Consider the heat equation \begin{align} \partial_t u - \Delta u &= 0 && \mbox{ in }Q=\Omega\times[0,T] \\ u &= 0 && \mbox{ on }\partial \Omega \times [0,T] \\ u(\cdot,0) &= u_0 && \mbox{ on }\Omega\times\{0\} \end{align} with $u_0 \in H^1(\Omega)$ and a smooth doamin $\Omega$. For the solution of $u$ it is known that $u$ is in $$ H^{2,1}(Q) := \{ u \in L^2(0,T,H^2(\Omega)); u_t \in L^2(Q); u=0 \text{ on } \partial \Omega \times [0,T] \} $$ and further we have the bound $\Vert u \Vert_{H^{2,1}(Q)} \leq C \Vert u_0 \Vert_{H^1(\Omega)}$ with $$ \Vert w \Vert_{H^{2,1}(Q)}^2 = \sum_{|\alpha|\leq 2} \Vert D^{\alpha} w \Vert_{L^2(Q)}^2 + \Vert \partial_t w \Vert_{L^2(Q)}^2 $$ and $D^{\alpha}$ the spatial derivatives. On the domain $Q_{\varepsilon} = \Omega \times [\varepsilon,T]$ the solution is smooth as soon as $\varepsilon > 0$.
Now may question: Can I get a bound of the form $$ \Vert u \Vert_{H^{k,k}(Q_{\varepsilon})} \leq C(\varepsilon) \Vert u_0 \Vert_{H^1(\Omega)} $$ with $k \geq 2$ and if yes how does $C(\varepsilon)$ behave for $\varepsilon \rightarrow 0$?
I don't expect that a complete answer is directly in your mind, but if you have ideas on how to approach this question please share your thoughts.
I have no ready solution. A first approach might be looking at the decay rates of Fourier coefficients; then $C(\epsilon)$ would be related to the "worst case harmonic" with the decay rate proportional to $\mathrm{sup}_{n\in \mathbb{N}}n^k e^{-\alpha n^2\epsilon}$ for the $k$ th order spatial derivatives.
Or one might start from the derivative of the fundamental solution and work from there.