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The principle of optimizing inequalities, or their equivalent operatortheoretic formulation, is well established in analysis. For an operator, thiscorresponds to extending its action to larger domains, hopefully to the largestpossible such domain (i.e, its \textit{optimal domain}). Some classicaloperators are already optimally defined (e.g., the Hilbert transform in$L^p(\mathbb{R})$, $1<p<\infty$) and others are not (e.g., the Hausdorff-Younginequality in $L^p(\mathbb{T})$, $1<p<2$, or Sobolev's inequality in variousspaces). In this paper a detailed investigation is undertaken of the finiteHilbert transform $T$ acting on rearrangement invariant spaces $X$ on $(-1,1)$,an operator whose singular kernel is neither positive nor does it possess anymonotonicity properties. For a large class of such spaces $X$ it is shown that$T$ is already optimally defined on $X$ (this is known for $L^p(-1,1)$ for all$1<p<\infty$, except $p=2$). The case $p=2$ is significantly different becausethe range of $T$ is a proper dense subspace of $L^2(-1,1)$. Nevertheless, by acompletely different approach, it is established that $T$ is also optimallydefined on $L^2(-1,1)$. Our methods are also used to show that the solution ofthe airfoil equation, which is well known for the spaces $L^p(-1,1)$ whenever$p\not=2$ (due to certain properties of $T$), can also be extended to the classof r.i.\ spaces $X$ considered in this paper.

Inversion and extension of the finite Hilbert transform on (-1,1)