}^n_+) \cap \{x_n = 0\}$ — an $|D_{x_n} w_k|$-null set if $w_k$ has no jump along $\{x_n = 0\}$ from inside (which it does not, since $w_k = 0$ outside $V_k$ is consistent with the BV pull-back having no boundary jump from outside). More directly: $\bigcap_h \{0 < x_n < h\} = \emptyset$, and the finite measure $|D_{x_n} w_k|$ satisfies $\lim_{h \to 0} |D_{x_n} w_k|(\{0 < x_n < h\}) = |D_{x_n} w_k|(\emptyset) = 0$ by countable additivity. *Pull back to $\partial\Omega$ and combine charts.* The chart-level Cauchyness of slices transfers to $\partial\Omega$ via the boundary Jacobian $\le L_k^{n-1}$ from Step 3, and the partition-of-unity sum involves only $N$ charts (each contributing a term that vanishes), so the global Cauchyness on $L^1(\partial\Omega; \mathcal{H}^{n-1})$ follows. The mollification error $\sim 1/j$ chosen at the start of the construction is summable and vanishes as $j \to \infty$. The conclusion is that the *boundary slices* $u_j|_{\partial\Omega}$ form a Cauchy sequence in $L^1(\partial\Omega; \mathcal{H}^{n-1})$ — without needing or claiming that $u_j$ is Cauchy in BV norm on $\Omega$ (which would be false in general). *Bonus: $L^1$-strict convergence on $\Omega$.* The bulk convergence $u_j \to u$ in $L^1(\Omega)$ and $|Du_j|(\Omega) \to |Du|(\Omega)$ also holds for this construction. In each chart, $w_k^{(h_j, \varepsilon_j)} \to w_k$ in $L^1(W_k \cap \mathbb{R}^n_+)$ as $h_j, \varepsilon_j \to 0$ (translation continuity in $L^1$ + mollification continuity in $L^1$), and the chart-level total variation contracts: $|D w_k^{(h_j, \varepsilon_j)}|(W_k') \le |D w_k|(W_k \cap \{x_n > h_j - \varepsilon_j\}) \to |D w_k|(W_k \cap \mathbb{R}^n_+)$. Combining via the partition of unity and using lower semicontinuity to match upper and lower bounds gives strict convergence on $\Omega$. (One may equivalently invoke 3131's strict-convergence construction directly and check that the shift-and-mollify version is, up to negligible adjustment, the same construction.) [/guided] [/step] [step:Define the trace as the $L^1(\partial\Omega)$ limit of boundary values of the approximants] By Step 5, $(u_j|_{\partial\Omega})$ is Cauchy in $L^1(\partial\Omega; \mathcal{H}^{n-1})$. Since $L^1(\partial\Omega; \mathcal{H}^{n-1})$ is a Banach space, the sequence converges, and we define \begin{align*} Tu := \lim_{j \to \infty} u_j|_{\partial\Omega} \in L^1(\partial\Omega; \mathcal{H}^{n-1}). \end{align*} [guided] The Cauchyness of the boundary slices established in Step 5 — obtained directly by the Fubini / horizontal-slice argument, not via BV-norm Cauchyness of the bulk approximants — places $(u_j|_{\partial\Omega})$ inside the Banach space $L^1(\partial\Omega; \mathcal{H}^{n-1})$ as a Cauchy sequence. Completeness of $L^1$ then provides a unique limit, which we name $Tu$: \begin{align*} Tu := \lim_{j \to \infty} u_j|_{\partial\Omega} \in L^1(\partial\Omega; \mathcal{H}^{n-1}). \end{align*} The limit is taken in the $L^1$ norm with respect to $\mathcal{H}^{n-1}$ on $\partial\Omega$, which is finite since $\partial\Omega$ is compact Lipschitz. Note that this construction gives $Tu$ as an element of $L^1(\partial\Omega; \mathcal{H}^{n-1})$ associated to a *specific* approximating sequence (the shift-and-mollify