The Bergman projection as a singular integral operator
A Calderón-Zygmund type kernel is a kernel \((x,y\) are variables in a metric space) such that \[ \bigl | K(x,y) \bigr | < {K \over \mu \biggl( P(x, \rho \bigl( x,y) \bigr) \biggr)}, \qquad \int_{\rho (x,y)>T \rho(x,y)} \bigl | K(x,y) - K(z,y) \bigr | dy \leq K. \] \((\rho\) a metric, \(P\) a \(\rho\)-ball). This can be extended for a pseudometric \((\rho (x,z) \leq C(\rho (x,y) + \rho (y,z))\) if the measure \(\mu\) satisfies certain conditions (leading to a so-called space of homogeneous type). Starting from a defining function for a domain \(D\) (i.e. \(D = \{q : r(q) < 0\})\) of \(\mathbb{C}^ n\) several pseudometrics are constructed using the derivatives of \(r\), or local subharmonic functions. Using estimates previously obtained, mostly be the same author, it is then proved that under certain conditions the Bergman kernel is a Calderón-Zygmund type operator for these pseudometrics (the measure \(\mu\) being ordinary Lebesgue measure). As a consequence, the Bergman projection is, in these cases, a bounded operator in \(L_ p(D)\), \(1<p<\infty\).
scientific article; zbMATH DE number 579776