Abstract generator-domain smoothing becomes spatial regularity only after domain identification

Assume Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain) for the cited integral and semigroup suppliers.

The homogeneous smoothing theorem Smoothing estimates for the semigroup generated by a sectorial operator says that for every $x\in X$ and $t>0$, $T(t)x\in D(A^m)$ for each $m\ge 1$. For a forced mild solution $u(t)=T(t)x+{\int }_0^{t}T(t-s)f(s)ds$, the positive-time $D(A^m)$ conclusion of Abstract parabolic smoothing for mild solutions uses the stated source hypothesis $f\in C^{m-1,\alpha }([0,b],D(A^{m-1}))$ in the graph norm; it is not asserted for arbitrary $X$-valued forcing. The graph-domain membership is a statement about an abstract operator on a Banach space; it names a Sobolev derivative only after an elliptic-regularity theorem identifies $D(A^m)$ with a concrete space. For the Dirichlet Laplacian on a bounded $C^2$ domain one has $D(A)=H^2\cap H_0^1$ by Global $H^2$ Dirichlet regularity, and for a $C^{2m}$ boundary $D(A^m)=\{u\in H^{2m}:{\mathrm{\Delta }}^{j}u\in H_0^1,0\le j<m\}$ by Higher-order boundary regularity for Dirichlet problems; without the boundary compatibility hypotheses, only the recursive graph domain is available (the identification clauses of The Dirichlet Laplacian generates an analytic heat semigroup). Likewise, a diagonal analytic semigroup on a sequence space smooths homogeneous orbits into powers of a sequence operator with no intrinsic spatial variables (the abstract sequence-space example of the companion page). This remark is not proof-bearing; it fixes the seam between the abstract theory and its PDE realisations.