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

Abstract

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 $Φ$-inequalities. These methods yield selection maps in Besov spaces $\dot B^{n/p}_{p,q}(\mathbb{R}^n)$ for every $1\leq q\leq p\leq 2$, and in Triebel--Lizorkin spaces $\dot F^{n/p}_{p,q}(\mathbb{R}^n)$ for every $1<p\leq q\leq 2$. The theorems hold for every dimension $n\geq 2$ and every form degree $1\leq l\leq n-1$. 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.