The finite Hilbert transform acting on $L^\infty$
Guillermo P. Curbera, Susumu Okada, Werner J. Ricker
TL;DR
The paper analyzes the finite Hilbert transform $T$ acting on $L^\infty(-1,1)$ with values in the Zygmund space $L_{\mathrm{exp}}(-1,1)$, complementing prior work on $T$ from $L\log L$ to $L^1$ and highlighting non-separability effects. It establishes boundedness $T: L^\infty\to L_{\mathrm{exp}}$ and develops the auxiliary operator $\widecheck{T}$ as a functional inverse on the range, linking $T$ and $\widecheck{T}$ via $T\widecheck{T}(f)=f-Q(f)$ and $\widecheck{T}T(f)=f$ for $f\in L^\infty$. The range of $T$ is characterized as a proper subset of $\ker(\varphi_{1/w})$, with $w(x)=\sqrt{1-x^2}$, and a full inverse theory is obtained: $g\in T(L^\infty)$ iff $\widecheck{T}(g)\in L^\infty$ and $\int g/w=0$, enabling inversion $f=\widecheck{T}(g)$ when $g=T(f)$. The paper also proves the optimality of the domain for $T$ taking values in $L_{\mathrm{exp}}$, showing $[T,L_{\mathrm{exp}}]=L^\infty$ and that no continuous linear extension to a larger domain exists, thereby clarifying the operator's domain-range structure in this non-separable setting.
Abstract
The action of the finite Hilbert transform defined on $L^\infty(-1,1)$ and taking its values in the Zygmund space $L_{\textnormal{exp}}(-1,1)$ is studied in detail. This is a reciprocal situation to the investigation recently undertaken in [11] of the finite Hilbert transform defined on the Zygumd space $L\textnormal{log} L(-1,1)$ and taking its values in $L^1(-1,1)$. The fact that both $L^\infty(-1,1)$ and $L_{\textnormal{exp}}(-1,1)$ fail to be separable generates new features not present in[11].