PRELIMINARIES

0. INTRODUCTORY REMARKS

Fix a smoothly bounded domain Q C

EiV+1.

In classical treatments

of the d operator (see [SWE]), one considers the complex

T2

d

v

T2

d

v

Lq * Lq+l y

Here

L2

denotes the q-forms of Cartan and de Rham having

L2(Q)

coefficients and the operator d is understood to be densely defined. One

considers the operator • = dd* + d*d. Here the adjoints are calculated

in the L2(Q) topology.

Now • makes sense on those forms ip such that ip G domef and

dtp G domd*. One can decompose the space Lq into (the closure of)

the image of D and its orthogonal complement. Then one exploits this

decomposition to construct a right inverse for O This inverse is easily

used to show that • and its accompanying boundary conditions form a

second order elliptic boundary value problem of the classical (coercive)

type.

In 1963, J.J. Kohn [KOH1] determined how to carry out the ana-

logue of these last calculations for the d operator of complex analysis

on a strongly pseudoconvex domain Q in C n . This is the so-called d-

Neumann problem. Of course this analysis, while similar in spirit, is

much more complicated. It gave rise to the important "Kohn canon-

ical solution" to the equation du — f. That is the solution u that is

orthogonal to holomorphic functions in the L2(Q) topology.

Experience in the function theory of several complex variables has

shown that it is useful to have many different canonical solutions to the

9-problem. For instance, in the strongly pseudoconvex case we prof-

itably study the Kohn solution by comparing it with the Henkin solu-

tion (not canonical, but nearly so), see [HEN], and the Phong solution

Date: April 10, 1995.

Second author supported in part by National Science Foundation Grant DMS-

DMS-9531967 and at MSRI by NSF Grant DMS-9022140.

Third author supported in part by the Consiglio Nazionale delle Ricerche.

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