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 optimal domain). Some classical operators are already optimally defined (e.g., the Hilbert transform in Lp(R) , 1 < p