# zbMATH — the first resource for mathematics

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

##### MSC:
 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