Marcinkiewicz multipliers on the real line are bounded functions of uniformly bounded variation on each Littlewood–Paley dyadic interval. Optimal weak-type endpoint estimates for the corresponding multiplier operators have been studied by Tao and Wright showing that these operators map locally $L \log^{1/2} L$ to weak $L^1$.

In this seminar, we consider higher-order Marcinkiewicz multipliers, that is, multipliers of uniformly bounded variation on each interval arising from a higher-order lacunary partition of the real line. We present optimal weak-type endpoint estimates for the associated multiplier operators. These are obtained as a consequence of more general endpoint estimates that we establish for a higher-order variant of a class of multipliers introduced by Coifman, Rubio de Francia, and Semmes and further studied by Tao and Wright. As a byproduct, this also yields optimal endpoint bounds for higher-order Hörmander–Mihlin multipliers. Central to our analysis is a family of generalized Zygmund–Bonami type inequalities which are related to a dual version of the Chang–Wilson–Wolff inequality.

The talk is based on joint work with Odysseas Bakas, Ioannis Parissis, and Marco Vitturi.