J. J. Kohn

Harmonic Integrals on Strongly Pseudo-Convex Manifolds: II

In this paper we prove the basic existence and regularity theorems for the a-Neumann problem (see Theorems 6.6 and 6.14). The results presented here were outlined by the author in [8]. In Part I of this work (see [7]) we established some of the fundamental properties of the aNeumann problem and, on the assumption of existence and regularity, we obtained several applications. A variant of the 8-Neumann problem was first formulated in [3]. D. C. Spencer and the author studied the problem by means of singular integral equations in [5]. The starting point for the authors work (see [6] and [7]) is the estimate (1.6), a special case of this estimate was first established by C. B. Morrey in [9]. In his thesis, (see [1]) M. E. Ash has derived estimates relative to moving frames. This method has enabled him to generalize our work (see also [2]). The introduction of moving frames is also very useful in the present work. The method of proving regularity by studying the families of norms depending on the parameters a and zwas suggested to the author by L. Hbrmander; essential use is made of some of the results (stated in Ch. 4) which are developed in his book (see [4]). The T-norms of Ch. 3 have been also introduced for a different purpose by Andreotti and Vesentini. They have obtained, by the argument of [6] for forms with values in a holomorphic vector bundle over a strongly pseudoconvex domain of Cn, an inequality which contains the inequality of Proposition 3.5 as a particular case. In Ch. 7 we show how the solution of the 8-Neumann problem implies the solution of several boundary value problems which were posed in [5]. The methods developed here and in [7] can be used to prove existence and regularity theorems for very general elliptic over determined systems. We shall return to this question in a future publication.

Boundaries of Complex Manifolds

If M is a component of the boundary of a complex n-dimensional manifold X, then M has real dimension 2n - 1 and at each point x ∈ M the complexified tangent space T x has a distinguished (n - 1)-dimensional subspace S x which is the intersection of T x with the holomorphic vectors at x. Thus, vector fields with values in S x are the “tangential” Cauchy-Biemann operators.

Superlogarithmic estimates on pseudoconvex domains and CR manifolds

This paper is concerned with proving superlogarithmic estimates for the operator r]b on pseudoconvex CR manifolds and using them to establish hypoellipticity of lb and of the (-Neumann problem. These estimates are established under the assumption that subellipticity degenerates in certain specified ways.

Hölder estimates on CR manifolds with a diagonalizable Levi form

(a) For all E > 0 the operator Cl;’ maps Lip(s, PO) into Lip(s + 2/m E, PO). Here Lip(s, P,) denotes the space of square-integrable forms (of degree (p, q) with 1 G q Otheoperators~,O~l,~~O~l, q ~‘~,,and0~‘~,* map Lip(s, P,) into Lip(s + l/m E, PO). (c) For all E>O the operators 8b8zO;‘, 8z8hO;1, q ;‘a,aa, Cl -’ 8* 8 8 q -’ 8:, and 8

Quantitative Estimates for Global Regularity

0;’ ab map Lip(s, P,) into Lip(s-&, P,,). Hebre aiag web asm’ acts on forms of type (p, q) with 1~ q 0. (e) Zf f is a square-integrable function on .A? with f E Lip@, P,) then S,(f) E Lip(s E, P,) for all E > 0 where S, is the orthogonal projection on square-integrable CR functions.

Lectures on Degenerate Elliptic Problems

Let Ω ⊂ ℂ n be a bounded pseudoconvex domain with a smooth boundary. We denote by L 2(Ω) the space of square-integrable functions on Ω and by H (Ω) the space of square-integrable holomorphic functions on Ω. Let B: L 2(Ω) → (Ω) denote the Bergman projection operator, which is the orthogonal projection of L 2(Ω) onto (Ω) . Here we will be concerned with the global regularity of B in terms of Sobolev norms, that is, the question of when B(H S (Ω)) ⊂ H S (Ω) where H s (Ω) denotes the Sobolev space of order s. Of course, if B preserves H S (Ω) locally (i.e., if B(s/loc(Ω)) ⊂ H s loc(Ω)), then B also preserves H s (Ω) globally. Aspects of the local question are very well understood, in particular when Ω is of finite D’Angelo type (see [Cal] and [D’A]). Local regularity can still occur when the D’Angelo type is infinite, as in the examples given in [Chr2] and [K2]. Local regularity fails whenever there is a complex curve V in the boundary of Ω. In that case, if P ∈ V, then for given s there exists an f ∈ L 2(Ω) such that ζf ∈ H S (Ω) for every smooth function ζ with support in a fixed small neighborhood of P and such that ζB(f) ∉ H S (Ω) whenever ζ = 1 in some neighborhood of P. In contrast, global regularity always holds for small s. That is, if Ω is pseudoconvex, then there exists η > 0 such that B(H S (Ω)) ⊂ H S (Ω) for s ⩽ η. Furthermore, there is a series of results showing global regularity under a variety of conditions (see [Ca2], [BC], [Ch], [BS1], and [BS2]).

