Lars Hörmander

Fourier integral operators

Pseudo-differential operators have been developed as a tool for the study of elliptic differential equations. Suitably extended versions are also applicable to hypoelliptic equations, but their value is rather limited in genuinely non-elliptic problems. In this paper we shall therefore discuss some more general classes of operators which are adapted to such applications. For these operators we shall develop a calculus which is almost as smooth as that of pseudo-differential operators. It also seems that one gains some more insight into the theory of pseudo-differential operators by considering them from the point of view of the wider classes of operators to be discussed here so we shall take the opportunity to include a short exposition.

The Spectral Function of an Elliptic Operator

In this paper we shall obtain the best possible estimates for the remainder term in the asymptotic formula for the spectral function of an arbitrary elliptic (pseudo-)differential operator. This is achieved by means of a complete description of the singularities of the Fourier transform of the spectral function for low frequencies.

On the theory of general partial differential operators

I I . Min ima l d i f fe ren t i a l ope ra to r s w i t h c o n s t a n t coeff ic ients . I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 N o t a t i o n s a n d f o r m a l p r o p e r t i e s of d i f fe ren t ia l ope ra to r s w i t h c o n s t a n t coeff ic ients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 E s t i m a t e s b y Lap lace t r a n s f o r m s . . . . . . . . . . . . . . . . . . . . . . 177 T h e d i f fe ren t ia l ope ra to r s w e a ke r t h a n a g i v e n one . . . . . . . . . . . . . 178 T h e a lgebra of e n e r g y in t eg ra l s . . . . . . . . . . . . . . . . . . . . . . 180 A n a l y t i c a l p rope r t i e s of e n e r g y in tegra l s . . . . . . . . . . . . . . . . . . 182 ]~s t imates b y e n e r g y in t eg ra l s . . . . . . . . . . . . . . . . . . . . . . . 183 Some special cases of T h e o r e m 2.2 . . . . . . . . . . . . . . . . . . . . . 185 T h e s t r u c t u r e of t he m i n i m a l d o m a i n . . . . . . . . . . . . . . . . . . . . 189 Some t h e o r e m s o n comple t e c o n t i n u i t y . . . . . . . . . . . . . . . . . . . 201 On some se t s of p o l y n o m i a l s . . . . . . . . . . . . . . . . . . . . . . . . 207 R e m a r k s on t h e case of n o n b o u n d e d d o m a i n s . . . . . . . . . . . . . . . 208 CHAPTER I I I . M a x i m a l d i f fe ren t ia l ope ra to r s w i t h c o n s t a n t coff ic ients . 3.0. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 3.1. C o m p a r i s o n of t h e d o m a i n s of m a x i m a l d i f ferent ia l ope ra to r s . . . . . . . . 211 3.2. T h e ex i s t ence of nul l so lu t i ons . . . . . . . . . . . . . . . . . . . . . . . 216 3.3. Di f fe ren t i a l ope ra to r s of local t y p e . . . . . . . . . . . . . . . . . . . . . 218 3.4. C o n s t r u c t i o n of a f u n d a m e n t a l so lu t ion of a comple t e ope ra to r of local t y p e . 222 3.5. P roof of T h e o r e m 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 1 1 553810. Acta Mathematica. 94. Imprim~t le 26 septembre 1955.

The boundary problems of physical geodesy

The purpose of this paper is to discuss the determination of the figure of the earth and its gravity field from astrogeodetic and gravimetric data. (By gravity we mean the resultant of the attractive force of the masses of the earth, also called gravitation, and the centrifugal force of the earths rotation.) Let us recall that at the surface of the earth: (a) astronomic observations allow one to determine the direction of the gravity vector G; (b) gravimetric measurements give the length I GI of the gravity vector; (c) levelling combined with gravimetric measurements gives the differential of the gravity potential W, and thus yields W apart from an additive constant. We assume the measured data G and W corrected for, say, the gravitational interaction with the moon, the sun and the planets, for the precession of the earth, and so on, so that we have the following idealized situation:

A uniqueness theorem of Beurling for Fourier transform pairs

There are many theorems known which state that a function and its Fourier transform cannot simultaneously be very small at infinity, such as various forms of the uncertainty principle and the basic results on quasianalytic functions. One such theorem is stated on page 372 in volume II of the collected works of Arne Beurling [1]. Although it is not in every respect the most precise result of its kind, it has a simplicity and generality which make it very attractive. The editors state that no proof has been preserved. However, in my files I have notes which I made when Arne Beurling explained this result to me during a private conversation some time during the years 1964---1968 when we were colleagues at the Institute for Advanced Study, I shall reproduce these notes here in English translation with onIy minor details added where my notes are too sketchy. Theorem. Let fELl (R) and assume that

Pseudo-differential operaors of type 1,1.

On donne des conditions necessaires et suffisantes pour la continuite des operateurs dans OpS 1,1 m de H (s+m) a H (s) quand s est negatif

On the uniqueness of the Cauchy problem under partial analyticity assumptions

A few years ago Robbiano [7] proved a striking and surprising uniqueness theorem for hyperbolic differential operators in R n+1 of the form

Lower bounds at infinity for solutions of differential equations with constant coefficients

P = D_t^2 - A\left( {x,\,D_x } \right)