L. FONTANA, S. KRANTZ, AND M. PELOSO
(determined in the
topology rather than the
The ultimate goal of the program that we are initiating in this pa-
per is to construct solutions to the d problem that are orthogonal to
holomorphic functions in a Sobolev space Ws inner product. There are
a priori reasons for knowing that this program is feasible. First, each
Sobolev space is a Hilbert space, so there must be a minimal solution
in the Sobolev topology. Second, Boas [BOA] has studied the space of
Ws holomorphic functions as a Hilbert space with reproducing kernel.
The associated Bergman projection operator is of course closely related
to the Neumann operator for the d problem.
The present paper carries out the first step of the proposed program.
We work out the Hodge theory for the exterior differentiation operator
d in the inner product induced by the Sobolev space Ws topology.
Of particular interest are the boundary conditions that arise when we
calculate the adjoint d* in this topology, and the elliptic boundary value
problem that arises when we consider • = dd* + d*d. We calculate a
complete existence and regularity theory.
In this paper, we restrict attention to the case 5 = 1. This is done
both for convenience and to keep the notation relatively simple—even
in this basic case the calculations are often unduly cumbersome. In
geometric applications, the case s = 1 is already of great interest. We
leave the detailed treatment of higher order s to a future paper.
In future work, we will carry out this program of analysis in the
Sobolev space topology for the 9-Neumann problem on a strongly pseu-
doconvex domain. Not only will this give rise to new canonical solutions
for the d problem, but it should give a new way to view the Sobolev reg-
ularity of the classical d problem, and of understanding the subelliptic
gain of 1/2 in regularity.
Essentially the paper is divided into two parts. In the first of these we
study the boundary value problem that arises from our Hodge theory
on the special domain given by the half space, and in the second one
we deal with the problem on a smoothly bounded domain. We would
like to point out that the second part has been written so that it can be
read independently from the first part. When we use results from the
half space case, we give precise reference to them.
In detail, the plan of the paper is as follows: Section 1 introduces
basic notation and definitions, while in Section 2 we formulate the
problem and state the main results. Sections 3 through 6 are devoted
the problem on the half space.
In Section 3 we calculate the operator d* on 1-forms and also cal-
culate its domain when the region under study is the upper half space