Bourgain--Brezis selection maps for Hodge systems in Besov and Triebel--Lizorkin spaces

We establish Bourgain–Brezis selection maps for Hodge systems in new ranges of Besov and Triebel–Lizorkin spaces. Our approach combines the trace-free Beurling–Ahlfors transform with a nonlinear duality method. To control concentrations, we use tools developed by Stolyarov in the study of Maz’ya’s $\mathrm{\Phi }$-inequalities. These methods yield selection maps in Besov spaces ${\dot{B}}_{p,q}^{n\mathrm{/}p}({\mathbb{R}}^n)$ for every $1\le q\le p\le 2$, and in Triebel–Lizorkin spaces ${\dot{F}}_{p,q}^{n\mathrm{/}p}({\mathbb{R}}^n)$ for every $1<p\le q\le 2$. The theorems hold for every dimension $n\ge 2$ and every form degree $1\le l\le n-1$. Although our results overlap with some earlier work, they considerably extend the known range of solutions to the Bourgain–Brezis problem and underscore the importance of the trace-free Beurling–Ahlfors transform in its resolution.