On Hardy spaces associated with certain Schrödinger operators in dimension 2

Abstract

We study the Hardy space $H^1$ associated with the Schrödinger operator $L=-\Delta +V$ on ${\mathbb{R}}^2$, where $V\ge 0$ is a compactly supported non-zero $C^2$-potential. We prove that this space, which is originally defined by means of the maximal function associated with the semigroup generated by $-L$, admits a special atomic decomposition with atoms satisfying a weighted cancellation condition with a weight of logarithmic growth.