A Sobolev mapping property of the Bergman kernel (Q1586853)

Let \(D\subset\mathbb C^n\) be a bounded pseudoconvex domain with Lipschitz boundary having an exhaustion function \(\rho\) such that \(-(-\rho)^\eta\) is plurisubharmonic. The main result of this note says that the bigger one can take the number \(\eta,\) the better regularity properties one has for the Bergman projection. More precisely, the authors show that the Bergman projection is bounded on the Sobolev space \(W_s\) for any \(s < \eta/2.\) A similar result holds for the operator \(K\) giving the \(L^2\)-minimal solution to the \(\overline\partial\)-problem.