AI 中文总结
该研究引入无迹Beurling--Ahlfors变换，利用其符号的反射奇性得到端点抵消估计，证明了任意维度与形式次数下Bourgain--Brezis猜想的希尔伯特情形，还得到多类新的临界估计及端点Hodge分解，并证明多数情形下有界选择算子非线性。
AI 中文摘要
我们证明，Hodge系统Bourgain--Brezis估计的对偶方法比此前认知的更具灵活性。对于1≤l≤n-1，我们引入无迹Beurling--Ahlfors变换S=((n-l)/n)P-(l/n)P^⊥，这是ℝⁿ中l-形式上广义Beurling--Ahlfors变换的一种标准规范化形式。其矩阵符号可分解为在适当正交反射下为奇函数的标量乘子，由此得到|D|^{-n}S从有限测度到L^∞的端点抵消估计。这种抵消特性让我们在任意维度、任意形式次数下都证明了Bourgain--Brezis猜想的希尔伯特情形。随后我们发展了多线性反射估计，得到了新的临界Triebel--Lizorkin和Besov型Bourgain--Brezis估计。特别地，对任意维度和形式次数，Ẇ^{n/p,p}中的Sobolev型Bourgain--Brezis猜想在p=2k/(2k-1)（k≥1）时成立，因此对任意接近1的指数都成立。我们还推导了端点Hodge分解和Hodge--Sobolev不等式。最后，除了临界空间已嵌入L^∞的端点Besov情形外，我们证明了相关的有界选择算子不能是线性的。
英文摘要
We show that the dual approach to Bourgain--Brezis estimates for Hodge systems is substantially more flexible than previously understood. For $1\leq l\leq n-1$, we introduce the trace-free Beurling--Ahlfors transform $S=\frac{n-l}{n}P-\frac lnP^\perp$, 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 $\dot W^{\frac np,p}$ holds for $p=\frac{2k}{2k-1}$, $k\geq1$, 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.
