Analysis Seminar: Optimal Sparse Bounds and Commutator Characterizations Without Doubling

Speaker: Brett Wick, Washington University in St. Louis

Title:  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 to be doubling. We first establish a pointwise sparse domination for dyadic paraproducts and related operators with symbols $b \in \mathrm{BMO}(\mu)$, improving on an earlier result of Lacey, where the symbol $b$ was assumed to satisfy a stronger Carleson-type condition that coincides with $\mathrm{BMO}(\mu)$ only in the doubling setting. This allows us to obtain sharpened weighted inequalities for the commutator of a dyadic Hilbert transform $H$ previously studied by T. Borges, J. Conde Alonso, J. Pipher, and N. A. Wagner. We also characterize the symbols for which the commutator $[H, b]$ is bounded on $L^p(\mu)$ for $1 < p < \infty$, showing that this class of symbols strictly depends on $p$ and is nested between symbols satisfying the $p$-Carleson packing condition and symbols in $\mathrm{BMO}$. This is joint work with F. D'Emilio, Y. Lin, and N. A. Wagner.

Date/Time: Friday, September 18, 11:30am

Location: Exploratory Hall, Room 4106 or Zoom