Hilbert boundary value problem in half-plane in the sense of \(L^1\)-convergence (Q1278913)

Let \(G^+,G^-\) be the upper and lower half-planes of the complex plane, respectively. Let \(Ay\) be the class of analytic outside the real axis functions \(\Phi\) satisfying the condition \(|\Phi(z)|<c| z|^m\), within each region \(|\text{Im} z|\geq y>0\). In this paper, the authors consider the following Hilbert type boundary value problem which consists in finding functions \(\Phi\in Ay\) satisfying the condition \[ \lim_{y\to 0^+} \bigl\|\Phi^+(x+iy)-a(x)\Phi^-(x-iy)-f(x)\bigr\|_1=0, \] where \(f\in L^1(-\infty, +\infty)\), \(\|\;\|_1\) is the norm of the space \(L^1(-\infty, +\infty)\), the function \(a\not\equiv 0\) is piecewise continuous in the Hölder sense. The conditions for solvability and the general solution of this problem are given.