On the canonical solution of $\protect \hspace{0.0pt}\protect \hspace{0.0pt}\protect \overline{\protect \hspace{0.0pt}\partial }$ on polydisks

Abstract

We observe that the recent result of implies that the canonical solution operator satisfies Sobolev estimates with a loss of n − 2 derivatives on the polydisk ∆ n and particularly is exact regular on ∆ 2 .

Sobolev regularity of the canonical solutions to ∂ ¯ on product domains

“…The L p regularity of the canonical solutions on product domains was already thoroughly understood through works of [5-7, 12, 15, 22, 25] and the references therein. In the Sobolev category, combined efforts in [2,11,18,23] have given the existence of a bounded solution operator of ∂ sending…”

“…□ Proposition 5 and (9) also immediately give the following Sobolev regularity of the Bergman projection operator P on general product domains. We mention that the Sobolev regularity of P on the polydisc was due to [8,11].…”