Uniform estimates with weights for the \(\overline{\partial}\)-equation. (English) Zbl 0923.32014

Berndtsson, Bo

J. Geom. Anal. 7, No. 2, 195-215 (1997).

The paper gives some \(L^\infty\) variants of Hörmander’s weighted \(L^2\) estimates for the \(\overline\partial\)-equation on strongly pseudoconvex domains. In more detail, in the context of the unit ball \(B\) of \(\mathbf C^n\), a \(\overline\partial\)-closed (0,1) form \(f\) in \(B\), a plurisubharmonic function \(\phi\) on \(B\), and the solution \(u\) of \(\overline\partial u=f\) which is of minimal norm in \(L^2(e^{-\phi}(1-| z| ^2)^N)\), for \(N>(n+1)^2\), estimates are given for \(\sup | u| e^{-\widetilde\phi/2}(1-| z| ^2)^{N/2}\) in terms of \(\sup (| f| +| \partial f|) e^{-\phi/2}(1-| z| ^2)^{N/2}\), where \(\widetilde\phi\) is a certain regularization of \(\phi\) and the norms \(| f| ,| \partial f| \) are taken with respect to the metric \(\partial\overline\partial\phi\) or to the Bergman metric on \(B\) (in the latter case, \(| \partial f| \) can even be omitted). In addition to strongly pseudoconvex domains, a variant is also established for the case of the entire complex space \(\mathbf C^n\). The proofs use a refined \(L^2\)-estimate for the canonical solution \(u\). Finally, a relation of these results to the corona problem in the ball is discussed.

Reviewer: M.Engliš (Praha)

MSC:

32W05	\(\overline\partial\) and \(\overline\partial\)-Neumann operators
32A35	\(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables

Keywords:

\(d\)-bar equation; corona problem; \(L^2\) estimates

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