---
abstract: 'We study the $\overline{\partial}$-Neumann problem using the Sobolev space inner product. We show that the problem can be solved on any smoothly bounded, pseudoconvex domain. We further formulate estimates and the basic results of a Sobolev Hodge theory.'
address:
- |
    -Luigi Fontana, Dipartimento di Matematica, Via Saldini 50, Università di Milano\
    20133 Milano (Italy)
- '-Steven G. Krantz, Mathematical Sciences Research Institute [and]{} Department of Mathematics, Washington University, St. Louis, MO 63130 (U.S.A.)'
- '-Marco M. Peloso, Dipartimento di Matematica, Politecnico di Torino, 10129 Torino (Italy)'
author:
- Luigi Fontana
- 'Steven G. Krantz'
- 'Marco M. Peloso'
title: 'The $\dbar$-Neumann problem in the Sobolev topology'
---

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Introduction
============

Let $\Omega$ be a smoothly bounded domain in ${{\Bbb C}^n}$. We write the coordinates $z_j =x_j +ix_{j+n}$, $j=1,\dots,n$, and the standard basis of vector fields $D_k := {\partial}/{\partial}x_k$, for $k=1,\dots,2n$. For $s$ a non-negative integer we define the Sobolev inner product $\la \cdot,\cdot\ra_s$ to be $$\label{Sobolev}
\la f,g\ra_s := \sum_{|\alpha|\le s} \CP \int_\Omega D^\alpha f 
\overline{D^\alpha g} .$$ Here, and throughout the paper, we use $D^\alpha$ to denote the $\alpha$-order derivative, where $\alpha$ 