Failure of the one-half-Hölder maximal regularity implication

Let $\mathfrak{A}: [0,T] \to \mathcal{L}(V,V^*)$ be the operator associated with a non-autonomous form $\mathfrak{a}$, and suppose that

$$\|\mathfrak{A}(t)-\mathfrak{A}(s)\|_{V\to V^*} \leq C|t-s|^{1/2} \qquad (t,s\in[0,T]).$$

The one-half-Hölder maximal regularity counterexample claim. This regularity does not imply maximal regularity of $\mathfrak{a}$ in $H$ in general, even when $\mathfrak{a}$ induces elliptic differential operators in divergence form with Dirichlet, Neumann, or mixed boundary conditions on an open set $\Omega$.

The claim identifies the criticality of one-half-Hölder time regularity: stronger temporal regularity yields maximal regularity results, whereas at the one-half-Hölder threshold maximal regularity can fail even for elliptic divergence-form operators with standard boundary conditions.