The trace-free Beurling--Ahlfors transform and the Bourgain--Brezis problem for Hodge systems

Abstract

We show that the dual approach to Bourgain--Brezis estimates for Hodge systems is substantially more flexible than previously understood. For $1\le l\le n-1$, we introduce the trace-free Beurling--Ahlfors transform $S=\frac{n-l}{n}P-\frac{l}{n}{P}^{\perp }$, a canonical normalization of the generalized Beurling--Ahlfors transform on $l$-forms in ${\mathbb{R}}^n$. Its matrix symbol decomposes into scalar multipliers that are odd under suitable orthogonal reflections, yielding an endpoint cancellation estimate from finite measures to $L^{\infty }$ for $|D{|}^{-n}S$. This cancellation allows us to complete the Hilbertian case of the Bourgain--Brezis conjecture in every dimension and for every form degree. We then develop multilinear reflection estimates and obtain new critical Triebel--Lizorkin and Besov Bourgain--Brezis estimates. In particular, for every dimension and form degree, the Sobolev Bourgain--Brezis conjecture in ${\dot{W}}^{\frac{n}{p},p}$ holds for $p=\frac{2k}{2k-1}$, $k\ge 1$, and hence for exponents arbitrarily close to $1$. We also derive endpoint Hodge decompositions and Hodge--Sobolev inequalities. Finally, except in the endpoint Besov case where the critical space already embeds into $L^{\infty }$, we prove that the associated bounded selections cannot be linear.