C*-algebras of Bergman Type Operators with Piecewise Slowly Oscillating Coefficients over Domains with Dini-Smooth Corners

Abstract

Given a simply connected domain $$U\subset \mathbb {C}$$ with a piecewise Dini-smooth boundary $$\partial U$$ which admits a finite set of Dini-smooth corners of openings lying in $$(0,2\pi ]$$ , we study the $$C^*$$ -algebra $$\mathfrak {B}_U=\{aI, B_U, \widetilde{B}_U:a\in \mathfrak {X}(\mathfrak {L})\}$$ generated by the operators of multiplication by functions in $$\mathfrak {X}(\mathfrak {L})$$ , and by the Bergman projection $$B_U$$ and anti-Bergman projection $$\widetilde{B}_U$$ acting on the Lebesgue space $$L^2(U)$$ . The $$C^*$$ -algebra $$\mathfrak {X}(\mathfrak {L})$$ is generated by all piecewise continuous functions on the closure $$\overline{U}$$ of U with discontinuities on a finite union $$\mathfrak {L}$$ of piecewise Dini-smooth curves that have one-sided tangents at every point, do not form cusps and are not tangent to $$\partial U$$ at the points of $$\mathfrak {L}\cap \partial U$$ , and by all bounded continuous functions on U that slowly oscillate at points of $$\partial U$$ . Making use of the Allan-Douglas local principle, the limit operators techniques and the Kehe Zhu results on the class $$Q=VMO_\partial (\mathbb {D}) \cap L^\infty (\mathbb {D})$$ , a Fredholm symbol calculus for the $$C^*$$ -algebra $$\mathfrak {B}_U$$ is constructed and a Fredholm criterion for the operators $$A\in \mathfrak {B}_U$$ is obtained.