Fefferman--Chang-type atomic decompositions for product Hardy spaces associated with Schrödinger operators

Abstract

We establish a geometric (Fefferman--Chang-type) atomic characterization of the two-parameter Hardy space associated with a pair of Schrödinger operators $L_i=-Δ+V_i$ on $\mathbb R^{n_i}$, $i=1,2$. The space is defined by the product conical square function. We assume comparability of neighbouring covering cubes, a local bound on the potentials, and decay of the semigroup mass at large scales. We prove that the Hardy norm is equivalent to a sum of classical local product Hardy norms at scales determined by the adapted coverings. This localization formula yields an atomic decomposition whose atoms satisfy size and partial cancellation conditions, with cancellation required only below the corresponding local scales. Both the atoms and their rectangular components remain supported inside fixed enlargements of the covering rectangles. The proof combines one-parameter Schrödinger Hardy space characterizations with standard vector-valued arguments and a localized product atomic decomposition with uniform control of scales and supports. The assumptions cover the model case of reverse Hölder potentials $V\in RH_q$, $q>n/2$, as well as potentials outside every $RH_q$ class, $q>1$, including half-space indicators and exponential potentials.