Classes Of hardy spaces associated with operators, duality theorem and applications

Let L be the infinitesimal generator of an analytic semigroup on L ²(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 BMO_l(ℝⁿ) space associated with the operator L were introduced and studied. In this paper we define a class of Hp/L(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/l(ℝⁿ) spaces. We then establish a duality theorem between the Hp L(ℝⁿ) spaces and the Morrey-Campanato spaces in that same paper. As applications, we obtain the boundedness of fractional integrals on Hp/L(ℝⁿ) and give the inclusion between the classical H ^p(ℝⁿ) spaces and the Hp/L(ℝⁿ) spaces associated with operators.