On a Hilbert problem from the \(L^ p\)-convergence viewpoint (Q1901431)

Let \(D^+\) be the unit disk, \(\Gamma= \partial D^+\), \(D^-= \mathbb{C}\setminus \overline {D}^+\) and let \(A(\Gamma)\) be the class of analytic in \(D^+ \cup D^-\) functions which have finite order in infinity. The author considers the following problem: find a function \(\Phi\in A\) satisfying the condition \[ \lim_{r\to 1-0} |\Phi^+ (rt)- a(t) \Phi^- (r^{-1} t)- f(t) |_p =0, \qquad p\geq 1, \] where \(f\in L^p (\Gamma)\), \(\Phi^\pm\) are the restrictions of \(\Phi\) on \(D^\pm\). A necessary and sufficient condition for normal solvability and Noetherness of this problem is given.