Bounded Bogovskii right inverse on a star-shaped domain

First take $f\in C_c^{\mathrm{\infty }}(U)$ with $\int f=0$. After translating the ball center to $0$, both $\omega$ and $x_k\omega$ are supported in $B(0,\rho )\subset U$. For either $\varphi =\omega$ or $\varphi =x_k\omega$ and any $g\in C_c^{\mathrm{\infty }}(U)$, every point in the support of a truncated $T_{\varphi ,j}g$ has the form $x=(1-t)y+tz$ with $y\in \mathrm{supp}g$, $z\in \mathrm{supp}\varphi$, and $0\le t\le 1$. Star-shapedness puts every such segment in $U$; compactness of $\mathrm{supp}g\times \mathrm{supp}\varphi \times [0,1]$ shows the resulting support is contained in a compact subset of $U$. Hence the distributional operators in [F2] are supported in $U$ and their ambient $L^2$ norms equal their $L^2(U)$ norms. By [F2]–[F3], for each $k,j$, $\mathrm{\parallel }{\mathrm{\partial }}_j({\mathcal{B}}_{U}f{)}_k{\mathrm{\parallel }}_2\le [A_j(x_k\omega )+R_U{A}_j(\omega )]\mathrm{\parallel }f{\mathrm{\parallel }}_2$, since $\mathrm{\mid }{y}_k\mathrm{\mid }\le R_U$ on $U$. Squaring and summing the finitely many derivative components gives $\mathrm{\parallel }\mathrm{\nabla }{\mathcal{B}}_{U}f{\mathrm{\parallel }}_2\le C_U\mathrm{\parallel }f{\mathrm{\parallel }}_2$.

By [F1], this field is supported in a compact subset $K⋐U$, and step 1.1 puts it in $H^1({\mathbb{R}}^d)$. Choose the normalized bump from [F6] and its scaled mollifier ${\rho }_{ϵ}$ from [F7]. For $ϵ<\frac{1}{2}\mathrm{dist}(K,{\mathbb{R}}^d\setminus U)$, [F8] makes ${\rho }_{ϵ}\ast {\mathcal{B}}_{U}f$ smooth with compact support in $U$. Its first derivatives are ${\rho }_{ϵ}\ast {\mathrm{\partial }}_j({\mathcal{B}}_{U}f)$ and converge in $L^2$ together with the field by [F9], so the approximants converge to ${\mathcal{B}}_{U}f$ in $H^1$. Hence ${\mathcal{B}}_{U}f\in H_0^1(U)$. The divergence identity in [F1] reduces to $\mathrm{div}{\mathcal{B}}_{U}f=f$.

For arbitrary $f\in L_0^2(U)$, choose $h_n\in C_c^{\mathrm{\infty }}(U)$ with $h_n\to f$ in $L^2$ using [F5], and put $f_n=h_n-\omega {\int }_U{h}_n$. Then $f_n\in C_c^{\mathrm{\infty }}(U)$, $\int f_n=0$, and $\mathrm{\parallel }{f}_n-f{\mathrm{\parallel }}_2\le (1+\mathrm{\mid }U{\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{\parallel }\omega {\mathrm{\parallel }}_2)\mathrm{\parallel }{h}_n-f{\mathrm{\parallel }}_2\to 0$ by Cauchy–Schwarz and $\int f=0$.

By step 1.1 applied to differences, ${\mathcal{B}}_U{f}_n$ is Cauchy in the gradient norm. [F4] makes it Cauchy in $H^1$; completeness and closedness in [F5] give a limit in $H_0^1(U)$. The limit is independent of the approximating sequence by the same bound, so it defines a linear operator and retains the gradient bound. Continuity of divergence $H^1\to L^2$ and step 2.1 give $\mathrm{div}{\mathcal{B}}_{U}f=f$.