Low-order identifications of the Dirichlet scale

Fix an integer $k\ge 0$ and write $W^{k,2}(\mathrm{\Omega })$ for the weak-derivative space to distinguish its norm from the restriction quotient norm. If $u=F{\mathrm{\mid }}_{\mathrm{\Omega }}$ for $F\in H^k({\mathbb{R}}^n)$, the whole-space identification and contractive weak restriction give $u\in W^{k,2}(\mathrm{\Omega })$ and $\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{k,2}(\mathrm{\Omega })}\le C_{n,k}\mathrm{\parallel }F{\mathrm{\parallel }}_{H^k({\mathbb{R}}^n)}$. Taking the infimum over all extensions $F$ gives $\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{k,2}(\mathrm{\Omega })}\le C_{n,k}\mathrm{\parallel }u{\mathrm{\parallel }}_{H^k(\mathrm{\Omega })}$. For $k=0$, conversely, zero extension is an $L^2$ extension and the whole-space order-zero identification supplies the reverse norm bound. It remains to prove that reverse inclusion and bound for each fixed $k\ge 1$.

Fix $k\ge 1$ and $u\in W^{k,2}(\mathrm{\Omega })$. Choose $R>0$ with $\mathrm{\Theta }⋐B_R(0)$ and set $G=B_R(0)\setminus \mathrm{\Theta }$. Its boundary is the disjoint union of the smooth obstacle boundary and the sphere, so it satisfies the bounded $C^k$ one-sided graph condition. Choose $\chi ,\rho \in C_c^{\mathrm{\infty }}(B_R(0))$ with $\chi =1$ on an open neighborhood $U$ of $\mathrm{\Theta }$ and $\rho =1$ near $\mathrm{supp}\chi$. Treat these as ambient smooth functions extended by zero. The weak product and restriction estimates give $v=(\chi u){\mathrm{\mid }}_G\in W^{k,2}(G)$ and $\mathrm{\parallel }v{\mathrm{\parallel }}_{W^{k,2}(G)}\le C_k\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{k,2}(\mathrm{\Omega })}$. Apply the bounded smooth-domain theorem with $p=2$, complex scalars and this fixed order to obtain $B_{k}v\in W^{k,2}({\mathbb{R}}^n)$ extending $v$, with the corresponding bounded norm.

The stated exterior geometry and Choice assumption are those required by the fractional-identification lemma. The canonical realization in F8 is the weighted Fourier Bessel convention used by that lemma. For $s\ge 0$, its inverse bracket weight is bounded, so $h=⟨\xi {⟩}^{-s}{J}_{s}U$ belongs to $L^2$. Taking $V=J_0^{-1}h$ gives $E_{s}U=E_{0}V$ by the embedding formula, and the order-zero case of F4 identifies this with a regular $L^2$ class. Thus its ordinary restriction has precisely the meaning in the Statement. Applying the fractional-identification lemma on the two specified index ranges gives both displayed Dirichlet-scale equalities and the asserted norm equivalences, with exactly that restriction-space convention.

Let $w=(1-\chi )u\in W^{k,2}(\mathrm{\Omega })$ and let $Z$ denote extension of a measurable class by zero on $\mathrm{\Theta }$. The weak Leibniz formula shows that $w$ and each $D^{\alpha }w$, $\mathrm{\mid }\alpha \mathrm{\mid }\le k$, vanish on $U\cap \mathrm{\Omega }$ and have $L^2$ norms bounded by $C_k\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{k,2}(\mathrm{\Omega })}$. Choose $\zeta \in C_c^{\mathrm{\infty }}(U)$ equal to one near $\mathrm{\Theta }$ and put $\vartheta =1-\zeta$. For any $\varphi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$, $\vartheta \varphi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$, and $\vartheta =1$ wherever $w$ or its indicated derivatives can be nonzero. Thus the weak test identity on $\mathrm{\Omega }$ gives ${\int }_{{\mathbb{R}}^n}Zw{D}^{\alpha }\varphi ={\int }_{\mathrm{\Omega }}w{D}^{\alpha }(\vartheta \varphi )=(-1{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}{\int }_{\mathrm{\Omega }}{D}^{\alpha }w\vartheta \varphi =(-1{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}{\int }_{{\mathbb{R}}^n}Z(D^{\alpha }w)\varphi$. Consequently $D^{\alpha }(Zw)=Z(D^{\alpha }w)$ for every $\mathrm{\mid }\alpha \mathrm{\mid }\le k$, and $\mathrm{\parallel }Zw{\mathrm{\parallel }}_{W^{k,2}({\mathbb{R}}^n)}\le C_k\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{k,2}(\mathrm{\Omega })}$. This uses compact support only for the test functions, and allows an unbounded tail.

Set ${\mathcal{E}}_{k}u=\rho B_{k}v+Zw$. The product estimates and the previous bounds give ${\mathcal{E}}_{k}u\in W^{k,2}({\mathbb{R}}^n)$ with norm at most $C_k\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{k,2}(\mathrm{\Omega })}$. On $G$ its first summand equals $\rho \chi u=\chi u$; on $\mathrm{\Omega }\setminus G$ both $\rho$ and $\chi$ vanish. Hence $({\mathcal{E}}_{k}u){\mathrm{\mid }}_{\mathrm{\Omega }}=\chi u+(1-\chi )u=u$ almost everywhere. The whole-space Fourier identification now makes ${\mathcal{E}}_{k}u$ an $H^k({\mathbb{R}}^n)$ extension, so the quotient norm satisfies $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^k(\mathrm{\Omega })}\le \mathrm{\parallel }{\mathcal{E}}_{k}u{\mathrm{\parallel }}_{H^k({\mathbb{R}}^n)}\le C_k^{\mathrm{\prime }}\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{k,2}(\mathrm{\Omega })}$. Together with step 1.1, this proves equality of the two spaces with equivalent norms for every integer $k\ge 0$. In particular the two $H^1$ norms give the same closure of $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$, as used in the Statement. No extension operator common to all orders is asserted or needed.

At $s=1$ and $s=2$, the scale definition identifies the spaces with $D(A^{1\mathrm{/}2})$ and $D(A)$ respectively. The realization supplies their weak Sobolev endpoint descriptions and norm comparisons, and step 3.1 identifies those weak spaces and their test-function closure with the restriction-space conventions here. The scale definition also identifies $H_D^{-s}$ isometrically with the antidual of $H_D^s$ through the extended pivot pairing for every $s\ge 0$. These give all remaining assertions.