Consider $f: \mathbb{R}^2 \to \mathbb{R}$ a smooth and compactly supported function. We know that the function $$ w(x) := - \frac 1 {2\pi}\int_{\mathbb{R}^2} f(y) \log|x-y| \ dy $$ is the only solution to the problem $$ -\Delta w = f, \text{ on } \mathbb{R}^2 $$ with the property that its gradient vanishes at infinity.
From Calderon--Zygmund theory, for a given $1 < p < \infty$, it follows that the inequality holds true: $$ \|\partial_i \partial_j w\|_{L^p(\mathbb{R}^2)} \leq C_p \|f\|_{L^p(\mathbb{R}^2)}. $$ Indeed, $\partial_i \partial_j u = R_{ij} u = \partial_i \partial_j (- \Delta^{-1})u$, the Riesz transform, which is a Calderon--Zygmund operator.
Usually textbooks prove the fact that Calderon-Zygmund operators send $L^\infty$ into $BMO$. But I would like something different. Namely, my question is: does the following inequality (*) hold true?
$$ \|\partial_i \partial_j w\|_{L^\infty(\mathbb{R}^2)} \leq C \|f\|_{L^\infty(\mathbb{R}^2)} (1 + \log( e+\|\partial^2 w\|_{L^\infty(\mathbb{R}^2)})) \qquad (*) $$ Here, we denoted $\|\partial^2 w\|_{L^\infty(\mathbb{R}^2)} := \sum_{i =1,2, \, j = 1,2}\|\partial_i \partial_j w\|$. This inequality loses a logarithm, and I have no clue in how to start proving it. I tried writing down the expression for $w$ in terms of principal value, and integrating by parts near the singularity, but it does not help. One could think of splitting the principal value integral in two parts, one in a ball near the singularity, say of size $\delta$, and then optimize in terms of $\delta$. But again, no clue in how to deal with the part near the singularity.
Any idea would be much appreciated.