one of Step 5). Step 7 will check that the limit is independent of the choice of $h_j, \varepsilon_j$ and agrees with any other strict approximation, justifying the notation $Tu$ as a function of $u$ alone. [/guided] [/step] [step:Verify well-definedness, strict continuity, and linearity of $T$] We work explicitly in the **strict topology** of $BV(\Omega)$: a sequence $u_j \in BV(\Omega)$ converges *strictly* to $u \in BV(\Omega)$ if $u_j \to u$ in $L^1(\Omega)$ and $|Du_j|(\Omega) \to |Du|(\Omega)$. The construction of Step 5 produces, for any $u \in BV(\Omega)$, a sequence $u_j \in C^\infty(\overline\Omega) \cap BV(\Omega)$ with $u_j \to u$ strictly *and* with $(u_j|_{\partial\Omega})$ Cauchy in $L^1(\partial\Omega; \mathcal{H}^{n-1})$. *Independence of the construction parameters.* Suppose $(u_j)$ and $(v_j)$ are two shift-and-mollify approximating sequences for the same $u \in BV(\Omega)$, with possibly different shift schedules $h_j^{(u)}, h_j^{(v)} \downarrow 0$ and mollifier scales $\varepsilon_j^{(u)}, \varepsilon_j^{(v)}$. Interleave them as $w_{2j} := u_j$, $w_{2j+1} := v_j$. The shift heights of the interleaved sequence still satisfy $h_m^{(w)} \downarrow 0$. The Fubini-slicing estimate of Step 5 gives, for any $m, m'$ in chart $k$, \begin{align*} \|w_m|_{\partial\Omega} - w_{m'}|_{\partial\Omega}\|_{L^1(V_k \cap \partial\Omega)} \le L_k^{n-1}\big(|D_{x_n} w_k|(\{h_{\min} < x_n < h_{\max}\}) + \tfrac{2}{\min(m, m')}\big), \end{align*} where $h_{\min}, h_{\max}$ are the shift heights of $w_m, w_{m'}$ in chart $k$ and the $\tfrac{2}{\min(m,m')}$ accounts for mollification errors. Since both shift schedules tend to zero, $\max(h_m^{(w)}, h_{m'}^{(w)}) \to 0$ as $m, m' \to \infty$, and the strip measure vanishes. Hence $(w_m|_{\partial\Omega})$ is Cauchy in $L^1(\partial\Omega; \mathcal{H}^{n-1})$, and the two subsequences $(u_j|_{\partial\Omega})$ and $(v_j|_{\partial\Omega})$ — both convergent inside this Cauchy parent — share the same limit. So $Tu := \lim_j u_j|_{\partial\Omega}$ is independent of the choice of shift-and-mollify parameters. *Strict continuity of $T$.* We assert that $T$ extends to a **strictly continuous** map $BV(\Omega) \to L^1(\partial\Omega; \mathcal{H}^{n-1})$. Concretely: if $v_n \to v$ strictly in $BV(\Omega)$ (where $v_n, v \in BV(\Omega)$, no smoothness assumed), then $Tv_n \to Tv$ in $L^1(\partial\Omega; \mathcal{H}^{n-1})$. To prove this, run the shift-and-mollify construction of Step 5 with a *diagonal* parameter selection: for each $n$, run Step 5 on $v_n$ with parameters $(h^{(n)}_n, \varepsilon^{(n)}_n)$ chosen small enough that \begin{align*} \|w^{(n)}|_{\partial\Omega} - Tv_n\|_{L^1(\partial\Omega)} < \tfrac{1}{n}, \end{align*} where $w^{(n)}$ is the shift-and-mollify approximant of $v_n$ at parameters $(h^{(n)}_n, \varepsilon^{(n)}_n)$. The diagonal sequence $(w^{(n)}) \subset C^\infty(\overline\Omega) \cap BV(\Omega)$ then converges strictly to $v$ (since $v_n \to v$ strictly and the within-step approximation is itself strictly close to $v_n$, with errors $\le 1/n$ on each strict ingredient by the parameter choice). Apply the shift-and-mollify Cauchyness of Step 5 directly to $(w^{(n)})$: it is a smooth strict-approximating sequence for $v$, so its