Uniqueness Results for the Riemann-Hilbert Problem with a Vanishing Coefficient

Abstract.

We study the Riemann-Hilbert problem of finding φ, ψ ∈ H^p such that their nontangential boundary values satisfy the equation

$$\varphi ^* = a\overline {\psi ^* } ,$$

where $a\,:\,\mathbb{R}\,\to\,\mathbb{C}$ is a given 2π-periodic continuous function. We prove the nonexistence of nontrivial solutions for a wide class of continuous vanishing complex-valued coefficients a.