Decomposition of the distribution on $BMO$ space

Amina Lahmar Benbernou, Sadek Gala

In this paper, we characterize $\vec f$ so that if the inequality \[ \bigg|ıt_{\mathbb R^d}ěc g\cdot(\bar uabla v-vabla\bar u)dx\bigg|eq C\|u\|_{\dot H^1}\|v\|_{\dot H^1} \] holds for all $u,v\in\mathcal D(\mathbb T^d)$, then $\vec f$ can be represented in the form \[ ěc f=abla g+peratorname{Div}H \] where $g\in BMO(\mathbb R^d)$, $H$ is a skew-symmetric matrix field such that $H\in BMO(\mathbb R^d)^{d^2}$.