Characterizations of Hardy spaces associated to higher order elliptic operators

In this paper, the authors first show that the classical Hardy space H1(Rn) can be characterized by the non-tangential maximal functions and the area integrals associated with the semigroups e−tP and e−tP, respectively, where P is an elliptic operator with real constant coefficients of homogeneous order 2m (m⩾1). Moreover, the authors also prove that H1(Rn) can be characterized by the Riesz transforms ∇mP−1/2 if and only if m is an odd integer. In the main part of this paper, the authors develop a theory of Hardy space associated with L, where L is a higher order divergence form elliptic operator with complex bounded measurable coefficients. The authors set up a molecular Hardy space HL1(Rn) and give its characterizations by area integrals related to the semigroups e−tL and e−tL, respectively. Finally, authors give the (HL1,L1) boundedness of Riesz transforms, square functions and maximal functions associated with L.