zbMATH — the first resource for mathematics

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

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
Full Text: DOI
[1] Beals, R; Greiner, P.C; Stanton, N, Lp and Lipschitz estimates for the \( \̄\)t6-equation and the \( \̄\)t6-Neumann problem, (1986), preprint
[2] Belanger, J, Hölder estimates for \( \̄\)t6 in \(C\)^2, ()
[3] Boas, H.P; Shaw, M.-C, Sobolev estimates for the lewy operator on weakly pseudo-convex boundaries, Math. ann., 274, 221-231, (1986) · Zbl 0588.32023
[4] Bonami, A; Lohoué, N, Projecteurs de Bergman et szegö pour une classe de domaines faiblement pseudo-convexes et estimations Lp, Compositio math., 46, 159-226, (1982) · Zbl 0538.32005
[5] Bruna, J; del Castillo, J, Hölder and L^p-estimates for the \( \̄\)t6-equation in some convex domains with real analytic boundary, Math. ann., 269, 527-539, (1984) · Zbl 0533.32010
[6] Burns, D, Global behaviour of some tangential Cauchy-Riemann equations, (), 51-56
[7] Catlin, D, Subelliptic estimates for the \( \̄\)t6-Neumann problem on pseudo-convex domains, Ann. of math., 126, 131-191, (1987) · Zbl 0627.32013
[8] \scD. Catlin, Estimates of invariant metrics on pseudo-convex domains of dimension two, preprint. · Zbl 0661.32030
[9] Chang, D.-C, On Lp and Hölder estimates for the \( \̄\)t6-Neumann problem on strongly pseudo-convex domains, ()
[10] \scD.-C. Chang, On Lp estimates for the Kohn solution of the \( \̄\)t6-equation on domains (z1, z2) ∈ \(C\)2: Im(z2) > |z1|2m, preprint.
[11] Chen, S.-C, Microlocal analysis on nilpotent Lie groups, ()
[12] Christ, M, On the \(\̄t6b- equation\) and szegö projection on a CR manifold, (), in press
[13] \scM. Christ, Regularity properties of the \(\̄t6b- equation\) on weakly pseudo-convex CR manifolds of dimension three, preprint.
[14] D’Angelo, J.P, A note on the Bergman kernel, Duke math. J., 45, 259-265, (1978) · Zbl 0384.32006
[15] Diaz, K.P, The szegö kernel as a singular integral kernel in a weakly pseudo-convex domain, ()
[16] Diederich, K; Fornæss, J.E; Wiegerinck, J, Sharp Hölder estimates for \( \̄\)t6 on ellipsoids, Manuscripta math., 56, 399-413, (1986) · Zbl 0602.32006
[17] Fefferman, C.L; Phong, D, Subelliptic eigenvalue problems, (), 590-606
[18] Fefferman, C.L; Sanchez-Calle, A, Fundamental solutions for second order subelliptic operators, Ann. of math., 124, 247-272, (1986) · Zbl 0613.35002
[19] Folland, G.B; Stein, E.M, Estimates for the \(\̄t6b\) complex and analysis on the Heisenberg group, Comm. pure appl. math., 27, 429-522, (1974) · Zbl 0293.35012
[20] Fornæss, J.E, Sup-norm estimates for \( \̄\)t6 in \(C\)^2, Ann. of math., 123, 335-345, (1986)
[21] Grauert, H; Lieb, I, Das ramirezsche integral und die Lösung der gleichung \(\̄t6f = α\) im bereich der beschränkten formen, Rice univ. stud., 56, 26-50, (1970) · Zbl 0217.39202
[22] Greiner, P.C, On subelliptic estimates of \( \̄\)t6-Neumann problem in \(C\)^2, J. differential geom., 9, 239-250, (1974) · Zbl 0284.35054
[23] Greiner, P.C; Kohn, J.J; Stein, E.M, Necessary and sufficient conditions for solvability of the lewy equation, (), 3287-3289 · Zbl 0308.35017
[24] Greiner, P.C; Stein, E.M, Estimates for the \( \̄\)t6-Neumann problem, Math. notes, 19, (1977)
[25] Greiner, P.C; Stein, E.M, On the solvability of some differential operators of type □_b, (), 106-165
[26] Harvey, R; Polking, J; Harvey, R; Polking, J, Fundamental solutions in complex analysis, I, II, Duke math. J., Duke math. J., 46, 301-340, (1979) · Zbl 0441.35044
[27] Henkin, G.M, Integral representations of functions in strictly pseudo-convex domains and applications to the \( \̄\)t6-problem, Mat. sb., Math. USSR-sb., 11, 273-281, (1970), English translation · Zbl 0216.10402
[28] Henkin, G.M; Leiterer, J, Theory of functions on complex manifolds, (1984), Akademie-Verlag Berlin
[29] Henkin, G.M; Romanov, A.V, Exact Hölder estimates for the solutions of the \( \̄\)t6-equation, Izv. akad. nauk SSSR ser. mat., Mat. USSR-izv., 5, 1180-1192, (1971), English transl. · Zbl 0248.35090
[30] Hörmander, L, L2-estimates and existence theorems for the \( \̄\)t6-operator, Acta math., 113, 89-152, (1965) · Zbl 0158.11002
[31] Hörmander, L, Hypoelliptic second order differential equations, Acta math., 119, 147-171, (1967) · Zbl 0156.10701
