Endpoint estimates for Fourier multipliers with Zygmund singularities

Abstract: The Hilbert transform maps $L^1$ functions into $L^{1,\infty}$ ones. In fact, this estimate holds true for any operator $T_m$ defined by a bounded Fourier multiplier $m$ with singularity only in the origin. Tao and Wright identified the space replacing $L^1$ in the endpoint estimate for $T_m$ when $m$ has singularities in a lacunary set of frequencies, in the sense of the Hörmander-Mihlin condition.

In this talk we will quantify how the endpoint estimate for $T_m$ for any arbitrary $m$ is characterized by the lack of additivity of its set of singularities $\Xi(m)$ . This property of $\Xi(m)$ 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.

analysis of PDEsclassical analysis and ODEsfunctional analysis

Audience: researchers in the topic

OARS Online Analysis Research Seminar

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