TITLE: Minimal assumptions such that the solution of Poisson equation is $C^2$
QUESTION [6 upvotes]: 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:

Are there some weaker assumptions that on $f$ that ensure that $u \in C^2(\mathbb{R}^d)$?

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).

REPLY [9 votes]: 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 by T. Gantumur, see Theorem 42 and Corollary 43 there.