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

Let $L$ be the infinitesimal generator of an analytic semigroup on $L^2({\mathbb{R}}^n)$ with suitable upper bounds on its heat kernels. In Auscher, Duong, and McIntosh (2005) and Duong and Yan (2005), a Hardy space $H^1_L({\mathbb{R}}^n)$ and a ${\rm BMO}_L({\mathbb{R}}^n)$ space associated with the operator $L$ were introduced and studied. In this paper we define a class of $H^p_L({\mathbb{R}}^n)$ 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 $H^p({\mathbb{R}}^n)$ spaces. We then establish a duality theorem between the $H^p_L({\mathbb{R}}^n)$ spaces and the Morrey-Campanato spaces in that same paper. As applications, we obtain the boundedness of fractional integrals on $H^p_L({\mathbb{R}}^n)$ and give the inclusion between the classical $H^p({\mathbb{R}}^n)$ spaces and the $H^p_L({\mathbb{R}}^n)$ spaces associated with operators.