boundary slices are $L^1(\partial\Omega)$-Cauchy and converge to $Tv$. Hence $Tv_n \to Tv$ in $L^1(\partial\Omega; \mathcal{H}^{n-1})$. We have now characterized $T$ as the **unique strictly-continuous extension** of the classical smooth-class restriction map $C^\infty(\overline\Omega) \cap BV(\Omega) \to L^1(\partial\Omega; \mathcal{H}^{n-1})$, $u \mapsto u|_{\partial\Omega}$. Existence is provided by Steps 5–6; uniqueness within the class of strictly-continuous extensions is the subject of Step 10. *Linearity.* Suppose $(u_j), (v_j)$ are shift-and-mollify approximations of $u, v \in BV(\Omega)$ respectively, with a *common* parameter schedule $(h_j, \varepsilon_j)$ (this is possible because the construction's only constraints are upper bounds on $\varepsilon_j$ depending on $u$ or $v$, plus the avoidance of an $\mathcal{L}^1$-null exceptional set of heights; intersecting the constraints for $u$ and $v$ still leaves positive room and full-measure heights). With a common schedule, $u_j + \lambda v_j$ is itself a shift-and-mollify approximation of $u + \lambda v$, since translation and convolution are linear: \begin{align*} \eta_\varepsilon * (w_k^u + \lambda w_k^v) = \eta_\varepsilon * w_k^u + \lambda \eta_\varepsilon * w_k^v, \end{align*} and the partition-of-unity recombination is linear. Restriction to $\partial\Omega$ is linear on smooth functions: $(u_j + \lambda v_j)|_{\partial\Omega} = u_j|_{\partial\Omega} + \lambda v_j|_{\partial\Omega}$. Passing to the $L^1(\partial\Omega; \mathcal{H}^{n-1})$ limit preserves linearity: \begin{align*} T(u + \lambda v) = \lim_j (u_j + \lambda v_j)|_{\partial\Omega} = Tu + \lambda Tv. \end{align*} [guided] *The strict topology.* The natural topology on $BV(\Omega)$ for trace theory is the *strict topology*, in which a sequence $u_j$ converges to $u$ if $u_j \to u$ in $L^1(\Omega)$ AND $|Du_j|(\Omega) \to |Du|(\Omega)$. This topology is strictly finer than weak-* convergence and strictly weaker than BV-norm convergence; it is the natural one because (a) smooth functions are dense in $BV(\Omega)$ in the strict topology (theorem 3131) but not in BV-norm (when $u$ has a non-trivial singular part), and (b) the trace operator is naturally continuous with respect to it. This is the standard strict topology of $BV$, cf. Ambrosio-Fusco-Pallara. *Independence of the shift-and-mollify parameters.* Step 5 specifies a class of approximants — shift-and-mollify with shrinking $h_j, \varepsilon_j$. Two members of this class for the same $u$ might have different parameter schedules. Their boundary limits agree because the Cauchy estimate at the heart of Step 5 controls $\|w_m|_{\partial\Omega} - w_{m'}|_{\partial\Omega}\|_{L^1(\partial\Omega)}$ by $|D_{x_n} w_k|(\{h_{\min} < x_n < h_{\max}\})$, where $h_{\min}, h_{\max}$ are the shift heights of the two approximants in chart $k$. Since both schedules go to zero, the strip measure vanishes by countable additivity, and the interleaved sequence is Cauchy. Two convergent subsequences of a Cauchy sequence share the same limit. *Strict continuity of $T$.