Annals of Communications in Mathematics

ABSTRACT.Let \( D \) be a connected bounded domain in \( \mathbb{R}^{2} \), \( S \) be its boundary which is closed, connected and smooth or \( S = (-\infty,\infty) \). Let\[\Phi(z) = \frac{1}{2\pi i}\int_{S} \frac{f(s)\, ds}{s - z}, \qquad f \in L^{1}(S), \; z = x + iy.\]The singular integral operator\[Af := \frac{1}{\pi i}\int_{S} \frac{f(s)\, ds}{s - t}, \qquad t \in S,\]is defined in a new way. This definition simplifies the proof of the existence of \( \Phi(t) \). Necessary and sufficient conditions are given for \( f \in L^{1}(S) \) to be a boundary value of an analytic function in \( D \). The Sokhotsky–Plemelj formulas are derived for \( f \in L^{1}(S) \). Our new definition allows one to treat singular boundary values of analytic functions.

ABSTRACT.Let \( D \) be a connected bounded domain in \( \mathbb{R}^{2} \), \( S \) be its boundary which is closed, connected and smooth. Let\[\Phi(z) = \frac{1}{2\pi i} \int_{S} \frac{\phi(s)\, ds}{s - z}, \qquad \phi \in X, \; z = x + iy,\]\( X \) is a Banach space of linear bounded functions on \( H^{\mu} \), a Banach space of distributions, and \( H^{\mu} \) is the Banach space of Hölder-continuous functions on \( S \) with the usual norm. As \( X \) one can use also the space Hölder continuous of bounded linear functionals on the Sobolev space \( H^{\ell} \) on \( S \). Distributional boundary values of \( \Phi(z) \) on \( S \) are studied in detail. The function \( \Phi(t) \), \( t \in S \), is defined in a new way. Necessary and sufficient conditions are given for \( \phi \in X \) to be a boundary value of an analytic function in \( D \). The Cauchy formula is generalized to the case when the boundary values of an analytic function in \( D \) are tempered distributions. The Sokhotsky–Plemelj formulas are derived for \( \phi \in X \).

ABSTRACT.The Riemann problem is stated as follows: find an analytic function in a domain \( D_{+} \cup D_{-} \) such that \( (*) \ \phi_{+}(t) = G(t)\phi_{-}(t) + g(t), \ t \in S \). Here \( S \) is the boundary of \( D_{+} \), \( D_{-} \) complements the complex plane to \( D_{+} \cup S \), the functions \( G = G(t) \) and \( g = g(t) \) belong to \( H^{\mu}(S) \), the space of Hölder-continuous functions. The theory of problem \( (*) \) is developed also for continuous \( G \). If \( G = 1 \), \( S \in C^{\infty} \), and \( g \) is a tempered distribution, then problem \( (*) \) has a solution in tempered distributions. It is proved that problem \( (*) \) for \( G \in L_{p}(S) \) and \( g \) a tempered distribution does not make sense. It is proved that if \( G \in C^{\infty}(S) \), \( G \ne 0 \) on \( S \), and \( g \) is a distribution of the class \( \mathcal{D}' \), then the Riemann problem makes sense and a method for solving this problem is given. It is proved that if \( S = \mathbb{R} = (-\infty,\infty) \) and \( G \in L_{p}(\mathbb{R}) \), where \( p \ge 1 \) is a fixed number, then \( \ln |G| \) does not belong to \( L_{q}(\mathbb{R}) \) for any \( q \ge 1 \).