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arXiv · 2212.08835

The finite Hilbert transform acting in the Zygmund space LlogL

Guillermo P. Curbera·Susumu Okada·Werner J. Ricker
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Abstract

The finite Hilbert transform T is a singular integral operator which maps the Zygmund space $LlogL:=LlogL(-1,1)$ continuously into $L^1:=L^1(-1,1)$. By extending the Parseval and Poincar\'e-Bertrand formulae to this setting, it is possible to establish an inversion result needed for solving the airfoil equation $T(f)=g$ whenever the data function $g$ lies in the range of $T$ within $L^1$ (shown to contain $LlogL$). Until now this was only known for $g$ belonging to the union of all $L^p$ spaces with $p>1$. It is established (due to a result of Stein) that $T$ cannot be extended to any domain space beyond $LlogL$ whilst still taking its values in $L^1$, i.e., $T:LlogL\to L^1$ is optimally defined.

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Guillermo P. Curbera, Susumu Okada, Werner J. Ricker. 2022-12-17. The finite Hilbert transform acting in the Zygmund space LlogL. https://arxiv.org/abs/2212.08835

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