On the atomic decomposition of 𝐻¹ and interpolation

Abstract

In [1] Coifman used the Fefferman-Stein theory of HP spaces [4] to decompose the functions of these spaces into basic building blocks (atoms), further clarifying their real variable nature. Coifman and Weiss have provided a comprehensive treatment of these ideas and many applications to harmonic analysis in [2]. In this note, we use the nontangential maximal function Nf to give an elementary proof of the decomposition of H' functions on the line and then characterize the Peetre K-functional for H' and L?? in terms of Nf. Let u be the harmonic extension [5] of f to the upper half plane R+. For x E R, denote by F2 = {(z, y) E R+: Ix zl < y} the cone with vertex at x. The nontangential maximal function of f is defined by Nf(x) = sup{Iu(z, y)I: (z, y) E F'}. We define the (real) H' norm of f to be the standard H' norm of u + iv, where v is the harmonic conjugate of u which satisfies v(O) = 0. A classical result of Hardy and Littlewood asserts that JINfIlL1 < clIfIlH1. For an interval I an H'-atom is any function aI such that f a, = 0 and Jail < III-'XI a.e.