Second Order Elliptic Operators -- Pseudo-Differential Operators -- Elliptic Operators on a Compact Manifold Without Boundary -- Boundary Problems for Elliptic Differential Operators -- Symplectic Geometry -- Some Classes of (Micro-)Hypoelliptic Operators -- The Strictly Hyperbolic Cauchy Problem -- The Mixed Dirichlet-Cauchy Problem for Second Order Operators.

From the reviews: "Volumes III and IV complete L. Hörmander's treatise on linear partial differential equations. They constitute the most complete and up-to-date account of this subject, by the author who has dominated it and made the most significant contributions in the last decades.....It is a superb book, which must be present in every mathematical library, and an indispensable tool for all - young and old - interested in the theory of partial differential operators." L. Boutet de Monvel in Bulletin of the American Mathematical Society, 1987 "This treatise is outstanding in every respect and must be counted among the great books in mathematics. It is certainly no easy reading (...) but a careful study is extremely rewarding for its wealth of ideas and techniques and the beauty of presentation." J. Brüning in Zentralblatt MATH, 1987.