“Endpoint estimates for Fourier multipliers with Zygmund singularities”
Mercoledì 12 Marzo 2025, ore 14:30 - Aula 1BC50 - Marco Fraccaroli (BCAM - Basque Center for Applied Mathematics, Bilbao)
Abstract
The Hilbert transform maps $L^1$ functions into $L^{1,∞}$ ones. In fact, this estimate holds true for any operator $\mathrm{T}_m$ defined by a Fourier multiplier m with singularity only in the origin. Tao and Wright identified the space replacing $L^1$ in the endpoint estimate for $\mathrm{T}_m$ when $m$ has singularities in a lacunary set of frequencies, in the sense of the Hörmander-Mihlin condition.
In this talk we will discuss the characterization of the endpoint estimate for $\mathrm{T}_m$ for any arbitrary $m$ by the lack of additivity of its set of singularities $\Xi$. This property of $\Xi$ is expressed in terms of a Zygmund-type inequality. The main ingredient in the proof of the estimate is a multi-frequency projection lemma based on Gabor expansion playing the role of Calderón-Zygmund decomposition.
The talk is based on joint work with Bakas, Ciccone, Di Plinio, Parissis, and Vitturi.
