Musielak–Orlicz Hardy spaces associated to operators satisfying Davies–Gaffney estimates and bounded holomorphic functional calculus

Abstract

Let $X$ be a metric space with doubling measure and $L$ be an operator which satisfies Davies–Gaffney heat kernel estimates and has a bounded $H_\infty$ functional calculus on $L^2(X)$. In this paper, we develop a theory of Musielak–Orlicz Hardy spaces associated to $L$, including a molecular decomposition, square function characterization and duality of Musielak–Orlicz Hardy spaces $H_{L,\omega}(X)$. Finally, we show that $L$ has a bounded holomorphic functional calculus on $H_{L,\omega}(X)$ and the Riesz transform is bounded from $H_{L,\omega}(X)$ to $L^1(\omega)$.