* This is the central pedagogical point of the construction. We claim that $T$, defined initially on $u \in BV(\Omega)$ via the shift-and-mollify approximation of Step 5, extends to a map $BV(\Omega) \to L^1(\partial\Omega; \mathcal{H}^{n-1})$ that is continuous in the strict topology: $v_n \to v$ strictly implies $Tv_n \to Tv$ in $L^1(\partial\Omega)$. The proof is a diagonal argument. For each $n$, the shift-and-mollify construction applied to $v_n$ produces a smooth approximant arbitrarily close to $Tv_n$ in $L^1(\partial\Omega)$. Choose parameters $(h_n, \varepsilon_n)$ small enough — depending on $n$ — that the approximant $w^{(n)} := $ (shift-and-mollify of $v_n$ at parameters $(h_n, \varepsilon_n)$) satisfies $\|w^{(n)}|_{\partial\Omega} - Tv_n\|_{L^1(\partial\Omega)} < 1/n$ AND that $\|w^{(n)} - v_n\|_{L^1(\Omega)} < 1/n$, $||Dw^{(n)}|(\Omega) - |Dv_n|(\Omega)| < 1/n$ (controllable by parameter choice — the shift-and-mollify construction is strictly close to $v_n$ at small parameters, by the Bonus paragraph at the end of Step 5). Then the diagonal sequence $(w^{(n)})$ satisfies: - $w^{(n)} \in C^\infty(\overline\Omega) \cap BV(\Omega)$, - $\|w^{(n)} - v\|_{L^1(\Omega)} \le \|w^{(n)} - v_n\|_{L^1(\Omega)} + \|v_n - v\|_{L^1(\Omega)} \to 0$, - $|Dw^{(n)}|(\Omega) \to |Dv|(\Omega)$ by the triangle bound $||Dw^{(n)}|(\Omega) - |Dv|(\Omega)| \le ||Dw^{(n)}|(\Omega) - |Dv_n|(\Omega)| + ||Dv_n|(\Omega) - |Dv|(\Omega)| \to 0$. So $(w^{(n)})$ is a smooth strict-approximating sequence for $v$. By Step 5's Cauchyness applied to this sequence (with the shift-and-mollify decomposition guaranteed by construction), $w^{(n)}|_{\partial\Omega} \to Tv$ in $L^1(\partial\Omega)$. Combining with $\|w^{(n)}|_{\partial\Omega} - Tv_n\|_{L^1(\partial\Omega)} < 1/n$ via the triangle inequality: \begin{align*} \|Tv_n - Tv\|_{L^1(\partial\Omega)} \le \|Tv_n - w^{(n)}|_{\partial\Omega}\|_{L^1(\partial\Omega)} + \|w^{(n)}|_{\partial\Omega} - Tv\|_{L^1(\partial\Omega)} \to 0. \end{align*} Hence $T$ is strictly continuous. *Why the strictly-continuous extension framing is the right one.* On $BV(\Omega)$, smooth functions are dense in the strict topology but not in the BV-norm topology (the latter would require BV-norm-Cauchy smooth approximants, which fail for $u$ with non-trivial singular part). The trace operator must therefore be defined by extension along strict-convergent sequences, and continuity in the strict topology is the natural property guaranteeing the extension is well-defined. This is the framing under which: - *Existence* is exactly the strict-continuous-extension property, proved by the shift-and-mollify Cauchyness of Step 5. - *BV-norm boundedness* (Step 8) drops out as a corollary: BV-norm convergence implies $L^1$-convergence and (via $|Du| \le \|u\|_{BV}$) controls $|Du|(\Omega)$, so a strictly continuous map is automatically BV-norm continuous, i.e., bounded. - *Uniqueness* among strictly-continuous extensions is the subject of Step 10. *Linearity.