# Nirenberg’s contributions to linear partial differential equations: Pseudo-differential operators and solvability

## Abstract

This article is a survey of Louis Nirenberg’s contributions to linear partial differential equations, focusing on his groundbreaking work on pseudo-differential operators and solvability.

## 1. Introduction

One cannot overestimate Louis Nirenberg’s impact on twentieth century mathematics, especially on the analysis of both linear and nonlinear partial differential equations. In this article, we shall concentrate on Nirenberg’s achievements in linear PDEs, in particular his development (with Kohn) of the calculus of pseudo-differential operators and microlocal analysis. These were developed as tools for the analysis of PDEs, but they have now become indispensible both for analysis and for other areas of mathematics. In connection with this, we shall also treat Nirenberg’s work (with Treves) on the solvability of partial and pseudo-differential operators. DOs

As a graduate student, I read Nirenberg’s papers Reference 25 and Reference 26 on pseudo-differential operators to learn the subject, and they have been a great inspiration to me. I also heared about Nirenberg’s work with Treves Reference 28 about solvability, especially their famous conjecture which came to play an important role in my research. Later I met Nirenberg several times, including at his Abel Prize celebration, but we never had any collaborations.

## 2. Background

To appreciate Nirenberg’s contributions in the development of one has to know the background which presented the need for these operators. The development of the theory of distributions by Laurent Schwartz DOs,Reference 29 at the end of the 1940’s revolutionized the analysis of PDEs. Distributions are generalizations of generalized functions, which extend the notion of functions and had been used as weak solutions of PDEs. By defining distributions as functionals on classes of smooth test functions, one could simplify the theory of PDEs and be relatively unrestricted in their use of the Fourier transform. For example, one could now define a fundamental solution to any linear PDE.

Distributions were not appreciated by all; see for example Bochner’s review Reference 3 of Schwartz’s book. In spite of the opposition, distributions lead to a quick development of the theory of PDEs and many new results. One example is the proof of existence of fundamental solutions to any constant coefficient PDEs by Ehrenpreis Reference 7 and Malgrange Reference 24 in 1954–55. By using distributions and Fourier transformation, the study of constant coefficient PDEs is often reduced to the study of polynomials and their zeros.

But nonconstant coefficient PDEs presented a more difficult problem. Here singular integral operators became a useful tool, which for example Calderón Reference 4 used to prove the uniqueness of the Cauchy problem in 1958. Singular integral operators on have the form:

where *symbol*

By work of Calderón and Zygmund, Horwath and Kohn, the symbol turned out to involve the partial Fourier transform of

which is then homogeneous of degree 0 in the

## 3. Pseudo-differential operators

Kohn and Nirenberg Reference 17 defined pseudo-differential operators in 1965 as

where the symbol

Here

These operators are called *classical* (or *polyhomogeneous*)

Kohn and Nirenberg’s paper also gave the calculus of

With

The simpler calculus and the easy use of symbols made *microlocalization*.

Kohn and Nirenberg immediately used the techniques of Reference 17 to study elliptic boundary problems in Reference 18. There were also many properties of

Kohn and Nirenberg’s highly influential paper started a revolution in the analysis of PDEs and initiated the field of *microlocal analysis*, where one can localize in cones in *singular support* *wave front set*

Hörmander Reference 13 also proved that for pseudo-differential operators with real principal symbols that are of *principal type*, the wave front sets of the solutions propagate along the bicharacteristics of the principal symbol, which generalizes geometrical optics. Principal type means that the principal symbol vanishes of first order at its zeros, called *characteristics*. The *bicharacteristics* are the flow-out of the Hamilton vector field of the principal symbol on the characteristics; see Equation 4.2.

This refinement simplifies the study of singularities of solutions to

*Fourier integral operators* were developed by Hörmander Reference 12 to obtain symplectic coordinate transformations of

But the development was also towards more general classes of pseudo-differential operators. The classical

where

## 4. Solvability

One area where microlocal analysis had a great impact is the solvability of PDOs and

This means that the equation

Thus, it has an important role in complex analysis in several variables.

The tangential Cauchy–Riemann operators always have a large kernel containing any analytic function of the coordinates

After Lewy’s example, Hörmander Reference 9 took up the quest for solvability in 1960 by showing that the lack of solvability is generic for PDEs. In fact, if

where

The condition that

If *principally normal* and Hörmander Reference 10 showed that these operators are locally solvable. One could also have special conditions on the repeated brackets when studying solvability, but then the situation gets rather complicated.

By using microlocal analysis and preparation theorems, one can reduce a classical

where *Principally normal* means that

## 5. The Nirenberg–Treves conjecture

But Nirenberg and Treves Reference 27 in 1963 changed the perspective and introduced conditions on the sign changes of the imaginary part of the principal symbol of a PDO. The most important is condition

Nirenberg and Treves showed that condition

Armed with the tools of pseudo-differential operators, Nirenberg and Treves Reference 28 in 1970 took up the study of the solvability of PDOs and

Nirenberg and Treves conjectured that condition (

When the principal symbol is analytic satisfying condition

where

Under these conditions, Nirenberg and Treves Reference 28 proved the following

if

But in 1973, Beals and Fefferman Reference 1 proved the sufficiency of condition

Hörmander Reference 14 developed this calculus further in 1979 into the *Weyl calculus*, defining symbols to be uniformly smooth with respect to suitable metrics on

The necessity of condition (

It was in general assumed that condition (

But a counterexample of Lerner Reference 21 in 1994 shows that in general condition (

But in 2004 the sufficiency of condition (

The estimate was then improved in Reference 5 to a loss of

In closing, the insights of Nirenberg led to a revolution in the analysis of PDEs, to the development of microlocal analysis, and to breakthroughs in the solvability of PDOs and

## About the author

Nils Dencker is professor of mathematics at Lund University, working in microlocal analysis and spectral theory of partial differential operators. He is a member of the Royal Swedish Academy of Sciences and received the Gårding Prize in 2003 and the Clay Research Award in 2005 for his resolution of the Nirenberg–Treves conjecture.