The finite Hilbert transform acting on $L^\infty $
Studia Mathematica
MSC: Primary 44A15; Secondary 46E30, 47A53, 47B34
DOI: 10.4064/sm260204-8-6
Published online: 29 October 2026
Abstract
The action of the finite Hilbert transform defined on $L^\infty (-1,1)$ and taking its values in the Zygmund space $L_{\operatorname{exp}}(-1,1)$ is studied in detail. This is a reciprocal situation to the investigation, recently undertaken by the present authors, of the finite Hilbert transform defined on the Zygmund space $L\log L(-1,1)$ and taking its values in $L^1(-1,1)$. The fact that both $L^\infty (-1,1)$ and $L_{\operatorname{exp}}(-1,1)$ fail to be separable generates new features not present in [Ann. Scuola Norm. Sup. Pisa 25 (2024), 1527–1557].
