A note on smoothing properties of the Bergman projection (Q2836247)

This paper deals with smoothing properties of the Bergman projection on certain Reinhardt domains acting on anti-holomorphic functions. Let \(f\) such that \(f\in A^2(\Omega)\), where \(A^2(\Omega)\) is the space of holomorphic functions on \(\Omega\) which also belong to \(L^2(\Omega)\). When \(\Omega\) is a Reinhardt domain with \(C^1\) boundary, the authors prove that the Bergman projection operator \(B\) acts as a smoothing operator on functions \(g\in A^2(\Omega)\), using the additional assumption that the domain does not intersect the axes. This result is comparable to a result of \textit{A. K. Herbig} et al. [Trans. Am. Math. Soc. 366, No. 2, 647--665 (2014; Zbl 1286.32003)] who showed that \(B\) acts as a smoothing operator on such functions for smoothly bounded domains which satisfy Condition \(R\). The theorem of the paper under consideration furthermore shows (on the class of Reinhardt domains considered) that the resulting function is not only smooth to the boundary but extends holomorphically to a strictly larger domain.NEWLINENEWLINEThe proofs are nice examples of illustrations of calculations involving the Bergman projection with the use of orthornormal bases of spaces \(A^2(\Omega)\). An input function is written in terms of the bases, the projection then applied, and the resulting function, and its extension property, are analyzed in terms of such bases.