Critical molecular theory and the maximal admissible class for Goldberg-type splittings of $h^p(\mathbb{R}^n)$, $0<p\le 1$

Abstract

We identify the largest class of functions certifying membership in $h^p(\mathbb{R}^n)$, $0<p\le 1$, through Goldberg's convolution splitting: a critical Hölder modulus and an $\ell^p$-Dini decay majorant, sharp on the regularity, decay, and cancellation axes. The class reaches the critical decay $|x|^{-n/p}$, governed by a molecular theory extending the classical Taibleson-Weiss theory to every radius, with the resulting embedding shown sharp.