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Hölder estimates on domains of complex dimension two and on three- dimensional CR manifolds. (English) Zbl 0649.35068
The $${\bar \partial}$$-Neumann problem on a domain in $${\mathbb{C}}^ n$$ is elliptic in the interior. So interior regularity estimates follow easily from classical considerations. At the boundary matters are more subtle: there is no coercive estimate.
The first estimates up to the boundary for this problem were obtained by J. J. Kohn in 1963. He discovered how to exploit the geometry of strong pseudoconvexity to obtain a substitute for the coercive estimate; his results are sharp in the $$L^ 2$$ Sobolev topology. Sharp results for the $${\bar \partial}$$ problem on strongly pseudoconvex domains in other topologies came about only after the advent of integral formulas in the period 1969-1971. Many of these results were obtained by Kerzman, Overlid, Grauert/Lieb, Greiner/Stein, Siu, and the reviewer. The results of Greiner/Stein were for the canonical solution (the one orthogonal to holomorphic functions) in a special “Levi metric”. This extra hypothesis on the metric was later removed by Rothschild/Stein, Beals/Greiner/Stanton, Lieb/Range, and Chang; these authors have also obtained additional refined estimates for the solution. Other results for strongly pseudoconvex domains, and for a variety of different solutions, have been obtained by Harvey/Polking, Phong/Stein, Chang, and Bonneau. It is worth noting that for much of this work major new analytic machinery had to be developed in order to deal with the new classes of singular integrals that arose.
The next level of complexity in the subject is domains of finite type. This type of domain differs sharply from strongly pseudoconvex domains in the following sense. While strongly pseudoconvex geometry is different from standard Euclidean geometry, it is still a condition that remains essentially the same as one moves from point to point (modulo a certain twisting of the coordinate frame which can ultimately be handled). On weak type domains, boundary geometry varies semi-continuously but otherwise in a rather complicated fashion from point to point. Thus the learning curve for treating these domains, even in $${\mathbb{C}}^ 2$$, has had a rather small first derivative.
In the late 1970’s, Range obtained sharp estimates for the equation $${\bar \partial}u=f$$ when the domain is one of a family of complex ellipsoids in $${\mathbb{C}}^ 2$$ and f is bounded; he obtained nearly sharp estimates for such ellipsoids in dimensions three and higher. The discovery that Range brought to light was that the amount of smoothing that one could expect near a point of type m is of order 1/m. At about the same time, the reviewer produced a family of examples to show that such an estimate is sharp. It took nearly another ten years before Diederich, Fornaess, and Wiegerinck found how to make Range’s estimates sharp in all dimensions. More recently, Cumenge and the reviewer have found other sharp estimates on ellipsoids in the $$L^ p$$ and Lipschitz topologies.
Until quite recently, probably the deepest work on finite type domains was that by Fornaess in which he obtained uniform estimates for $${\bar \partial}u=f$$ on a large class of finite type domains; Belanger subsequently obtained nearly sharp Lipschitz estimates on the same class of domains.
It should be mentioned that part of the interest, and also the difficulty, of the estimates described above is that they can all be localized at the boundary. Kohn made the remarkable discovery in 1973 that global regularity is a much easier phenomenon to understand: global regularity up to the boundary holds at least for some solution of $${\bar \partial}u=f$$ whenever f has coefficients which are smooth up to the boundary and the domain is smooth, bounded, and pseudoconvex. It is still an open problem whether the canonical solution to the $${\bar \partial}$$- Neumann problem has this property.
In a similar vein it should be mentioned that Polking has recently proved that on any convex domain (or any geometrically pseudoconvex domain in the sense of Thullen) $$L^ p$$ estimates hold for the $${\bar \partial}$$- Neumann problem.
We would also be remiss not to mention the deep and influential work of Hörmander, who obtained existence and regularity results on all pseudoconvex domains by working in a metric which makes the domain complete (in a sense compatible with Levi geometry).
The purpose of this lengthy background material is to establish the slow progress of this subject. There has been a paucity of tools for doing analysis in a setting where the essential geometry changes from point to point.
But now matters have changed and there are exciting new developments. The paper under review, alongside independent work by M. Christ, by Chang/Nagel/Stein, by Nagel/Stein/Wainger, and by Nagel/Stein/Rosay/Wainger, develops powerful new techniques for doing analysis on finite type domains in $${\mathbb{C}}^ 2$$. While all three approaches have certain elements in common - for instance scaling arguments and singular integrals - they differ in marked respects. The special feature of the work under review is the use of microlocalization; to the reviewer’s knowledge this is one of the first explicit uses of this technique in the theory of the $${\bar \partial}$$-Neumann problem.
The main results of the paper under review include sharp Lipschitz estimates for the $${\bar \partial}$$-Neumann problem on domains of finite type in $${\mathbb{C}}^ 2$$, sharp estimates for $$\square_ b$$ on such domains, and estimates for the Szegö and Bergman projections.
While the results being described here for domains in $${\mathbb{C}}^ 2$$ represent heartening progress and should lead to a wealth of new insights, it should be stressed that the situation in dimensions three and higher is vastly more complex. The scaling argument show in effect that the geometry near a finite type point is, in effect, a product geometry. Such is not the case already in dimension three, and there is compelling evidence that there will be not one but infinitely many different models for finite type boundary points in that setting.
We conclude by noting that the paper under review contains a thorough history of the subject with rather complete references.
Reviewer: S.Krantz

##### MSC:
 35N15 $$\overline\partial$$-Neumann problems and formal complexes in context of PDEs 32W05 $$\overline\partial$$ and $$\overline\partial$$-Neumann operators 35B45 A priori estimates in context of PDEs 35D10 Regularity of generalized solutions of PDE (MSC2000) 35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
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