Let $m \in \mathbb{N}\setminus \{0,1\}$, $\alpha \in ]0,1[$.
Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ of class $C^{m,\alpha}$.


It is known that if $f \in C^{\frac{m-2+\alpha}{2},m-2+\alpha}([0,T]\times \mathrm{cl}\,\Omega)$, $g \in C^{\frac{m+\alpha}{2};m+\alpha}([0,T]\times\partial\Omega)$, $u\_0 \in C^{m,\alpha}(\mathrm{cl}\, \Omega)$ (satisfying some compatibility conditions at $t=0$), then there exists a unique solution $u$ in $C^{\frac{m+\alpha}{2};m+\alpha}([0,T]\times \mathrm{cl}\,\Omega)$ of
\begin{cases}
\partial\_t u -\Delta u = f &\mbox{ in }[0,T]\times \mathrm{cl}\,\Omega,\\
u=g & \mbox{ on } [0,T] \times \partial\Omega,\\
u(0,\cdot) = u\_0 & \mbox{ in }\mathrm{cl} \, \Omega.
\end{cases}


For the elliptic case holds a very similar result, but in this case we allow $m$ to be also $1$, that is $m \in \mathbb{N}\setminus \{0\}$. If $f \in C^{m-2,\alpha}(\mathrm{cl}\,\Omega)$, $g \in C^{m,\alpha}(\partial\Omega)$, then there exists a unique solution $u$ in $C^{m,\alpha}(\mathrm{cl}\,\Omega)$ of
\begin{cases}
\Delta u = f &\mbox{ in }\mathrm{cl}\,\Omega,\\
u=g & \mbox{ on } \partial\Omega.
\end{cases}
In this case, for $m=1$, the space $C^{-1,\alpha}(\mathrm{cl}\,\Omega)$ is the space of distributions which euqals the divergence of an element in $C^{0,\alpha}(\mathrm{cl}\,\Omega,\mathbb{C}^n)$, and the Laplacian is to be intended in the weak sense.


Then my question is the following:


There exists an analog of the case $m=1$ for the heat (parabolic) equation?



Schauder regularity heat equation
For a partial answer, see Theorem 6.48 in *Second order parabolic differential equations*, 1996, by Gary Lieberman.


There is an additional assumption that $f$ belongs to the Morrey space $M^{1,n+1+\alpha}$ defined page 130 of the book. In particular, $L^{\infty}\subset M^{1,n+1+\alpha}\subset L^1$.


I do not know if you can remove this additional assumption or not.


