Optimal Sparse Bounds and Commutator Characterizations Without Doubling

Abstract: We examine dyadic paraproducts and commutators in the non-homogeneous setting, where the underlying Borel measure $\mu$ is not assumed doubling. We first establish a pointwise sparse domination for dyadic paraproducts and related operators with symbols $b \in BMO(\mu)$, improving upon a result of Lacey, where $b$ satisfied a stronger Carleson-type condition coinciding with BMO only in the doubling case. As an application, we derive sharpened weighted inequalities for the commutator of a dyadic Hilbert transform H previously studied by Borges, Conde Alonso, Pipher, and Wagner. We also characterize the symbols for which $[H,b]$ is bounded on $L^p$ for $1

analysis of PDEsclassical analysis and ODEsfunctional analysis

Audience: researchers in the topic

OARS Online Analysis Research Seminar

Series comments: Visit our homepage for further information. Some recordings available on YouTube