  The principle of optimizing inequalities, or their equivalent operator
theoretic formulation, is well established in analysis. For an operator, this
corresponds to extending its action to larger domains, hopefully to the largest
possible such domain (i.e, its \textit{optimal domain}). Some classical
operators 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-Young
inequality in $L^p(\mathbb{T})$, $1<p<2$, or Sobolev's inequality in various
spaces). In this paper a detailed investigation is undertaken of the finite
Hilbert transform $T$ acting on rearrangement invariant spaces $X$ on $(-1,1)$,
an operator whose singular kernel is neither positive nor does it possess any
monotonicity 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 because
the range of $T$ is a proper dense subspace of $L^2(-1,1)$. Nevertheless, by a
completely different approach, it is established that $T$ is also optimally
defined on $L^2(-1,1)$. Our methods are also used to show that the solution of
the 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 class
of r.i.\ spaces $X$ considered in this paper.
