Logarithmic bump conditions for Calder\'on-Zygmund Operators on spaces of homogeneous type

TL;DR

This paper proves new two-weight inequalities for Calderón-Zygmund operators on spaces of homogeneous type, using bump conditions that generalize Euclidean results with simpler proofs and a unified approach.

Contribution

It introduces separated logarithmic bump conditions for two-weight inequalities, extending previous Euclidean results to more general spaces with simplified proofs.

Findings

Established two-weight inequalities under double bump conditions

Proved separated bump results as a consequence of double bump theorem

Generalized Euclidean case results to spaces of homogeneous type

Abstract

We establish two-weight norm inequalities for singular integral operators defined on spaces of homogeneous type. We do so first when the weights satisfy a double bump condition and then when the weights satisfy separated logarithmic bump conditions. Our results generalize recent work on the Euclidean case, but our proofs are simpler even in this setting. The other interesting feature of our approach is that we are able to prove the separated bump results (which always imply the corresponding double bump results) as a consequence of the double bump theorem.