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\(L^p\)-estimates for the Bergman projection on strictly pseudoconvex nonsmooth domains. (English) Zbl 1147.32004

An open set \(D\subset \mathbb C ^n \) is called a strictly pseudoconvex non-smooth domain if in the definition of strictly pseudoconvex domain we drop the assumption \(dr\neq 0\) on \(\partial D\), where \(r\) is a defining function for \(D\). For such \(D\) consider weighted \(L^p\) spaces
\[ L^{p,k} (D) =\biggl\{ f: \|f\|^p _{p,k} =\int _D | \partial r | ^k | f| ^p\, dV <\infty \biggr\} . \] The main theorem of the paper says that for \(D\) strictly pseudoconvex non-smooth, with \(r\) having a finite number of critical points on \(\partial D\) the Bergman projection is a continuous operator from \(L^p\) into \(L^{p,k}\) for \(p\in [2, \infty )\) and \(k=k(p)\) large enough. Also, an estimate for optimal \(k(p)\) is obtained: \(k(p)\leq p(n+1)\). An interesting corollary follows for the solutions of the \(\overline{\partial}\)-equation. If \(D\) and \(k\) are as above, then for \(\overline{\partial}\)-closed \(f\in L^2 (D)\) the solution of the equation
\[ \overline{\partial} u=f \] minimizing the \(L^p\) norm satisfies \[ \| u\| _{p,k} \leq \text{const.} \| f\| _p, \;\;p\geq 2. \]

MSC:

32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32T15 Strongly pseudoconvex domains
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