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

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.