Eduard Curca
University Alexandru Ioan Cuza, Iasi, Romania

In 2002 Bourgain and Brézis proved that given a vector field $v\in \mathcal{S}^{\prime }(\mathbb{R}^{d})\cap \dot{W}^{1,d}( \mathbb{R}^{d})$ there exists a vector field $u\in L^{\infty }(\mathbb{R} ^{d})\cap \dot{W}^{1,d}(\mathbb{R}^{d})$ such that $\operatorname{div}u=\operatorname{div}v$ . We prove several results of a similar nature in which we take into consideration the Fourier support of the solutions. For instance, in the case $d\geq 3$ we prove the following: for any vector field $v\in \mathcal{S} ^{\prime }(\mathbb{R}^{d})\cap \dot{B}_{q}^{d/p,p}(\mathbb{R}^{d})$ (where $ p\in \lbrack 2,\infty )$ and $q\in (1,2) $), with $supp$ $\widehat{v} \subseteq \mathbb{R}^{d}\backslash (-\infty ,0)^{d}$, there exists a vector field $u\in L^{\infty }(\mathbb{R}^{d})\cap \dot{B}_{2}^{d/p,p}(\mathbb{R} ^{d})$, with $supp$ $\widehat{v}\subseteq \mathbb{R}^{d}\backslash (-\infty ,0)^{d}$, such that \begin{equation*} \operatorname{div}u=\operatorname{div}v, \end{equation*} and \begin{equation*} \left\Vert u\right\Vert _{L^{\infty }\cap \dot{B}_{2}^{d/p,p}}\lesssim \left\Vert v\right\Vert _{\dot{B}_{q}^{d/p,p}}. \end{equation*} Our arguments rely on a version of the complex interpolation method combined with some ideas of Bourgain and Brézis.