Microlocal analysis of the Bochner-Martinelli integral (Q2642717)

Treating the Bochner-Martinelli singular integral on the boundary of a domain in \(\mathbb{C}^{n}\) as a pseudodifferential operator, the authors compute the principal symbol: the value of the symbol at a point in the boundary surface and a (cotangent) vector~\(\xi\) equals half the projection of~\(\xi/| \xi| \) onto the special tangential direction at the given point. Then they study the \(C^{*}\)-algebra generated by the Bochner-Martinelli singular integral operator, its adjoint, and the multiplication operators corresponding to continuous functions on the boundary surface. They prove that this algebra is irreducible, and they characterize the quotient of this algebra by the compact operators. They also generalize a theorem of \textit{A.~V. Romanov} [Funct. Anal. Appl. 12, 232--234 (1979; Zbl 0427.47035)] from the ball to strictly pseudoconvex domains: let \(M_{-}\) denote the operator that takes a square-integrable function on the boundary surface to the limiting value (from the inside) of its Bochner-Martinelli integral; then the iterates of~\(M_{-}\) converge in the strong operator topology to the sum of the Szegő projection and a compact operator.