Lusin characterisation of hardy spaces associated with hermite operators

Abstract

Let d ∈ {3, 4, 5,...} and p ∈ (0, 1]. We consider the Hermite operator L = −Δ + |x|2 on its maximal domain in L2(ℝd). Let HpL(Rd) be the completion of {f∈L2(Rd):MLf∈Lp(Rd)} with respect to the quasi-norm ∥⋅∥HpL=∥ML⋅∥Lp, where MLf(⋅)=supt>0∣∣e−tLf(⋅)∣∣ for all f ∈ L2(ℝd). We characterise HpL(Rd) in terms of Lusin integrals associated with the Hermite operator for p∈(dd+1,1].