Atomic decompositions for Hardy spaces related to Schr\"odinger operators

Abstract

Let L_U = -Delta+U be a Schr\"odinger operator on R^d, where U\in L^1_{loc}(R^d) is a non-negative potential and d\geq 3. The Hardy space H^1(L_U) is defined in terms of the maximal function for the semigroup K_{t,U} = exp(-t L_U), namely H^1(L_U) = {f\in L^1(R^d): \|f\|_{H^1(L_U)}:= \|sup_{t>0} |K_{t,U} f| \|_{L^1(R^d)} < \infty. Assume that U=V+W, where V\geq 0 satisfies the global Kato condition sup_{x\in R^d} \int_{R^d} V(y)|x-y|^{2-d} < \infty. We prove that, under certain assumptions on W\geq 0, the space H^1(L_U) admits an atomic decomposition of local type. An atom a for H^1(L_U) is either of the form a(x)=|Q|^{-1}\chi_Q(x), where Q are special cubes determined by W, or a satisfies the cancellation condition \int a(x)w(x) dx = 0, where w is an (-Delta+V)-harmonic function given by w(x) = lim_{t\to \infty} K_{t,V} 1(x). Furthermore, we show that, in some cases, the cancellation condition \int_{R^d} a(x)w(x) dx = 0 can be replaced by the classical one \int_{R^d} a(x) dx = 0. However, we construct another example, such that the atomic spaces with these two cancellation conditions are not equivalent as Banach spaces.