Microlocalization of CR structures

These lectures will be concerned with equations (and systems of equations) which are “close” to elliptic. By this we mean that they are limits of elliptic ones; such as, for example, the heat equation can be expressed as the following limit:

Loss of Derivatives

Let M be a compact C∞ manifold of dimension 2n+1. A CR structure on M is given by a subbundle, denoted by \( {\text T}\begin{array}{*{20}c} {1, 0} \\ {\text b} \\ \end{array} ({\text M}) \), of the complexified tangent bundle ℂT(M), which satisfies the following properties: (a) The dimension of the fibers of \( {\text T}\begin{array}{*{20}c} {1, 0} \\ {\text b} \\ \end{array} ({\text M}) \) is n. (b) \( {\text T}\begin{array}{*{20}c} {1, 0} \\ {\text b} \\ \end{array} ({\text M})\, \cap \,\overline {{\text T}\begin{array}{*{20}c} {1, 0} \\ {\text b} \end{array} ({\text M})} \, = \,\{ 0\} . \) (c) If L and L′ are vector fields defined on an open set UCM with values in \( {\text T}\begin{array}{*{20}c} {1, 0} \\ {\text b} \\ \end{array} ({\text M}) \) then [L, L′] also has values in \( {\text T}\begin{array}{*{20}c} {1, 0} \\ {\text b} \\ \end{array} ({\text M}) \).

Propagation of Singularities for the Cauchy-Riemann Equations

In 1957, Hans Lewy (see Lewy [L]) obtained a remarkable result. Namely, he found a first-order partial differential operator L such that there exists a function \(f \in {C}^{\infty }({\mathbb{R}}^{3})\) so that the equation Lu = f does not have any distribution solutions u on any open set, equivalently the associated laplacian \(Eu = L{L}^{{_\ast}}u = f\) does not have any distribution solution. This operator comes from the study of the induced Cauchy-Riemann equation on the sphere in \({\mathbb{C}}^{2}\). Roughly speaking, nonexistence of distribution solutions means that no derivative of u can be uniformly estimated by some derivatives of f, that is, “E loses infinitely many derivatives.” In Kohn (Ann. Math. 162:943–986, 2005), the operator E was approximated by a sequence of operators {E k }, each of which loses exactly k − 1 derivatives but nevertheless is locally solvable and hypoelliptic. Here we study these phenomena for operators of the form ∑X i ∗ X i , where the X i are complex-valued vector fields and the corresponding approximating operators lose a finite number of derivatives.

Remembering Gian-Carlo Rota

These lectures are intended as an introduction to the study of several complex variables from the point of view of partial differential equations. More specifically here we take the approach of the calculus of variations known as the ā-Neumann problem. Most of the material covered here is contained in Polland and Kohn, [4], Hormander [11] and in the more recent work of the author (see [16], [17] and [20]). We consistently use the Laplace operator as in Kohn [l4], since we believe that this method is particularly suitable for the study of regularity and for the study of the induced Cauchy-Riemann equations. Our main emphasis is in finding regular solutions of the inhomogenious Cauchy-Riemann equations. We wish to call attention to the extensive research on this problem by different methods from the ones mentioned above (see Ramirez [29], Grauert and Lieb[8], Kerzman [13], ovrelid [28], Henkin [9], Folland and Stein [5]). It would take us too far afield to present these matters here. Another closely related subject which we cannot take up here is the theory of approximations by holomorphic functions (see R. Nlrenberg and o. Wells [27], R. Nlrenberg [26], Hormander and Wermer [12], etc.).