* Given $u, v \in BV(\Omega)$, choose shift-and-mollify approximations $(u_j), (v_j)$ with a common parameter schedule $(h_j, \varepsilon_j)$. The construction's only constraints are upper bounds on $\varepsilon_j$ depending on $u$ or $v$, plus avoidance of $\mathcal{L}^1$-null exceptional sets of heights; intersecting the constraints for $u$ and $v$ still leaves positive room and full-measure heights, so a common schedule exists. With a common schedule, $u_j + \lambda v_j$ is itself a shift-and-mollify approximant of $u + \lambda v$ at the same parameters, since the chart-by-chart construction is linear: pull-back via $\gamma_k$ is linear, $\psi_k$-cutoff is linear, vertical translation is linear, and convolution with $\eta_{\varepsilon_j}$ is linear. Restriction to $\partial\Omega$ is linear on smooth functions: \begin{align*} (u_j + \lambda v_j)|_{\partial\Omega} = u_j|_{\partial\Omega} + \lambda v_j|_{\partial\Omega}. \end{align*} Passing to the $L^1(\partial\Omega; \mathcal{H}^{n-1})$ limit preserves linearity: \begin{align*} T(u + \lambda v) = \lim_j (u_j + \lambda v_j)|_{\partial\Omega} = \lim_j u_j|_{\partial\Omega} + \lambda \lim_j v_j|_{\partial\Omega} = Tu + \lambda Tv. \end{align*} [/guided] [/step] [step:Establish the operator estimate by passing to the limit in the smooth-class inequality] For each shift-and-mollify approximant $u_j$, Step 4 gives \begin{align*} \int_{\partial\Omega} |u_j| \, d\mathcal{H}^{n-1} \le C_\Omega \|u_j\|_{BV(\Omega)} = C_\Omega \big(\|u_j\|_{L^1(\Omega)} + |Du_j|(\Omega)\big). \end{align*} The left side converges to $\int_{\partial\Omega} |Tu| \, d\mathcal{H}^{n-1}$ by the $L^1(\partial\Omega; \mathcal{H}^{n-1})$ convergence $u_j|_{\partial\Omega} \to Tu$ established in Step 6 (continuity of the $L^1$ norm under $L^1$ convergence). The right side converges to $C_\Omega(\|u\|_{L^1(\Omega)} + |Du|(\Omega)) = C_\Omega \|u\|_{BV(\Omega)}$ by strict convergence of the shift-and-mollify approximation. Passing to the limit, \begin{align*} \int_{\partial\Omega} |Tu| \, d\mathcal{H}^{n-1} \le C_\Omega \|u\|_{BV(\Omega)}, \end{align*} which is the operator bound with $C = C_\Omega$, the same constant produced in Step 4. [guided] The smooth-class boundary inequality of Step 4 says, for each smooth approximant $u_j$, \begin{align*} \int_{\partial\Omega} |u_j| \, d\mathcal{H}^{n-1} \le C_\Omega \|u_j\|_{BV(\Omega)}. \end{align*} We pass to the limit on both sides separately. Left side: $u_j|_{\partial\Omega} \to Tu$ in $L^1(\partial\Omega; \mathcal{H}^{n-1})$ by Step 6. The map $f \mapsto \|f\|_{L^1(\partial\Omega; \mathcal{H}^{n-1})}$ is continuous in $L^1$-norm by the reverse triangle inequality, so \begin{align*} \int_{\partial\Omega} |u_j| \, d\mathcal{H}^{n-1} \to \int_{\partial\Omega} |Tu| \, d\mathcal{H}^{n-1}. \end{align*} Right side: $\|u_j\|_{L^1(\Omega)} \to \|u\|_{L^1(\Omega)}$ by $L^1$-convergence, and $|Du_j|(\Omega) \to |Du|(\Omega)$ by strict convergence (Step 5). Therefore \begin{align*} \|u_j\|_{BV(\Omega)} = \|u_j\|_{L^1(\Omega)} + |Du_j|(\Omega) \to \|u\|_{L^1(\Omega)} + |Du|(\Omega) = \|u\|_{BV(\Omega)}. \end{align*} Passing to the limit in both sides of the smooth-class inequality: \begin{align*} \int_{\partial\Omega} |Tu| \, d\mathcal{H}^{n-1} \le C_\Omega \|u\|_{BV(\Omega)}, \end{align*} which is the desired operator estimate, with the constant $C = C_\Omega$ from Step 4 (the same constant on both sides; no relabelling occurs in the limit). The operator $T: BV(\Omega) \to L^1(\partial\Omega; \mathcal{H}^{n-1})$ is therefore bounded with $\|T\| \le C_\Omega$. [/guided] [/step] [step:Verify agreement on continuous functions] Suppose $u \in C(\overline{\Omega}) \cap BV(\Omega)$. We