differential equations, in geometry, and in function theory. Recent work of Christ [@Christ] has shown that the [*canonical solution*]{}—the solution that is minimal in $L^2$ norm—that arises from Kohn’s work in the $L^2$ topology is not as well behaved as one might have hoped. The program presented in this paper endeavors to seek other canonical solutions that may serve when Kohn’s solution will not. This work is also interesting from the point of view of partial differential equations—particularly boundary value problems—and in the study of the energy integral in geometry. We mention that H.  Boas [@Boas1] and [@Boas2] studied properties and regularity of the Hilbert space orthogonal projection of ${W^s(\Omega)}$ onto the subspace consisting of the holomorphic functions.

The present paper is the first of a series of papers that we devote to the study of the $\dbars$-Neumann problem. We begin by showing that problem [**(3,s)**]{} can always be solved on any smoothly bounded pseudoconvex domain $\Omega$. This result does not depend on the particular choice of Sobolev inner product. Then we investigate the $\dbars$-Neumann problem more closely by determining a description of the Hilbert space adjoint $\dbar^*$ of $\dbar$, and the boundary conditions arising from requiring that $u$ and 