The finite Hilbert transform acting on $L^\infty$

TL;DR

This paper investigates the finite Hilbert transform's behavior on $L^ extnormal{infty}$ functions, revealing new features due to non-separability of involved spaces, contrasting prior work on related spaces.

Contribution

It provides a detailed analysis of the finite Hilbert transform on $L^ extnormal{infty}$ and the Zygmund space, highlighting novel properties arising from non-separable spaces.

Findings

Finite Hilbert transform maps $L^ extnormal{infty}$ into $L_{ extnormal{exp}}$

Non-separability of spaces introduces new mathematical features

Contrasts with previous results on $L extnormal{log} L$ and $L^1$ spaces

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].