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# Seminar on Singularities of Solutions of Linear Partial Differential Equations. (AM-91)

EDITED BY LARS HÖRMANDER
Pages: 296
Stable URL: http://www.jstor.org/stable/j.ctt1bd6jvc
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1. Front Matter (pp. i-iv)
3. PREFACE (pp. vii-2)
Lars Hörmander
4. SPECTRAL ANALYSIS OF SINGULARITIES (pp. 3-50)
Lars Hörmander

The existence of solutions of a linear partial differential equation is closely related to the singularities which solutions of the adjoint equation can have. We shall therefore study singularities of solutions first and discuss existence theorems afterwards as applications.

By the support supp u of a function or distribution u one means the smallest closed set such that u vanishes in the complement. Similarly the singular support sing supp u is the smallest closed set such that u is a Cfunction in the complement. However, it is possible to make a harmonic analysis of u at the singularities which...

5. FOURIER INTEGRAL OPERATORS WITH COMPLEX PHASE FUNCTIONS (pp. 51-64)
J. Sjöstrand

In his lectures L. Hörmander introduced local Fourier integral operators in order to transform pseudo-differential operators. Another important application of Fourier integral operators is the construction of solutions to homogeneous pseudo-differential equations; Pu ≡ 0 mod C. Such constructions sometimes also lead to parametrices (i.e. inverses modulo smoothing operators). In this context, it is often important to study Fourier integral operators globally as was done first by Hörmander [4] and Duistermaat-Hörmander [2]. If the leading symbol of the pseudo-differential equation under study takes complex values it is often necessary to consider complex valued phase functions. We shall here outline the...

6. HYPOELLIPTIC OPERATORS WITH DOUBLE CHARACTERISTICS (pp. 65-80)
A. Menikoff

Let P(x, D) be a pseudo-differential operator of order m in Ω ⊂ Rn. We shall say that P is hypoelliptic if

$\mathrm{sing\; supp\; u = sing\; supp\; pu,\; u\; \epsilon \; {\cal{{D}'}}\left ( \Omega \right )}.$

Further we shall say that P ishypoelliptic with loss ofµderivativesif$\mathrm{u}\; \epsilon \; {\cal{{D}'}}\left ( \Omega \right ),\; \mathrm{Pu}\; \epsilon \; \mathrm{H}_{\left ( \mathrm{s} \right )}^{\mathrm{loc}}$implies$\mathrm{u}\; \epsilon \; \mathrm{H}_{\left ( \mathrm{s+m-\mu } \right )}^{\mathrm{loc}}$. In this definition µ measures the deviation from the ideal case of an elliptic operator which is hypoelliptic with no loss of derivatives. An easy application of the closed graph theorem gives

Lemma 1.1.IfPis hypoelliptic with loss ofµderivatives, then for any compact setK ⊂ Ωands, sʹ ϵ Rthere is a...

7. DIFFERENTIAL BOUNDARY VALUE PROBLEMS OF PRINCIPAL TYPE (pp. 81-112)
R. B. Melrose

In these lectures our main concern is with the behavior, in particular the singularities, of solutions to boundary value problems for linear partial differential operators. Thus, we are given a Cmanifold M (Hausdorff, paracompact) with boundary ∂M. On M is prescribed a linear partial differential operator P. We shall make the following assumptions on P:

(1.1) P is second order and of real principal type.

(1.2) ∂M is nowhere characteristic for P.

Both these conditions are restrictions only on the principal symbol, p, of P. Thus, (1.1) requires that p ϵ C(T*M) be real-valued and that its zero surface...

8. PROPAGATION OF SINGULARITIES FOR A CLASS OF OPERATORS WITH DOUBLE CHARACTERISTICS (pp. 113-126)
Nicholas Hanges

In this lecture we will study pseudo-differential operators P with real principal symbol. It is assumed that the characteristic variety of P is the union of two smooth hypersurfaces with noninvolutive intersection. The construction of microlocal parametrices enables us to study the propagation of singularities.

The approach outlined here is that of Hanges [8], and we refer to that paper for complete details. Working independently, Ivrii [11], [12] and Melrose [14], [15] have obtained essentially the same results as us. Also see Alinhac [0], for the study of a closely related system, Kashiwara, Kawai and Oshima [19], [20] (or Miwa...

