The divergence equation with L ∞ source

Abstract

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.