The notation $H^k$ and the reserved zero-boundary symbol

• John K. Hunter, Notes on Partial Differential Equations, Chapter 3 §3.5, Definition 3.23, printed p. 59 (PDF p. 63). Hunter writes $W^{k,2}(\mathrm{\Omega })=H^k(\mathrm{\Omega })$ after defining the integer-order Sobolev space. This item applies that notation to the real or complex almost-everywhere Sobolev classes fixed in this library.

Assume Countable Choice, as in Integer-order Sobolev spaces and their norms and The Sobolev norm descends to equivalence classes. For an open set $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$, $n\ge 1$, an integer $k\ge 0$, and $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$, set $H^k(\mathrm{\Omega };\mathbb{K}):=W^{k,2}(\mathrm{\Omega };\mathbb{K})\mathrm{.}$ The elements are the same almost-everywhere classes, and their weak derivatives and norm are exactly those already defined for $W^{k,2}$. When the scalar field is fixed, write $H^k(\mathrm{\Omega })$.

The notation $H_0^k(\mathrm{\Omega })$ is reserved for the later definition by closure of compactly supported smooth functions in the Sobolev norm. This item assigns no boundary values to an equivalence class and asserts no characterization by pointwise vanishing or by traces.