show $Tu = u|_{\partial\Omega}$. By the Tietze extension theorem, $u$ extends to a continuous function on $\mathbb{R}^n$; multiplying by a smooth cutoff that equals $1$ on a neighbourhood of $\overline\Omega$ and is supported in a slightly larger open neighbourhood produces a compactly supported continuous extension defined on an open neighbourhood of $\overline\Omega$. Then the shift-and-mollify approximation construction of Step 5 produces $u_j \to u$ both in $L^1(\Omega)$ strictly and uniformly on $\overline\Omega$: within each chart, the shift $w_k(\cdot, \cdot + h_j)$ converges uniformly to $w_k$ on compact subsets where $w_k$ is continuous (by uniform continuity), and the further mollification $\eta_{\varepsilon_j} * w_k(\cdot, \cdot + h_j)$ also converges uniformly to $w_k$. The partition-of-unity recombination preserves uniform convergence. Hence $u_j|_{\partial\Omega} \to u|_{\partial\Omega}$ uniformly, and a fortiori in $L^1(\partial\Omega; \mathcal{H}^{n-1})$ since $\mathcal{H}^{n-1}(\partial\Omega) < \infty$. By uniqueness of $L^1$ limits, $Tu = u|_{\partial\Omega}$ in $L^1(\partial\Omega; \mathcal{H}^{n-1})$. [guided] The agreement-on-continuous-functions property is what ties $T$ to its name as a "trace" — it should genuinely restrict continuous functions to their boundary values. For $u \in C(\overline\Omega) \cap BV(\Omega)$, extend $u$ to a continuous function on a slight thickening of $\overline\Omega$ (this uses the Tietze extension theorem on the closed set $\overline\Omega \subset \mathbb{R}^n$, then truncate to a compactly supported continuous extension). The shift-and-mollify construction of Step 5, applied to this extension, produces approximants $u_j$ which converge to $u$ uniformly on $\overline\Omega$ (in addition to strictly in $BV$). Why uniform convergence holds: in each chart $W_k$, the pull-back $w_k$ is continuous on the closure $\overline{W_k}$ (continuity is preserved under bi-Lipschitz pull-back) and hence uniformly continuous there (compact closure). The vertical shift $w_k(\cdot, \cdot + h_j) \to w_k$ uniformly on $\overline{W_k'}$ as $h_j \to 0$ by uniform continuity. The subsequent mollification $\eta_{\varepsilon_j} * w_k(\cdot, \cdot + h_j) \to w_k(\cdot, \cdot + h_j)$ uniformly on compact subsets as $\varepsilon_j \to 0$ (continuous functions are uniformly approximated by their mollifications on compact sets). Combining, $u_j \to u$ uniformly chart-by-chart. Since the partition of unity $\{\psi_k\}$ is finite and smooth, uniform convergence on each chart yields uniform convergence on $\overline\Omega$ after recombination. Restricting to $\partial\Omega$: $u_j|_{\partial\Omega} \to u|_{\partial\Omega}$ uniformly. Since $\partial\Omega$ is compact Lipschitz, $\mathcal{H}^{n-1}(\partial\Omega) < \infty$, so uniform convergence implies $L^1(\partial\Omega; \mathcal{H}^{n-1})$ convergence. By Step 6, $u_j|_{\partial\Omega} \to Tu$ in $L^1(\partial\Omega; \mathcal{H}^{n-1})$. Limits in $L^1$ are unique, so \begin{align*} Tu = u|_{\partial\Omega} \quad \text{in } L^1(\partial\Omega; \mathcal{H}^{n-1}). \end{align*} [/guided] [/step] [step:Establish uniqueness of $T$ as the strictly-continuous extension] We characterize $T$ as the **unique strictly-continuous