[32] Kerzman, N, Hölder and Lp estimates for solutions of \(\̄t6u = f\) in strongly pseudo-convex domains, Comm. pure appl. math., 24, 301-379, (1971) · Zbl 0217.13202
[33] Kerzman, N; Stein, E.M, The szegö kernel in terms of Cauchy-fantappie kernels, Duke math. J., 45, 197-224, (1978) · Zbl 0387.32009
[34] Kohn, J.J; Kohn, J.J, Harmonic integrals on strongly pseudo-convex manifolds, I, II, Ann. of math., Ann. of math., 79, 450-472, (1964) · Zbl 0178.11305
[35] Kohn, J.J, Boundaries of complex manifolds, (), 81-94 · Zbl 0166.36003
[36] Kohn, J.J, Estimates for \(\̄t6b\) on pseudo-convex CR manifolds, (), 207-217
[37] \scJ. J. Kohn, Microlocal analysis on pseudo-convex domains and CR manifolds, preprint. · Zbl 0743.32010
[38] Kohn, J.J, Boundary behaviour of \( \̄\)t6 on weakly pseudo-convex manifolds of dimension two, J. differential geom., 6, 523-542, (1972) · Zbl 0256.35060
[39] Kohn, J.J, The range of the tangential Cauchy-Riemann operator, Duke math. J., 53, 525-545, (1986) · Zbl 0609.32015
[40] Kohn, J.J, Global regularity for \( \̄\)t6 on weakly pseudo-convex manifolds, Trans. amer. math. soc., 181, 272-292, (1973) · Zbl 0276.35071
[41] Kohn, J.J, Pseudo-differential operators and non-elliptic problems, (), 157-165
[42] Kohn, J.J; Nirenberg, L, Non-coercive boundary value problems, Comm. pure appl. math., 18, 443-492, (1965) · Zbl 0125.33302
[43] Kohn, J.J; Nirenberg, L, A pseudo-convex domain not admitting a holomorphic support function, Math. ann., 201, 265-268, (1973) · Zbl 0248.32013
[44] Kohn, J.J; Rossi, H, On the extension of holomorphic functions from the boundary of a complex manifold, Ann. of. math., 81, 451-472, (1965) · Zbl 0166.33802
[45] Krantz, S, Intrinsic Lipschitz classes on manifolds application to complex function theory and estimates for the \( \̄\)t6 and \(\̄t6b\) equations, Manuscripta math., 24, 351-378, (1978) · Zbl 0382.32012
[46] Krantz, S, Function theory of several complex variables, (1982), Wiley New York · Zbl 0471.32008
[47] Lewy, H, On the local character of the solutions of an atypical linear differential equation in three variables and a related theorem for regular functions of two complex variables, Ann. of math., 64, 514-522, (1956) · Zbl 0074.06204
[48] Lewy, H, An example of a smooth linear partial differential equation without solution, Ann. of. math., 66, 155-158, (1957) · Zbl 0078.08104
[49] Lieb, I; Range, R.M, Estimates for a class of integral operators and applications to the \( \̄\)t6-Neumann problem, Invent. math., 85, 415-438, (1986) · Zbl 0569.32008
[50] \scM. Machedon, Estimates for the parametrix of the Kohn Laplacian on certain weakly pseudo-convex domains, preprint. · Zbl 0651.35017
[51] Nagel, A; Rosay, J.-P; Stein, E.M; Wainger, S, Estimates for the Bergman and szegö kernels in certain weakly pseudo-convex domains, Bulletin of the A.M.S., 18, 55-59, (1988) · Zbl 0642.32014
[52] Nagel, A; Stein, E.M; Wainger, S, Balls and metrics defined by vector fields. I. basic properties, Acta math., 155, 103-147, (1985) · Zbl 0578.32044
[53] Phong, D; Stein, E.M, Estimates for the Bergman and szegö projections on strongly pseudo-convex domains, Duke math. J., 44, 416-435, (1977)
[54] Range, R.M, On Hölder estimates for \(\̄t6u = f\) on weakly pseudo-convex domains, (), 247-267
[55] Range, R.M, Holomorphic functions and integral representations in several complex variables, (1986), Springer-Verlag New York · Zbl 0591.32002
[56] Rossi, H, Attaching analytic spaces to a space along a pseudo-convex boundary, (), 242-256
[57] Rothschild, L.P; Stein, E.M, Hypoelliptic differential operators and nilpotent groups, Acta math., 137, 247-320, (1976) · Zbl 0346.35030
[58] Shaw, M.-C, L2 estimates and existence theorems for the Cauchy-Riemann complex, Invent. math., 82, 133-150, (1985) · Zbl 0581.35057
[59] \scM.-C. Shaw, Hölder and Lp estimates for \(\̄t6b\) on weakly pseudo-convex boundaries in \(C\)2, preprint.
[60] Sibony, N, Un example de domain pseudo-convexe regulier ou l’équation \(\̄t6u = f\) n’admet pas de solution bornée pour f bornée, Invent. math., 62, 25-242, (1980) · Zbl 0429.32026
[61] Stein, E.M, Singular integrals and differentiability properties of functions, () · Zbl 0207.13501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.