Complex integer-order Sobolev notation

Assume Countable Choice: every sequence of nonempty sets has a choice function. Let $D\subseteq {\mathbb{R}}^n$ be open, with $n\ge 1$, and let $k$ be a nonnegative integer. All functions below are complex-valued and identified when equal almost everywhere. For $1\le p\le \mathrm{\infty }$, $W^{k,p}(D;\mathbb{C})$ is the space of $L^p$ classes whose weak derivatives $D^{\alpha }u$ of every multi-index $\alpha \in {\mathbb{N}}_0^n$ with $\mathrm{\mid }\alpha \mathrm{\mid }\le k$ belong to $L^p(D;\mathbb{C})$. A weak derivative is defined by integration against smooth compactly supported test functions, with the usual factor $(-1{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}$; $D^{0}u=u$.

Write $H^k(D)=W^{k,2}(D;\mathbb{C})$, with norm $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^k(D)}=({\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }\le k}\mathrm{\parallel }{D}^{\alpha }u{\mathrm{\parallel }}_{L^2(D)}^2{)}^{1\mathrm{/}2}$. Write $W^{k,\mathrm{\infty }}(D)$ for $W^{k,\mathrm{\infty }}(D;\mathbb{C})$, equipped with $\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{k,\mathrm{\infty }}(D)}={\max }_{\mathrm{\mid }\alpha \mathrm{\mid }\le k}\mathrm{\parallel }{D}^{\alpha }u{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(D)}$. In particular, the order-zero derivative requires $u$ itself to be essentially bounded.

Let $C_c^{\mathrm{\infty }}(D;\mathbb{C})$ denote the complex-valued smooth functions on $D$ whose supports are compact subsets of $D$. Define $H_0^k(D)=W_0^{k,2}(D;\mathbb{C})$ as their closure in the $H^k(D)$ norm: $u\in H_0^k(D)$ exactly when, for every $\epsilon >0$, there is $\phi \in C_c^{\mathrm{\infty }}(D;\mathbb{C})$ with $\mathrm{\parallel }u-\phi {\mathrm{\parallel }}_{H^k(D)}<\epsilon$. This definition asserts no trace characterization.

The Axiom of Countable Choice (AC_ω)

Integer-order Sobolev spaces and their norms

Zero-boundary Sobolev space as a norm closure

The complex Sobolev supplier defines the derivative classes and displayed finite-sum and maximum conventions under Countable Choice; the assumption is used in its representative-independence and weak-derivative uniqueness interface. The Sobolev norm descends to equivalence classes establishes that these expressions are norms on the classes. Zero-boundary Sobolev space as a norm closure supplies the inclusion of the test functions in the Sobolev space and the resulting metric closure, so the stated approximation condition defines a subset without requiring boundary traces.

Fractional or negative ordinary Sobolev spaces, Dirichlet spectral spaces, identifications between domains, norm equivalences and trace characterizations require separate statements.