Atomic decompositions for Hardy spaces related to Schrödinger operators
Volume 239 / 2017
Abstract
Let $\mathbf {L}^{U}= -\boldsymbol \Delta +U$ be a Schrödinger operator on ${\mathbb {R}^d}$, where $U\in L^1_{\rm loc}({\mathbb {R}^d})$ is a non-negative potential and $d\geq 3$. The Hardy space $H^1(\mathbf {L}^{U})$ is defined in terms of the maximal function of the semigroup $\mathbf {K}_{t}^{U} = \exp(-t\mathbf {L}^{U})$, namely $$H^1(\mathbf {L}^{U}) = \left \{f\in {L^1({\mathbb {R}^d})}:\| f \| _{H^1(\mathbf {L}^{U})}:= \left \| \sup_{t \gt 0} | \mathbf {K}_{t}^{U}f | \right\|_{L^1({\mathbb {R}^d})} \lt \infty \right\}.$$ Assume that $U=V+W$, where $V\geq 0$ satisfies the global Kato condition $$\sup_{x\in {\mathbb {R}^d}} \int _{{\mathbb {R}^d}} V(y)|x-y|^{2-d} \,dy \lt \infty .$$ We prove that, under certain assumptions on $W\geq 0$, the space $H^1(\mathbf {L}^{U})$ admits an atomic decomposition of local type. An atom $a$ for $H^1(\mathbf {L}^{U})$ either is of the form $a(x)=|Q|^{-1}\chi _Q(x)$, where $Q$ are special cubes determined by $W$, or satisfies the cancellation condition $\int _{\mathbb {R}^d}a(x)\omega (x)\, dx=0$, where $\omega $ is given by $\omega (x) = \lim_{t\to \infty } \mathbf {K}_{t}^{V}\mathbf {1}(x)$. Furthermore, we show that, in some cases, the above cancellation condition can be replaced by $\int _{\mathbb {R}^d}a(x)\, dx = 0$. However, we construct an example where the atomic spaces with these two cancellation conditions are not equivalent as Banach spaces.
