Hardy spaces associated to critical functions and applications to T1 theorems

Let X be a metric space equipped with a measure satisfying the doubling and reverse doubling conditions. In this paper, we develop the theory of new localized Hardy spaces H^ρ_p(X) for n/n+1<p≤1 associated to critical functions ρ defined on X where n is the doubling order. Our results include the atomic decomposition characterization and the maximal function characterization associated to an approximation of the identity. We then study the T₁ criteria for singular integrals to be bounded on our Hardy spaces.