Extension and Integral Representation of the finite Hilbert Transform In Rearrangement Invariant Spaces

The finite Hilbert transform $T$ is a classical (singular) kernel operator which is continuous in every rearrangement invariant space $X$ over $(-1,1)$ having non-trivial Boyd indices. For $X=L^p$, $1<p<\mathrm{\infty }$, this operator has been intensively investigated since the 1940’s (also under the guise of the ``airfoil equation’'). Recently, the extension and inversion of $T\text{\hspace{0.17em}⁣}:X\to X$ for more general $X$ has been studied in G. P. Curbera, S. Okada, W. J. Ricker, Inversion and extension of the finite Hilbert transform on $(-1,1)$, Ann. Mat. Pura Appl. 198 (2019), 1835-1860, where it is shown that there exists a larger space $[T,X]$, optimal in a well defined sense, which contains $X$ continuously and such that $T$ can be extended to a continuous linear operator $T\text{\hspace{0.17em}⁣}:[T,X]\to X$. The purpose of this paper is to continue this investigation of $T$ via a consideration of the $X$-valued vector measure $m_X\text{\hspace{0.17em}⁣}:A\mapsto T({\chi }_A)$ induced by $T$ and its associated integration operator $f\mapsto {\int }_{-1}^{1}fd{m}_X$. In particular, we present integral representations of $T\text{\hspace{0.17em}⁣}:X\to X$ based on the $L^1$-space of $m_X$ and other related spaces of integrable functions.