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. 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 \( \phi \in X \).

AbstractIn this paper, the main goal is to study an ordered Γ-semihypergroup H in the context of the characterizations of the associated Γ-semihypergroup B(H) of all bi-Γ-hyperideals of H. We show that an ordered Γ-semihypergroup H is a Clifford ordered Γ-semihypergroup if and only if B(H) is a semilattice. We also show that a Γsemihypergroup B(H) is a normal band if and only if the ordered Γ-semihypergroup H is simultaneously regular and intra regular. Furthermore, for each subclass S with many bands, we prove that for an ordered Γ-semihypergroup H, the conditional inclusion B(H) ∈ S holds true.