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## Über dieses Buch

S. Agmon: Asymptotic formulas with remainder estimates for eigenvalues of elliptic operators.- J. Bokobza-Haggiag: Une définition globale des opérateurs pseudo-différentiels sur une variété différentiable.- L. Boutet de Monvel: Pseudo-differential operators and analytic function.- A. Calderon: A priori estimates for singular integral operators.- B.F. Jones: Characterization of spaces of Bessel potentials related to the heat equation.- J.J. Kohn: Pseudo-differential operators and non-elliptic problems.- R.T. Seeley: Topics in pseudo-differential operators.- I.M. E. Shamir: Boundary value problems for elliptic convolution systems.- Singer: Elliptic operators on manifolds.

## Inhaltsverzeichnis

### Asymptotic Formulas with Remainder Estimates for Eingevalues of Elliptic Operators

We propose to discuss in this lecture a number of results related to the problem of eigenvalue distribution of elliptic operators. We start with some classical results. Let Δ be the Laplacian in Rn and consider the eigenvalue problem:
$$\begin{array}{*{20}c}{ - \Delta = \lambda {\text{u}}} & {{\text{in}}\,\,\,\Omega \,,}\\{{\text{u}} = 0} & {{\text{on}}\,\,\partial \Omega \,,}\\\end{array}$$
(1)
where Ω is a bounded open set in Rn. Let {λj} be the sequence of eigenvalues of (1), each repeated according to its multiplicity and set
$$N\left( t \right) = \sum\limits_{\lambda _j < t} 1.$$
(2)
S. Agmon

### Une Definition Globale des Operateurs Pseudo-Differentiels sur une Variete Differentiable

Nous introduisons dans ce qui suit une définition globale des opérateurs pseudo-différentiels sur une variété différentiate et un calcul symbolique qui permet d'établir une correspondance linéaire bijective entre les opérateurs pseudo-différentiels modulo les opérateurs régularisants d'une part et une classe de symboles modulo les symboles qui sont à décroissance rapide sur les fibres de l'espace cotangent d'autre part.
L'idée de ce calcul est basée sur le fait que la formule
$$\left( {A\phi } \right)\left( x \right) = \int {f\left( {x,\xi } \right)d\,\xi \,\int {e^{ - 2i\pi \left( {y - x} \right).\xi } } \phi \left( y \right)dy}$$
qui définit un opérateur pseudo-differentiel sur ℝn, si f a certaines propriétés de régularité et de croissance à l'infini, prend un sens sur une variété si l'on y remplace y-x par un vecteur tangent en x à la variété, soit v(x,y), “infinitésimalement égal” à y-x, et si l'on prend quelques précautions suppiémentaires destinées à faire converger l'intégrale et à lui assurer un sens intrinsfèque.
J. Bokobza-Haggiag

### Pseudo-Differential Operators and Analytic Functions

δ O- Introduction Le but de ce chapitre est de décrire une classe d'opérateurs pseudo-différentiels, sur les variétés analytiques réelles, qui se comporte bien vis à vis des fonctions analytiques. En gros ces opérateurs ont les propriétés suivantes
1. Un opérat r pseudo-différentiel analytique est est en particulier un opérateur pseudo-différentiel ordinaire, et le calcul symbolique (do Calderon, Kohn, Nirenberg) marche encore.
L. Boutet de Monvel

### A Priori Estimates for Singular Integral Operators

In this paper we develop the theory of singular integral operators with finitely differentiable symbols and apply it to the derivation of a priori inequalities for these and pseudo-differential operators. In section 1 we treat singular integral operators and introduce the pseudo-differential operators as was done before they were given their name, namely as compositions of singular integral operators with powers of the operator ∧.
In our opinion this is the correct point of view since in the finitely differentiable case these do not form an algebra, but merely a module over the algebra of singular integral operators. The differentiability assumptions made here are designed to yield self-adjoint algebras which are specially suited for the treatment of the L2 theory, and are not too far removed from the best possible. Relaxing these conditions substantially, as was done in [l], causes the loss of self-adjointness. In section 2 we discuss the action of our operators on rapidly oscillating functions with small support and obtain, as a byproduct, a new representation for the algebras under consideration which illuminates the negative, results on inequalities which are discussed in section. 4.
A. P. CalderÖn

### Characterization of Spaces of Bessel Potentials Related to the Heat Equation

1. Background. The operator ∧ω is often dealt with in the talks in this conference. For complex ω it is defined by the formula in the Fourier transform space,
$$\widehat{\Lambda ^\omega}{\rm{f}(\xi)} = \left( {1 + |\xi |^2 } \right)^{\omega /2} \widehat f{\rm{f}(\xi)}.$$
where |ξ| is the Euclidean length of ξ. Thus, if Δ is the Laplace operator, it follows that formally
$$\Lambda ^\omega = \left({1 - \Delta} \right)^{\omega/2}.$$
B. Frank Jones

### Pseudo-Differential Operators and Non-Elliptic Problems

We will discuss here problems which lead to integro-differential forms which involve derivatives of first order. Such a problem is called elliptic when the L2-norms of all first derivatives can be estimated by the corresponding form. In [4] it is shown that it suffices to estimate the ‖ ‖ε-norm (with 0<ε≤1) in order to establish smoothness of solutions, discretness of spectrum and other properties such problems have in common with the elliptic case. Here we consider the case when the L2-norms of only some derivatives are bounded.
The following is a special case of an estimate proved by Hörmander in 1. The proof that we give here uses only elementary properties of pseudo-differential operators.
J. J. Kohn

### Topics in pseudo-differential operators

The subject of pseudo-differential operators has sprung up in the last few years out of the earlier work of Giraud, Mihlin, and Calderon and Zygmund, and is still in the process of development. There are so many contributors to this development that the references given at the article are restricted to those papers actually referred to.
The first four chapters give the elementary theory of pseudo-differential operators, together with some fairly direct applications to elliptic problems on compact manifolds. The last two chapters sketch two more complicated applications, one to the study of the powers of an elliptic operators, and the other to boundary problems.
R. Seeley

### Boundary Value Problems for Elliptic Convolutions Systems

Whe shall study here generalized boundary-value problems (including potentials) for homogeneous elliptic systems of convolution equations in a half-space. The main result is Theorem 3, giving the necessary and sufficient conditions for a problem to be well-posed. Actually we establish an isomorphism between the space of unknowns and the data space. We use a pure L2-theory for systems of order 0. But we outline the way to obtain LP - theory and also the way to treat systems with various degrees of homogeneity - by introducing suitable Hs spaces for the various components. At the end we add several other remarks. More details will appear in [8], which is a natural continuation of [7].
Eliahu Shamir

### Elliptic Operators on Manifolds

In these talks I shall give examples of some applications of elliptic operators on manifolds centering around the fixed point formula and the index theorem. In his lectures at this session, R. T. Seeley will review pseudo-differential operators on manifolds and the elementary properties of the index. This will allow me to assume much of the analytical facts concerning elliptic operators. Since the primary discipline of the majority of the participants is analysis, I will spend more time with the geometry and topology but motivate the proofs of the main theorems from the analytic viewpoint. By and large analysts find the geometrical and topological background needed here formidable. It is my hope that these expository talks will help orient the listener and allow him to read the relevant papers listed at the end more easily.
I. M. Singer
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