On finite Hilbert transforms

Abstract:

Let E be a bounded measurable subset of the real line. The finite Hilbert transform is the operator ${H_E}$ defined on one of the spaces ${L^p}(E)(1 < p < \infty )$ by \[ {H_E}f(x) = {(\pi i)^{ - 1}}\int _E {f(t){{(t - x)}^{ - 1}}\;dt;} \] here, the singular integral is interpreted as a Cauchy principal value. The main result establishes that for ${H_E}$ to be Fredholm on ${L^p}(E)$, when $p \ne 2$, it is necessary and sufficient that E be equal almost everywhere to a finite union of intervals. The sufficiency of this condition was established in 1960 by H. Widom. In the case where E is not a finite union of intervals and $p < 2$ it is shown that the operator ${H_E}$ has an infinite dimensional null space. The method of proof is constructive.