Take a weak solution $u$ of the Poisson equation on $\mathbb{R}^d$
$$ \Delta u = f $$
By standard elliptic regularity theory we have (for some $\alpha\in (0,1]$) $f\in C^{0, \alpha}\_{\text{loc}}(\mathbb{R}^d)$, then $u \in C^{2, \alpha}\_{\text{loc}}(\mathbb{R}^d)$. Now my (naive) question is:



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> Are there some weaker assumptions that on $f$ that ensure that $u \in C^2(\mathbb{R}^d)$?
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By weaker assumptions I mean something like $f\in C^0(\mathbb{R}^d)$ plus some condition which is not of Hölder type (I am aware that $f\in C^0(\mathbb{R}^d)$ is not sufficient).



Minimal assumptions such that the solution of Poisson equation is $C^2$
*Dini continuity* may be what you are looking for: if $f = \Delta u$ is Dini continuous, that is, $$\int\_0^1 \frac{\omega\_f(t)}{t} \, dt < \infty,$$ then $u$ is $C^2$. This is a rather old result, but I do not know the reference. A quick Google search leads to [*Poisson's equation*](http://www.math.mcgill.ca/gantumur/math580f13/poisson.pdf) by T. Gantumur, see Theorem 42 and Corollary 43 there.


