CONTENTS

PRELIMINARIES 1

0. Introductory Remarks 1

1. Basic Notation and Definitions 3

2. Formulation of the Problem and Statement of the Main

Results 5

THE PROBLEM ON THE HALF SPACE 8

3. The Operator d* on 1-Forms and Its Domain 8

The Operators K and d* 11

4. Boutet de Monvel-Type Analysis of the Boundary Value

Problem 17

5. The Explicit Solution in the Case of Functions 23

The a priori Estimate 28

The Existence Theorem 31

6. Analysis of the Problem on the Half Space for q-Forms 37

THE CASE OF SMOOTHLY BOUNDED DOMAINS 52

7. Formulation of the Problem on a Smoothly Bounded Domain 52

8. A Special Coordinate System 59

The Tangential Laplacian 62

The Operator GQ on Functions 62

9. The Existence Theorem 64

The Coercive Estimate 67

10. The Regularity Theorem in the Case of Functions 74

The Interior Estimate 75

The Boundary Estimate 76

11. Estimates for q-Forms 79

The Regularity Theorem 79

Proof of Theorem 11.3 82

12. The Decomposition Theorem and Conclusions 89

Proof of the Main Result 89

Decomposition of W* 90

13. Final Remarks 98

References 98