Intrinsic nature of the Stein-Weiss $H^1$-inequality

Abstract

This paper explores the intrinsic nature of the celebrated Stein-Weiss $H^1$-inequality $$\Vert I_{s}u{\Vert }_{L^{\frac{n}{n-s}}}\lesssim \Vert u{\Vert }_{L^1}+\Vert \vec{R}u{\Vert }_{L^1}=\Vert u{\Vert }_{H^1}$$ through the tracing and duality laws based on Riesz's singular integral operator $I_s$. We discover that $f\in I_s([{\mathring{H}}_{-}^{s,1}{]}^{\ast })$ if and only if $\exists \vec{g}=(g_1,...,g_n)\in (L^{\infty }{)}^n$ such that $f=\vec{R}\cdot \vec{g}={\sum }_{j=1}^n{R}_j{g}_j$ 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 $\vec{R}\cdot (L^{\infty }{)}^n$ of Fefferman-Stein's decomposition (established in their 1972 {\it Acta Math} paper \cite{FS}) for $\mathrm{BMO}=L^{\infty }+\vec{R}\cdot (L^{\infty }{)}^n$ and yet indicates that $I_s([{\mathring{H}}_{-}^{s,1}{]}^{\ast })$ 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}).

38 pages