9. SUBELLIPTIC OPERATORS (pp. 127-208)
Lars Hörmander

Let Ω be an open set in Rn(or a manifold of dimension n) and let P be a pseudo-differential operator of order m in Ω with principal symbol p. It is well known that P is elliptic if and only if

$\left ( 1.1 \right )\; \; \mathrm{u}\; \epsilon \; {\cal{{D}'}}\left ( \Omega \right ),\; \mathrm{Pu}\; \epsilon \; \mathrm{H}_{\left ( \mathrm{s} \right )}^{\mathrm{loc}}\left ( \Omega \right )\Rightarrow \mathrm{u}\; \epsilon \; \mathrm{H}_{\left ( \mathrm{s +m}\right )}^{\mathrm{loc}}\left ( \Omega \right ).$

Here$\mathrm{H_{\left (s \right )}^{loc}}\left ( \Omega \right )$is the space of distributions f in Ω such that the Fourier transform of φf is in L2with respect to the measure$\left ( 1+\left | \xi \right |^{2} \right )^{\mathrm{s}}\mathrm{d}\xi$if$\phi\; \epsilon \;\mathrm{C}_{0}^{\infty }\left ( \Omega \right )$. Since$\mathrm{u}\; \epsilon \; \mathrm{H}_{\left ( \mathrm{s+m} \right )}^{\mathrm{loc}}\left ( \Omega \right )$implies$\mathrm{Pu}\; \epsilon \; \mathrm{H}_{\left ( \mathrm{s} \right )}^{\mathrm{loc}}\left ( \Omega \right )$it is clear that (1.1) is the best possible regularity result.

One calls P subelliptic with loss of δ derivatives if 0 < δ < 1...

10. LACUNAS AND TRANSMISSIONS (pp. 209-218)
Louis Boutet de Monvel

The purpose of this lecture is to describe the method of L. Gårding [1] for the study of lacunas of solutions of hyperbolic equations. For this a first step is the study of symmetries (transmissions) of Fourier integral distributions; such symmetries have been studied systematically by A. Hirschowitz and A. Piriou [2], and it is their point of view I present here. Transmissions also appear in pseudo-differential elliptic boundary value problems.

Let X be an open set in Rn(or a Cmanifold). A lacuna of a distribution f on X is an open set in which f vanishes. Pseudodifferential...

11. SOME CLASSICAL THEOREMS IN SPECTRAL THEORY REVISITED (pp. 219-260)
Victor Guillemin

Let D be a smooth strictly convex region in R2containing the origin. Let λD = {λx, x ϵ D}. A classical theorem of Van der Corput says that the number of lattice points in λD is equal to λ2volume D+O(λ2/3). This theorem is the simplest and most transparent of the “Weyl-type” theorems to which I am consecrating these lectures. In the first lecture, I will discuss this theorem and generalizations of it by Randol and Colin de Verdiere. For me this result is intriguing because it already involves, in an elementary way, difficulties with “periodic bicharacteristics” of the...

12. SZEGÖ’S THEOREM AND A COMPLETE SYMBOLIC CALCULUS FOR PSEUDO-DIFFERENTIAL OPERATORS (pp. 261-283)
Harold Widom

The classical Szegö theorem [10; 3, §5.2] states that for any bounded real-valued function φ on the unit circle, with Fourier coefficients denoted$\hat{\phi }_{\mathrm{j}}$, the eigenvalues of the Toeplitz matrix

$\mathrm{T_{N} = \left ( \hat{\phi }_{i-j} \right )\; \; 0\leq i,\; j\leq N}$

are asymptotically distributed as the values of φ in the sense that for any continuous function F

$\mathrm{\lim_{N\rightarrow \infty }N^{-1}\sum_{\lambda\; \epsilon\; spec\; T_{N}}F\left ( \lambda \right ) = \frac{1}{2\pi }\int_{0}^{2\pi }F\left [ \phi \left ( e^{i\theta } \right ) \right ]d\theta} .$

Szegö proved the general result from the special case F(λ) = log λ (assuming of course that φ > 0), the assertion then being equivalent to

$\mathrm{log\; det\; T_{N} = N\; \widehat{log\; \phi }\left ( 0 \right )+o\left ( N \right )}.$

This has since been generalized in many directions. Szegö himself [11; 3, §5.5] showed that if φ is sufficiently smooth then the error...

13. Back Matter (pp. 284-285)