Intrinsic nature of the Stein-Weiss $H^1$-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 Rieszs singular integral operator $I_s$. We discover that $fin I_sbig([mathring{H}^{s,1}_{-}]^astbig)$ if and only if $exists vec{g}=(g_1,...,g_n)in big(L^inftybig)^n$ such that $f=vec{R}cdotvec{g}=sum_{j=1}^n R_jg_j$ 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 $vec{R}cdotbig(L^inftybig)^n$ of Fefferman-Steins decomposition (established in their 1972 {it Acta Math} paper cite{FS}) for $mathrm{BMO}=L^infty+vec{R}cdotbig(L^inftybig)^n$ and yet indicates that $I_sbig([mathring{H}^{s,1}_-]^astbig)$ is indeed a solution to Bourgain-Brezis problem under $nge 2$: ``What are the function spaces $X, W^{1,n}subset Xsubset mathrm{BMO}$, such that every $Fin X$ has a decomposition $F=sum_{j=1}^n R_j Y_j$ where $Y_jin L^infty$? (posed in their 2003 {it J. Amer. Math. Soc.} paper cite{BB}).