Endpoint boundedness of commutators on spaces of homogeneous type

Abstract

Let be a space of homogeneous type in the sense of Coifman and Weiss. Let be a family of sublinear operators that include the well-known Calderón–Zygmund operator. The authors prove that the commutator [b, T] is bounded from a Hardy-type subspace of to , where denotes the Lebesgue space of all integrable functions, the atomic Hardy space of Coifman and Weiss with the dual space , b is a non-constant -function and . Indeed, the space is the largest subspace in that possesses this property. The approach taken in this article adopts the bilinear decomposition theory for the product of functions in and .

Keywords:

• Space of homogeneous type
• bilinear decomposition
• commutator
• Calderón–Zygmund operator
• Hardy space
• BMO

AMS Subject Classifications:

• Primary: 47B47
• Secondary: 42B20
• 42B30
• 42C40
• 30L99

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This project is supported by the National Natural Science Foundation of China [grant number 11471042], [grant number 11571039], [grant number 11671185]. Der-Chen Chang is partially supported by an NSF [grant number DMS-1408839]; a McDevitt Endowment Fund at Georgetown University.