Parabolicity, Volterra Calculus, and Conical Singularities, Softcover reprint of the original 1st ed. 2002
A Volume of Advances in Partial Differential Equations
Advances in Partial Differential Equations Series

Partial differential equations constitute an integral part of mathematics. They lie at the interface of areas as diverse as differential geometry, functional analysis, or the theory of Lie groups and have numerous applications in the applied sciences. A wealth of methods has been devised for their analysis. Over the past decades, operator algebras in connection with ideas and structures from geometry, topology, and theoretical physics have contributed a large variety of particularly useful tools. One typical example is the analysis on singular configurations, where elliptic equations have been studied successfully within the framework of operator algebras with symbolic structures adapted to the geometry of the underlying space. More recently, these techniques have proven to be useful also for studying parabolic and hyperbolic equations. Moreover, it turned out that many seemingly smooth, noncompact situations can be handled with the ideas from singular analysis. The three papers at the beginning of this volume highlight this aspect. They deal with parabolic equations, a topic relevant for many applications. The first article prepares the ground by presenting a calculus for pseudo differential operators with an anisotropic analytic parameter. In the subsequent paper, an algebra of Mellin operators on the infinite space-time cylinder is constructed. It is shown how timelike infinity can be treated as a conical singularity.

Volterra Families of Pseudodifferential Operators.- 1. Basic notation and general conventions.- 2. General parameter-dependent symbols.- 3. Parameter-dependent Volterra symbols.- 4. The calculus of pseudodifferential operators.- 5. Ellipticity and parabolicity.- References.- The Calculus of Volterra Mellin Pseudodifferential Operators with Operator-valued Symbols.- 1. Preliminaries on function spaces and the Mellin transform.- 2. The calculus of Volterra symbols.- 3. The calculus of Volterra Mellin operators.- 4. Kernel cut-off and Mellin quantization.- 5. Parabolicity and Volterra parametrices.- References.- On the Inverse of Parabolic Systems of Partial Differential Equations of General Form in an Infinite Space-Time Cylinder.- 1. Preliminary material.- 2. Abstract Volterra pseudodifferential calculus.- 3. Parameter-dependent Volterra calculus on a closed manifold.- 4. Weighted Sobolev spaces.- 5. Calculi built upon parameter-dependent operators.- 6. Volterra cone calculus.- 7. Remarks on the classical theory of parabolic PDE.- References.- On the Factorization of Meromorphic Mellin Symbols.- 1. Introduction.- 2. Preliminaries.- 3. Logarithms of pseudodifferential operators.- 4. The kernel cut-off technique.- 5. Proof of the main theorem.- References.- Coordinate Invariance of the Cone Algebra with Asymptotics.- 1. Cone operators on the half-axis.- 2. Operators on higher-dimensional cones.- References.