Hardy spaces meet harmonic weights revisited

We investigate Hardy spaces $H^1_L(X)$ corresponding to self-adjoint operators $L$. Our main aim is to obtain a description of $H^1_L(X)$ in terms of atomic decompositions similar to such characterisation of the classical Hardy spaces $H^1(\mathbb{R}^d)$. Under suitable assumptions, such a description was obtained by Yan and the authors in [Trans. Amer. Math. Soc. 375 (2022), no. 9, 6417-6451], where the atoms associated with an $L$-harmonic function are considered. Here we continue this study and modify the previous definition of atoms. The modified approach allows us to investigate settings, when the generating operator is related to a system of linearly independent harmonic functions. In this context, the cancellation condition for atoms is adjusted to fit this system. In an explicit example, we consider a symmetric manifold with ends $\mathbb{R}^d \# \mathbb{R}^d$. For this manifold the space of bounded harmonic functions is two-dimensional. Any element from the Hardy space $H^1_L(X)$ has to be orthogonal to all of the harmonic functions in the system.