The Fefferman-Stein decomposition of smooth functions and its application to \(H^ p({\mathbb{R}}^ n)\)
In [Acta Math. 129, 137-193 (1972; Zbl 0257.46078)], \textit{C. Fefferman} and \textit{E. M. Stein} proved that any function of bounded mean oscillation can be written as the sum of an \(L^{\infty}\) function and the Riesz transforms of \(L^{\infty}\) functions. In [Acta Math. 148, 215-241 (1982; Zbl 0514.46018)], \textit{A. Uchiyama} gave a constructive method for producing this decomposition of BMO functions. In the present paper A. Uchiyama continues his program by proving a so-called Fefferman- Stein decomposition for essentially arbitrary functions. This new result is quantitative in nature and contains the usual Fefferman-Stein decomposition as a simple corollary. The paper is too technical to discuss here in any detail. But it should be mentioned that results of Coifman/Rochberg on functions of bounded lower oscillation, as well as some constructions of P. Jones concerning bounded solutions of the \({\bar \partial}\) operator in a half space are used. Also Uchiyama's techniques with vector-valued singular integrals, as in the Acta paper, are used systematically. Some new kinds of maximal inequalities for vector-valued singular integrals are obtained in the process.
scientific article; zbMATH DE number 3895623