A note on an inequality of Bourgain and Brezis

• A new and elementary proof to an inequality of Bourgain and Brezis is given.
• Avoid using the Littlewood–Paley decomposition.
• An equivalent version of the inequality is given.

In this note, we give a new and elementary approach to an inequality of Bourgain and Brezis (2007) for L1L1 vector fields, avoid using the Littlewood–Paley decomposition that Bourgain and Brezis used. And we also give an equivalent version of the inequality.