Intrinsic nature of the Stein-Weiss H¹-inequality

This paper explores the intrinsic nature of the celebrated Stein-Weiss $H^1$-inequality $$ \|I_s u\|_{L^\frac{n}{n-s}}\lesssim \|u\|_{L^1}+\|\vec{R}u\|_{L^{1}}=\|u\|_{H^1} $$ through the tracing and duality laws based on Riesz's singular integral operator $I_s$. We discover that $f\in I_s\big([\mathring{H}^{s,1}_{-}]^\ast\big)$ if and only if $\exists\ \vec{g}=(g_1,...,g_n)\in \big(L^\infty\big)^n$ such that $f=\vec{R}\cdot\vec{g}=\sum_{j=1}^n R_jg_j$ in $\mathrm{BMO}$ (the John-Nirenberg space introduced in their 1961 {\it Comm. Pure Appl. Math.} paper \cite{JN}) where $\vec{R}=(R_1,...,R_n)$ is the vector-valued Riesz transform - this characterizes the Riesz transform part $\vec{R}\cdot\big(L^\infty\big)^n$ of Fefferman-Stein's decomposition (established in their 1972 {\it Acta Math} paper \cite{FS}) for $\mathrm{BMO}=L^\infty+\vec{R}\cdot\big(L^\infty\big)^n$ and yet indicates that $I_s\big([\mathring{H}^{s,1}_-]^\ast\big)$ is indeed a solution to Bourgain-Brezis' problem under $n\ge 2$: ``What are the function spaces $X, W^{1,n}\subset X\subset \mathrm{BMO}$, such that every $F\in X$ has a decomposition $F=\sum_{j=1}^n R_j Y_j$ where $Y_j\in L^\infty$?" (posed in their 2003 {\it J. Amer. Math. Soc.} paper \cite{BB}).