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

• Published in: Integral equations and operator theory 2019-10, Vol.91 (5), p.1-30, Article 47
• Main Authors: Espinoza-Loyola, Enrique, Karlovich, Yuri I.
• Format: Article
• Language: English
• Subjects: Algebra; Analysis; Continuity (mathematics); Corners; Domains; Mathematical analysis; Mathematics; Mathematics and Statistics; Multiplication; Operators (mathematics); Smooth boundaries; Tangents
• ISSN: 0378-620X; 1420-8989
• DOI: 10.1007/s00020-019-2545-z

Given a simply connected domain U ⊂ C with a piecewise Dini-smooth boundary ∂ U which admits a finite set of Dini-smooth corners of openings lying in ( 0 , 2 π ] , we study the C ∗ -algebra B U = { a I , B U , B ~ U : a ∈ X ( L ) } generated by the operators of multiplication by functions in X ( L ) , and by the Bergman projection B U and anti-Bergman projection B ~ U acting on the Lebesgue space L 2 ( U ) . The C ∗ -algebra X ( L ) is generated by all piecewise continuous functions on the closure U ¯ of U with discontinuities on a finite union L of piecewise Dini-smooth curves that have one-sided tangents at every point, do not form cusps and are not tangent to ∂ U at the points of L ∩ ∂ U , and by all bounded continuous functions on U that slowly oscillate at points of ∂ U . Making use of the Allan-Douglas local principle, the limit operators techniques and the Kehe Zhu results on the class Q = V M O ∂ ( D ) ∩ L ∞ ( D ) , a Fredholm symbol calculus for the C ∗ -algebra B U is constructed and a Fredholm criterion for the operators A ∈ B U is obtained.