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 $u \in C^2(\mathbb{R}^d)$?

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).