Compactly supported chart mollification in uniform and first Sobolev norms

By [F8], choose a bump $b$ equal to one on ${\bar{B}}_{1\mathrm{/}2}(0)$ with support in $B_1(0)$. Its integral is positive and finite, so $b\mathrm{/}\int b$ is an admissible kernel, proving the existence assertion with the real smooth-function convention of [F17]; the rest of the proof applies to any fixed kernel in the Statement. By [F16], the set $K$ in the Statement is $\mathrm{supp}\tilde{f}$. If $K$ is empty, all asserted functions are zero. Otherwise compactness gives ${\eta }_0>0$ such that $K+{\bar{B}}_{{\eta }_0}(0)$ is compact and contained in $V$; when $V={\mathbb{R}}^n$, take any ${\eta }_0>0$. The zero extension $\tilde{f}$ is globally Lipschitz: for $x\in V$ and $y\mathrm{\notin }V$, approach the first exit point of the segment from $x$ to $y$ from within $V$. The function is zero near that exit point because $K$ stays away from the complement, so the Lipschitz inequality gives $\mathrm{\mid }f(x)\mathrm{\mid }\le L\mathrm{\mid }x-y\mathrm{\mid }$. Pairs in $V$ satisfy the original bound and pairs outside have zero difference. By [F1], $\tilde{f}$ has bounded weak derivatives, zero almost everywhere off $K$, so the function and these derivatives are integrable and [F6] gives $\tilde{f}\in W^{1,1}({\mathbb{R}}^n)$. The support of ${\rho }_{\eta }\ast \tilde{f}$ lies in $K+{\bar{B}}_{\eta }(0)$, hence in the fixed compact subset above for $0<\eta \le {\eta }_0$.

By [F7], [F10] and the substitution in [F13], $f_{\eta }(x)-f(x)=\int \rho (z)(f(x-\eta z)-f(x))dz$. The integrand is integrable since $f$ is bounded and $\rho$ has compact support. Thus [F9] and [F12] give $\mathrm{\parallel }{f}_{\eta }-f{\mathrm{\parallel }}_{\mathrm{\infty }}\le L\eta \int \mathrm{\mid }z\mathrm{\mid }\rho (z)dz$, proving uniform convergence. For the Sobolev estimate take $g=f$ or a representative of one of its weak first derivatives. For a derivative, choose the following Borel representative in the sense of [F14]: take the limit of the continuous difference quotients $m(f(x+e_i\mathrm{/}m)-f(x))$ where the sequence converges and put zero elsewhere, where $e_i$ is the $i$th coordinate vector. The convergence set is Borel by the countable Cauchy criterion, and the resulting function is Borel measurable. It agrees almost everywhere with $D_{i}f$ by [F1]. The function $f$ itself is continuous. Consequently $F_{\eta }(x,z)=\rho (z)(g(x-\eta z)-g(x))$ is product-measurable for the Borel sigma-algebras: the product Borel sigma-algebra is the Euclidean Borel sigma-algebra because open rectangles from countable bases generate the product topology; compose the Borel functions with the continuous affine maps and take their difference and product. Each Euclidean Borel Lebesgue measure space is sigma-finite, being covered by bounded cubes. By [F13] and the $L^1$ norm triangle inequality and homogeneity from [F15], $\int \mathrm{\parallel }{F}_{\eta }(\cdot ,z){\mathrm{\parallel }}_{1}dz\le 2\mathrm{\parallel }g{\mathrm{\parallel }}_1\int \rho (z)dz=2\mathrm{\parallel }g{\mathrm{\parallel }}_1<\mathrm{\infty }$. All hypotheses of [F11] are therefore satisfied.

Applying [F11] gives $H_{\eta }(x)=\int \mathrm{\mid }{F}_{\eta }(x,z)\mathrm{\mid }dz\in L^1$. Hence $F_{\eta }(x,\cdot )$ is integrable for almost every $x$, and the convolution substitution [F10], [F13], its unit mass [F7] and [F12] imply $\mathrm{\mid }({\rho }_{\eta }\ast g)(x)-g(x)\mathrm{\mid }\le H_{\eta }(x)$ almost everywhere. It follows that $\mathrm{\parallel }{\rho }_{\eta }\ast g-g{\mathrm{\parallel }}_1\le \int \rho (z)\mathrm{\parallel }g(\cdot -\eta z)-g{\mathrm{\parallel }}_{1}dz$. Given $\epsilon >0$, [F3] supplies $\delta >0$ such that $\mathrm{\mid }h\mathrm{\mid }<\delta$ implies $\mathrm{\parallel }g(\cdot -h)-g{\mathrm{\parallel }}_1<\epsilon$. If $0<\eta <\delta$, every $z$ in the unit-ball support of $\rho$ satisfies this bound with $h=\eta z$, so the last integral is at most $\epsilon \int \rho =\epsilon$. This proves convergence for $f$ and each weak first derivative. Apply [F2] with $\mathrm{\Omega }={\mathbb{R}}^n$, $k=1$, $p=1$ and the kernel specified in the Statement; its hypotheses hold by step 1.1, so commutation of convolution with weak derivatives and [F6] prove the $W^{1,1}$ convergence. The smoothness supplier [F4] applies because $f$ is locally integrable and the fixed kernel is smooth, compactly supported and of mass one. Together with the support conclusion in step 1.1 and the uniform convergence in step 2.1, this proves every assertion.