The Riemann-Hilbert problem in the class of Cauchy type integrals with densities of grand Lebesgue spaces (Q325103)

Let \(D\) be a domain in the complex plane whose boundary \(\Gamma\) is a simple rectifiable curve. The authors investigate the solvability and give the solutions, when they exist, of the Riemann-Hilbert problem \(\text{Re}\,(\lambda(t)\phi^{+}(t))=b(t)\) in the set of functions representable as Cauchy-type integrals NEWLINENEWLINE\[NEWLINE \phi(z)=\frac{1}{2\pi i}\int_{\Gamma}\frac{\varphi(t)}{t-z}\,dt, \quad z\in D,NEWLINE\]NEWLINE with densities \(\varphi\) from the grand Lebesgue space \(L^{p),\theta}\), \(1<p<\infty\), \(\theta>0\), under the following assumptions: \(\lambda\in C(\Gamma)\), \(\lambda(t)\neq 0\), \(t\in\Gamma\), \(b\in L^{p),\theta}\) is a real-valued function.