Generalization of I. Vekua's integral representations of holomorphic functions and their application to the Riemann-Hilbert-Poincaré problem (Q657309)

Vekua's integral representations hold for holomorphic functions, whose \(m\)-th derivative is Hölder-continuous in closed domains bounded by Lyapunov curves. The authors extend the representations to densities from variable exponent Lebesgue spaces \(L^{p(\cdot)} (\Gamma,\omega)\) with power weights \(\omega\). An integration curve \(\Gamma\) can have cusp points for certain \(p\) and \(\omega\). This makes it possible to generalize Vekua's results to Riemann-Hilbert-Poincaré problems. It is established that the solvability essentially depends on the geometry of \(\Gamma\) and of the functions \(p\) and \(\omega\).