2014/09/16 by Preisner, Marcin
#35J10 (primary) #42B25 #42B30 #42B35 (secondary) #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1409.4612
Let LU = -Delta+U be a Schrödinger operator on Rd, where U∈ L1loc(Rd) is a non-negative potential and d≥ 3. The Hardy space H1(LU) is defined in terms of the maximal function for the semigroup Kt,U = exp(-t LU), namely H1(LU) = f∈ L1(Rd): ‖f‖H1(LU):= ‖supt>0 |Kt,U f| ‖L1(Rd) < ∞. Assume that U=V+W, where V≥ 0 satisfies the global Kato condition supx∈ Rd ∫Rd V(y)|x-y|2-d < ∞. We prove that, under certain assumptions on W≥ 0, the space H1(LU) admits an atomic decomposition of local type. An atom a for H1(LU) is either of the form a(x)=|Q|-1χQ(x), where Q are special cubes determined by W, or a satisfies the cancellation condition ∫ a(x)w(x) dx = 0, where w is an (-Delta+V)-harmonic function given by w(x) = limt→ ∞ Kt,V 1(x). Furthermore, we show that, in some cases, the cancellation condition ∫Rd a(x)w(x) dx = 0 can be replaced by the classical one ∫Rd 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.