extension** of the classical restriction map. Precisely, suppose $T_1, T_2: BV(\Omega) \to L^1(\partial\Omega; \mathcal{H}^{n-1})$ are two linear operators satisfying \begin{enumerate} \item[(i)] *Agreement on the smooth class*: $T_i u = u|_{\partial\Omega}$ for every $u \in C^\infty(\overline\Omega) \cap BV(\Omega)$, $i = 1, 2$; \item[(ii)] *Strict continuity*: whenever $v_j \to v$ strictly in $BV(\Omega)$ (i.e., $v_j \to v$ in $L^1(\Omega)$ and $|Dv_j|(\Omega) \to |Dv|(\Omega)$), one has $T_i v_j \to T_i v$ in $L^1(\partial\Omega; \mathcal{H}^{n-1})$, for $i = 1, 2$. \end{enumerate} Then $T_1 = T_2$ on all of $BV(\Omega)$, and both equal the operator $T$ constructed in Steps 5–7. The proof is two lines. Fix $u \in BV(\Omega)$. By Step 5, there exists a sequence $u_j \in C^\infty(\overline\Omega) \cap BV(\Omega)$ with $u_j \to u$ strictly in $BV(\Omega)$. By hypothesis (i), $T_1 u_j = u_j|_{\partial\Omega} = T_2 u_j$ for every $j$. By hypothesis (ii) applied to the strict-convergent sequence $u_j \to u$, \begin{align*} T_1 u = \lim_{j \to \infty} T_1 u_j = \lim_{j \to \infty} u_j|_{\partial\Omega} = \lim_{j \to \infty} T_2 u_j = T_2 u, \end{align*} where each limit is taken in $L^1(\partial\Omega; \mathcal{H}^{n-1})$. Hence $T_1 u = T_2 u$ for every $u \in BV(\Omega)$, so $T_1 = T_2$. Our constructed $T$ from Step 7 is one such operator: it is linear, agrees with the smooth restriction (the construction defines $Tu$ as the $L^1$-limit of $u_j|_{\partial\Omega}$ along smooth strict-approximants of $u$, which on the smooth class itself reduces to the classical restriction), and is strictly continuous by the well-definedness argument of Step 7. Therefore $T$ is the unique strictly-continuous extension. **Boundedness as a corollary.** Strict continuity together with the smooth-class boundary inequality (Step 4) implies BV-norm boundedness: for $u \in BV(\Omega)$ and any smooth strict-approximating $u_j \to u$, applying Step 4 to $u_j$ and passing to the strict-continuous limit (Step 8) gives $\|Tu\|_{L^1(\partial\Omega)} \le C_\Omega \|u\|_{BV(\Omega)}$. So the bounded-linear-operator formulation $T: BV(\Omega) \to L^1(\partial\Omega; \mathcal{H}^{n-1})$ with operator norm $\|T\| \le C_\Omega$ is a *consequence* of the strictly-continuous-extension construction, not an independent hypothesis. [guided] The uniqueness statement asks: how much freedom is there in choosing the trace operator? Once we agree that $T$ should (i) restrict continuous functions classically and (ii) be continuous in the natural topology on $BV(\Omega)$ for which smooth functions are dense, the answer is none — $T$ is uniquely determined. *The natural topology.* On $BV(\Omega)$, neither weak-* convergence nor BV-norm convergence is the right topology for trace theory. Weak-* is too weak: oscillating sequences can have wildly varying boundary slices while converging weakly-*. BV-norm is too strong: smooth functions are not BV-norm-dense in $BV(\Omega)$ when $u$ has a non-trivial singular part (the $W^{1,1}$ closure of smooth functions does not contain such $u$). The intermediate topology that gets the trace theory right is the **strict topology**: $u_j \to u$ strictly if \begin{align*} \|u_j - u\|_{L^1(\Omega)} \to 0 \quad \text{and} \quad |Du_j|(\Omega) \to |Du|(\Omega). \end{align*} This is the standard strict topology of $BV$ (cf. Ambrosio-Fusco-Pallara). *Density.