Hardy spaces associated to the discrete Laplacians on graphs and boundedness of singular integrals

Abstract:

Let $\Gamma$ be a graph with a weight $\sigma$. Let $d$ and $\mu$ be the distance and the measure associated with $\sigma$ such that $(\Gamma , d, \mu )$ is a doubling space. Let $p$ be the natural reversible Markov kernel associated with $\sigma$ and $\mu$ and $P$ be the associated operator defined by $Pf(x) = \sum _{y} p(x, y)f(y)$. Denote by $L=I-P$ the discrete Laplacian on $\Gamma$. In this paper we develop the theory of Hardy spaces associated to the discrete Laplacian $H^p_L$ for $0<p\leq 1$. We obtain square function characterization and atomic decompositions for functions in the Hardy spaces $H^p_L$, then establish the dual spaces of the Hardy spaces $H^p_L, 0<p\leq 1$. Without the assumption of Poincaré inequality, we show the boundedness of certain singular integrals on $\Gamma$ such as square functions, spectral multipliers and Riesz transforms on the Hardy spaces $H^p_L$, $0<p\leq 1$.