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Estimates for \({\bar\partial}_ b\) on pseudoconvex CR manifolds. (English) Zbl 0571.58027
Pseudodifferential operators and applications, Proc. Symp., Notre Dame/Indiana 1984, Proc. Symp. Pure Math. 43, 207-217 (1985).
[For the entire collection see Zbl 0562.00004.]
The author gives some estimates of the form \[ \| \xi u\|_{S+\epsilon}\leq C_ S(\| \xi '{\bar \partial}_ bu\|_ S+\| {\bar \partial}_ bu\|). \] Here \({\bar \partial}_ b\) is (the \(L^ 2\)-closure of) the operator \({\bar \partial}_ b: C^{\infty}(u)\to B^{0,1}(u)\) defined by \(<{\bar \partial}_ bu,\bar L>=\bar Lu\), where \(B^{0,1}(u)\) is the space of \(C^{\infty}\)-sections over an open set \(U\subset M\) of \(B^{0,1}(M)\). By \(B^{0,1}(M)\) the dual bundle of \(T^{0,1}(M)\) is denoted, M is the pseudoconvex CR- manifold under consideration, and \(\xi\),\(\xi\) ’ are suitable cut-off- functions. Concerning the assumptions of M and the range of \(\epsilon\) the author has to distinguish the case \(\dim_{{\mathbb{R}}} M=3\) from the case \(\dim_{{\mathbb{R}}} M=2n-1\), \(n>2\).
Reviewer: N.Jacob

58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
35N15 \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs
32T99 Pseudoconvex domains
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
Zbl 0562.00004