The divergence equation with $L^\infty$ source
Résumé
A well-known fact is that there exists $g\in
L^{\infty}(\mathbf{T}^{2})$ with zero integral, such that the equation
\begin{equation}
div f=g \tag{$\ast$}
\end{equation}
has no solution $f=(f_{1},f_{2})\in W^{1,\infty}(\mathbf{T}^{2})$. This was
proved by Preiss (1997), using an involved geometric argument, and, independently, by McMullen (1998), via Ornstein's non-inequality. We
improve this result: roughly speaking, we prove that, there exists $g\in
L^{\infty}$ for which ($\ast$) has no solution such that $%
\partial_{2}f_{2}\in L^{\infty}$ and $f$ is "slightly better" than $L^{1}$.
Our proof relies on Riesz products in the spirit of the approach of
Wojciechowski (1998) for the study of ($\ast$) with source $g\in L^{1}$. The
proof we give is elementary, self-contained and completely avoids the use of
Ornstein's non-inequality.
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