Boundedness of singular integrals in Hardy spaces on spaces of homogeneous type (Q1025696)

Let \(\mathcal{X}\) be a set with a positive Borel regular measure \(\mu\) and a quasi-metric \(d\) satisfying that there exists \(C_1\geq 1\) such that for all \(x,y,z\in \mathcal{X}\), \[ d(x,y)\leq C_1 (d(x,z)+d(y,z)). \] The triple \((\mathcal{X},d,\mu)\) is said to be a space of homogeneous type in the sense of Coifman and Weiss if \(\mu\) is doubling, namely, there exists \(C_2\geq 1 \) such that for all \(x\in\mathcal{X}\) and \(r>0\), \[ \mu (B_d(x,2r))\leq C_2 \mu (B_d(x,r)), \] where \(B_d(x,r)=\{y\in \mathcal{X}:d(x,y)<r\}\). The authors first give a detailed proof on the coincidence between atomic Hardy spaces of Coifman and Weiss on a space of homogeneous type with those Hardy spaces on the same underlying space with the original distance replaced by the measure distance. Then the authors present some general criteria which guarantee the boundedness of considered linear operators from a Hardy space to some Lebesgue space or Hardy space, provided that it maps all atoms into uniformly bounded elements of that Lebesgue space or Hardy space. Third, the authors obtain the boundedness in Hardy spaces of singular integrals with kernels only having weak regularity by characterizing these Hardy spaces with a new kind of molecules, which is deeply related to the kernels of considered singular integrals. Finally, as an application, the authors obtain the boundedness in Hardy spaces of Monge-Ampère singular integral operators.