Critical molecular theory and the maximal admissible class for Goldberg-type splittings of $h^p(\mathbb{R}^n)$, $0<p\le 1$
We extend the classical Taibleson-Weiss molecular theory to arbitrary radii, reaching the critical decay |x|^{-n/p}, with the resulting embedding shown sharp. This molecular theory governs the largest class of functions certifying membership in h^p(R^n), 0 < p <= 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.
math.CA↗