* In the strict topology, $C^\infty(\overline\Omega) \cap BV(\Omega)$ is dense in $BV(\Omega)$. For an arbitrary $u \in BV(\Omega)$, theorem 3131 (BV smooth approximation) provides $C^\infty(\Omega) \cap BV$-approximants strictly converging to $u$; the additional regularity up to $\overline\Omega$ is exactly what Step 5's shift-and-mollify construction adds. So Step 5 produces the smooth-strict-approximating sequence needed for uniqueness. *The strictly-continuous extension.* A linear map $T$ from a topological vector space to a Banach space is determined by its values on a dense subset, provided $T$ is continuous in the topology in which the subset is dense. Here: - The dense subset is $C^\infty(\overline\Omega) \cap BV(\Omega)$, dense in the strict topology of $BV(\Omega)$. - The Banach space is $L^1(\partial\Omega; \mathcal{H}^{n-1})$. - The topology on $BV(\Omega)$ is the strict topology. Two strictly-continuous linear extensions of the same map on a strictly-dense subset must coincide. This is exactly the two-line argument at the start of the step. *Why "strict continuity" rather than "BV-norm boundedness."* One might hope to weaken hypothesis (ii) to "$T_i$ is bounded as a linear map $BV(\Omega) \to L^1(\partial\Omega)$ (with respect to the BV norm)." This weaker hypothesis is *not* sufficient: BV-norm boundedness gives continuity along BV-norm-convergent sequences, but smooth approximants $u_j \to u$ converge to $u$ strictly, not in BV-norm (when $u$ has a singular part). So a BV-bounded operator agreeing with the smooth restriction is not automatically strictly-continuous, and the chain $T_1 u = \lim_j T_1 u_j$ would fail to close. Strict continuity is genuinely required. *The boundedness corollary.* Conversely, strict continuity *implies* BV-norm boundedness, with the explicit constant $C_\Omega$ from Step 4: for any $u \in BV(\Omega)$, take a smooth strict-approximating $u_j \to u$, apply the smooth-class inequality $\|u_j|_{\partial\Omega}\|_{L^1(\partial\Omega)} \le C_\Omega \|u_j\|_{BV(\Omega)}$, and pass to the strict-continuous limit on the left (using $\|u_j\|_{BV} \to \|u\|_{BV}$ from strict convergence on the right). This is exactly Step 8, and it yields $\|Tu\|_{L^1(\partial\Omega)} \le C_\Omega \|u\|_{BV(\Omega)}$. So the BV-norm operator estimate is a *derived* property of the strictly-continuous-extension construction, not part of the defining data. *Summary of what has been proved.* Steps 1–6 construct an operator $T: BV(\Omega) \to L^1(\partial\Omega; \mathcal{H}^{n-1})$ via shift-and-mollify approximation. Step 7 establishes that $T$ is well-defined (independent of the strict-approximating sequence) and linear, and characterizes $T$ as the unique strictly-continuous extension of the smooth restriction map. Step 8 derives the BV-norm boundedness as a corollary. Step 9 verifies agreement on continuous functions. This step establishes the uniqueness statement: any other linear operator $BV(\Omega) \to L^1(\partial\Omega; \mathcal{H}^{n-1})$ satisfying agreement-on-smooth-class and strict-continuity coincides with $T$. [/guided] [/step]