Classes of Hardy spaces associated with operators, duality theorem and applications

Let L be the infinitesimal generator of an analytic semigroup on L²( Rⁿ) with suitable upper bounds on its heat kernels. In Auscher, Duong, and McIntosh (2005) and Duong and Yan (2005), a Hardy space H¹L( Rⁿ) and a BMOL( Rⁿ) space associated with the operator L were introduced and studied. In this paper we define a class of HᵖL( Rⁿ) spaces associated with the operator L for a range of $p<1$ acting on certain spaces of Morrey-Campanato functions defined in New Morrey-Campanato spaces associated with operators and applications by Duong and Yan (2005), and they generalize the classical Hᵖ( Rⁿ) spaces. We then establish a duality theorem between the HᵖL( Rⁿ) spaces and the Morrey-Campanato spaces in that same paper. As applications, we obtain the boundedness of fractional integrals on HᵖL( Rⁿ) and give the inclusion between the classical Hᵖ( Rⁿ) spaces and the HᵖL( Rⁿ) spaces associated with operators.