Unitarity of singular integral operators on Riemann surfaces (Q1333093)

On a compact Riemann surface of genus \(g\), one deals with the Hilbert transform \((Hf)(x)= \int^ \infty_{- \infty} {f(t) dt\over t- x}\), and the Calderon-Zygmund integral \(\iint_ M {f(\tau) d\sigma_ t\over (\tau- z)^ 2}\). These are unitary operators on \(L^ 2\), a feature that plays an important role in harmonic analysis. The author is mainly concerned with the case of compact Riemann surfaces with a measure \(d\sigma\), different from zero on the whole surface. By using multivalued Cauchy kernels and related algebraic properties of Abelian integrals, one establishes connections with boundary value problems that are encountered in soliton theory. References are made to the author's book [The Riemann boundary problem on Riemann surfaces, Dordrecht (1988; Zbl 0695.30040)] and the book by \textit{V. E. Zakharov}, \textit{S. V. Manakov}, \textit{S. Novikov} and \textit{L. P. Pitaevskij} [Soliton theory; the inverse problem method, Plenum Press, New York (1984; Zbl 0598.35002)].