Endpoint Boundedness of Linear Commutators on Local Hardy Spaces Over Metric Measure Spaces of Homogeneous Type

Abstract

Let $$({{\mathcal {X}}},d,\mu )$$ be a metric measure space of homogeneous type in the sense of Coifman and Weiss. In this article, the authors prove that the commutator, generated by any $$b\in \mathrm {BMO}({\mathcal {X}})$$ and any Calderon–Zygmund operator, is bounded from the Hardy type space $$H^1_b({\mathcal {X}})$$ to the local Hardy space $$H^1_{\rho }({\mathcal {X}})$$ associated with an admissible function $$\rho $$ , where $$H^1_b({\mathcal {X}})$$ is the largest subspace of the Hardy space $$H^1({\mathcal {X}})$$ that ensures the boundedness of commutators from $$H^1_b({\mathcal {X}})$$ to $$L^1({\mathcal {X}})$$ . Moreover, the authors investigate the relations between the Hardy space $$H^1_L({\mathbb {R}}^n)$$ associated with the Schrodinger operator L and the local Hardy space $$h^1({\mathbb {R}}^n)$$ . The major novelties of this article are that the main result even essentially improves the corresponding Euclidean case and, throughout this article, $$\mu $$ is not assumed to satisfy the reverse doubling condition.