Analysis Seminar: scalar and vector valued dyadic operators in the non-homogeneous setting

Abstract: In the classical setting, dyadic analysis has proven to be a powerful framework for studying weighted inequalities for both scalar and vector valued Calderón–Zygmund operators. While the scalar theory is now well understood, sharp matrix-weighted inequalities are not fully resolved. The best-known matrix-weighted L^p estimates (when Lebesgue is the underlying measure) were obtained in 2018 by Cruz-Uribe, Isralowitz, and Moen, and were shown to be sharp— though only for p=2 and for specific Calderón–Zygmund operators—by Domelevo, Petermichl, Treil, and Volberg in 2024. We first present a new proof of these best-known estimates in the non-homogeneous setting, i.e., when the underlying measure is not assumed to be doubling. This extension, previously out of reach using existing techniques, relies instead on a generalization of the weighted Carleson embedding theorem.

Second, we explore the non-homogeneous setting in greater detail. A striking discrepancy emerges between dyadic and continuous models, as the necessary structural assumptions on the underlying space appear different and unrelated. We describe recent progress on dyadic operators—including shifts, paraproducts, martingale transforms, and their commutators with BMO symbols—and highlight several unexpected phenomena. These include weighted estimates that depend on the operator's complexity, failures of BMO to characterize commutators, and p-dependent continuity properties. These results shed light on a landscape where familiar intuitions from the classical, homogeneous setting no longer fully apply.

Host: Alan Chang

Note: This talk is given in partial fulfillment of the Third Year Candidacy Requirement.