On the possibility of extending a function from a part of a domain to a generalized analytic function on this domain (Q2713987)

Let \(D\) be a simply connected bounded domain whose boundary consists of a smooth curve \(\gamma\) and either an interval or some part of a circle. Given a Hölder function \(\varphi(z)\) defined on \(\gamma\), the problem is to find a generalized analytic function \(W(z)\) on \(D\), i.e., \(W(z)\) meets the equation NEWLINE\[NEWLINE \partial_{\bar{z}}W(z) + A(z)W(z)+B(z)\overline{W(z)}=0 \tag{1} NEWLINE\]NEWLINE and is such that \(W(z)|_{\gamma}=\varphi(z)\). Here the functions \(A(z)\) and \(B(z)\) also satisfy the Hölder condition on \(D\) and some additional conditions. The author establishes necessary and sufficient conditions on the function \(\varphi\) for this problem to have a positive solution. The conditions are given in terms of the behavior of some special integral operators with respect to a parameter. Explicit representations for the function \(W\) are also presented.