Inversion and extension of the finite Hilbert transform on \((-1,1)\) (Q2332916)

The authors study inversion and extension of the finite Hilbert transform \(T\) on rearrangement invariant (r.i.) spaces \(X\) on \((-1,1)\). Further, it is shown that the solution of the airfoil equation can also be extended to the class of r.i. spaces \(X\). Note that 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 \(1<p<\infty, \ p\neq 2\)). The case \(p=2\) is significantly different because the range of \(T\) is a proper dense subspace of \(L^2(-1,1)\). It is established that \(T:L^2(-1,1)\rightarrow L^2(-1,1)\) does not have a continuous \(L^2(-1,1)\)-valued extension to any larger Banach function space.